Designs in Product Association Schemes - Semantic Scholar
Designs in Product Association Schemes William J. Martin1 Department of Mathematics & Statistics University of Winnipeg Winnipeg, Manitoba April 23, 1998 Abstract
Let (Y; A) be an association scheme with primitive idempotents E ; E ; . . . ; Ed . For T f1; . . . ; dg, a Delsarte T -design in (Y; A) is a subset D of Y whose characteristic vector is annihilated by the idempotents Ej (j 2 T ). The case most studied is that in which (Y; A) is Q-polynomial and T = f1; . . . ; tg. For many such examples, a combinatorial characterization is known, giving an equivalence between Delsarte T -designs and 0
1
poset t-designs in what we call here \Q-posets". For example, combinatorial t-designs (i.e., block designs) can be described via the truncated Boolean lattice while orthogonal arrays can be described via the Hamming lattice. For 1 i m, let (Yi; Ai ) be a Q-polynomial association scheme. Assume that Delsarte t-designs in each (Yi; Ai ) are characterised as poset t-designs in a Q-poset Pi attached to that scheme. With these assumptions, we consider the product association scheme (Yi ; A). The primitive idempotents for this scheme naturally inherit the partial order structure of a product of chains. Our main result, Theorem 2.3, characterises Delsarte T -designs in (Yi; A) as poset designs in the product poset Pi where T is any downset in the product of chains. Using this characterisation, we immediately obtain linear programming bounds for a wide variety of combinatorial objects. On the other hand, if we assume each component scheme is Q-polynomial, we obtain bounds on the size and degree of such a T -design analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.
1 Overview and Background We are interested in applications of the theory of association schemes to problems in coding theory and design theory. This investigation is motivated by two applications. In [10], The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through grant OGP0155422 1
1
Levenshtein introduces split orthogonal arrays (see Example 2.4 below) and applies the theory of Krawtchouk polynomials to obtain bounds on the size of such objects. It was clear to the author early on that similar results hold for mixed-level orthogonal arrays. Independently, Sloane and Stufken [17] derived a linear programming bound for these objects. Our approach shows how the two studies can be carried out simultaneously and elegantly if one considers the product of Hamming association schemes and the associated product of Hamming lattices. In fact, much more can be accomplished if one abstracts the essential features of the relationship between the Hamming association scheme and (incidence matrices of) the Hamming lattice. With this motivation, we introduce in Section 2.1 the concept of a Q-poset attached to an association scheme. (While the de nition can be applied to arbitrary association schemes, the case where the scheme is Q-polynomial is our focus.) The main result of the paper is Theorem 2.3 in Section 2.2 which uses these partially ordered sets to characterise Delsarte T -designs in a product association scheme where each component scheme is assumed to have an attached Q-poset and where T is a downset in a product of chains. The balance of the paper explores bounds on T -designs in product schemes where each component is Q-polynomial. The general results include: a linear programming bound (which falls out immediately from Theorem 2.3 using the standard theory); a Delsarte bound (Theorem 3.2) analogous to the Rao bound for orthogonal arrays and the RayChaudhuri/Wilson bound for combinatorial t-designs; a degree bound (Theorem 3.3). Numerous examples are included to demonstrate the power and wide applicability of these bounds.
1.1 Association Schemes
All material in this section is standard (see [4, 1, 2, 8]). Let X be a nite set of cardinality v > 0 and let A = fA ; A ; . . . ; Ad g be a set of v v symmetric 01-matrices with rows and columns indexed by X and with linear span A Rvv. We say that (X; A) is a (symmetric) association scheme if A satis es (i) A = I ; (ii) Pdi Ai = J (the all-ones matrix); (iii) for 0 i; j d, AiAj 2 A . To each matrix Ai (i 6= 0) is associated its graph Gi with vertex set X : Ai is the adjacency matrix of Gi . The above de nition is customarily given in the language of relations. We will use graph and matrix terminology interchangeably. The vector space A spanned by the matrices in A is, by condition (iii), an algebra; this is the Bose-Mesner algebra of the association scheme. Extensive background material on association schemes can be found in the references. 0
0
=0
2
1
A. The Hamming Scheme
In terms of applications, two important association schemes are the Hamming scheme and the Johnson scheme. The Hamming scheme H (n; q) has as its vertices the set X of all words of length n over an alphabet Q having q elements. The graphs G ; . . . ; Gn are de ned by Hamming distance: if x; y 2 X , then (x; y) 2 Gk where k is the number of positions in which the words x and y di er. It is well-known that the mapping (x; y) 7! k is a metric. So the entire association scheme is determined by the Hamming graph G . Thus the Hamming scheme is a metric (hence, P -polynomial) association scheme. (Metric association schemes are the same as distance-regular graphs.) As examples, for q = 2, G is the n-cube and for n = 2, G is the grid graph Kq Kq . Several authors have associated to this scheme a partially ordered set (P ; ), commonly called the Hamming lattice (see [5, 19, 14, 8]). Adjoin to the alphabet Q a new symbol \". The elements of P are the words of length n over Q [ fg. For x; y 2 P , we have x y if and only if, for all i, yi = implies xi = . This poset has rank function ` given by `(x) = jfi : xi 6= gj. So the elements of X are precisely the members of P having maximum rank n. It is immediately evident that a subset D X is an orthogonal array of strength t in H (n; q) if and only if, in (P ; ), there exists such that every element of rank t is dominated by exactly members of D. 1
1
1
1
B. The Johnson Scheme Let V be a set of n points and let k satisfy 1 k n=2. The Johnson scheme J (n; k) has vertex set X consisting of all k-element subsets of V and (x; y) 2 Gi (1 i k) if jx \ yj = k ? i. Again, we have a metric association scheme; G is called the Johnson graph. To this association scheme, we attach the truncated boolean lattice (P ; ) where P = fS V : jS j kg with rank function `(S ) = jS j. A combinatorial t-design is simply 1
any subset D of X having the property that every element S of rank t is dominated by exactly members of D for some xed . The Johnson graphs and Hamming graphs are examples of distance-regular graphs and these two families are well-studied. In particular, Delsarte [5] analysed the relationship between these schemes and the above-mentioned posets (as well as their q-analogues) in the language of regular semilattices.
1.2 Linear Programming
Let (X; A) be an association scheme with adjacency matrices A = fA ; . . . ; Ad g and BoseMesner algebra A . Since the Ai are 01-matrices with pairwise disjoint supports, this algebra is also closed under Schur (entrywise) multiplication, denoted . The matrices Ai form a basis of mutually orthogonal idempotents for A with respect to this multiplication (Ai Aj = i;j Ai). There is also a unique basis fE ; E ; . . . ; Ed g of mutually orthogonal idempotents with respect to ordinary matrix multiplication (which we call the \primitive idempotents"). 0
0
1
3
The change-of-basis matrices P and Q, de ned by
Ai =
d X j =0
PjiEj
and
d X 1 Ej = v Qij Ai i
(1)
=0
satisfy the orthogonality relations PQ = vI and MP = QT K (2) where K is a diagonal matrix with Kii equal to the valency of (the regular graph) Gi and M is a diagonal matrix with Mjj = fj := rank Ej . We refer to these simply as the \P -matrix" and \Q-matrix" of the scheme. For the convenience of the reader mainly interested in applications, we record here the change-of-basis matrices for the two examples described above. For the Hamming scheme H (n; q), we have Pji = Ki (j ) where
Ki (x) =
Xi
(?1)` (q ? 1)i?`
xn ? x
(3) ` i ? ` ` is the Krawtchouk polynomial familiar to coding theorists and we have Q = P . (This is a self-dual association scheme.) For the Johnson scheme J (n; k), we have Pji = Ei (j ) where =0
Ei (x) =
Xi
(?1)`
xk ? xn ? k ? x
(4) i?` i?` Hahn polynomial of degree i. The rank of Ej in this case is given by fj = ?dual ?isnthe n , the valency of G is given by k = ?k?n?k, and Q is obtained from P using the ? i i i i j? j orthogonality relations above. Given any subset D of the vertices of an association scheme (X; A) on v points, we build the characteristic vector of D, denoted D , which is a 01-vector of length v with y-entry equal to 1 if and only if y 2 D. We are also interested in the inner distribution vector of D: a = [a ; a ; . . . ; ad] is de ned by (5) ai = jD1 j TD Ai D which, in the case of a distance-regular graph, gives the average number of elements of D at distance i from any element of D. For linear codes, a is simply the list of coecients of the weight enumerator. For combinatorial t-designs, ai measures the average number of blocks meeting a given block in k ? i points. Consider the product aQ, which is called the MacWilliams transform of a. By Equation (1b), we have (aQ)j = v TD Ej D jDj and, since Ej = Ej , this quantity is always non-negative (see also [2, Sec. 2.5]): `=0
`
1
0
1
2
4
Theorem 1.1 (Delsarte's Linear Programming Bound [4, Thm. 3.3]) For any sub-
set D of the vertices X of an association scheme, the MacWilliams transform of the inner distribution vector is non-negative: aQ 0. 2
For T f1; . . . dg, a Delsarte T -design is a subset D X satisfying Ej D = 0 for j 2 T . This is clearly equivalent to the condition (aQ)j = 0 for all j 2 T . In the case where T = f1; . . . ; tg, we will simply say D is a Delsarte t-design.
2 Combinatorial characterizations of Delsarte T -designs 2.1 Q-posets
Let (P ; ) be a partially ordered set (poset). If x y, we say y dominates x. If x y and there is no z with x z y, we say y covers x. For a non-negative integer d, write [d] = f0; 1; . . . ; dg. A rank function on (P ; ) is a surjection ` : P ! [d] satisfying the following condition: `(x) = i y x `(y) = i + 1. A ranked poset is then simply a poset (P ; ) with a speci ed rank function. If (P ; ) is a ranked poset with rank function `, then P is partitioned into bres P ; P ; . . . ; P d where P i = fx 2 P : `(x) = ig consists of those objects of rank i. For i 2 [d], let Wi be the incidence matrix of the i bre P i versus the top bre P d. The rows of Wi are indexed by elements of P i and the columns are indexed by elements of P d . As indicated above, a number of design-theoretic objects have a common de nition in the language of posets. Let (P ; ) be a ranked poset as above. Extending Delsarte's de nition [5], let us call D P d a poset t-design in (P ; ) if there exist constants ; ; . . . ; t such that, for 0 i t and for all x 2 P i , if
0
and
covers
, then
1
th
0
1
jfy 2 D : x ygj = i: For example, a combinatorial t-design is equivalent to a poset t-design in the truncated Boolean lattice and an orthogonal array of strength t is a poset t-design in the Hamming lattice. Let (X; A) be a symmetric association scheme having d classes and a xed ordering E ; . . . ; Ed on its primitive idempotents. Denote by Vj the j eigenspace, rowsp Ej , of the scheme. We say a ranked poset (P ; ) is a Q-poset for (X; A) (with respect to the ordering 0; 1; . . . ; d) if the following axioms are satis ed: (1) the top bre P d is the set X ; (2) for 0 i d, Wi has constant row sum; th
0
5
(3) for 0 i d, Vi rowsp Wi V V . . . Vi. Note: Unless explicitly stated to the contrary, the only ordering assumed on the primitive 0
1
idempotents of an association scheme is the natural ordering E ; E ; . . . ; Ed . This ordering, usually dictated by the application, is to be assumed in any statement regarding attached Q-posets or the Q-polynomial property. Remark: Viewing the rows of the incidence matrices Wi as subsets of X , our axioms (2) and (3) can be stated as conditions on certain collections Et (0 t d) of \antidesigns". Following Roos [16], we call a subset C of X an antidesign with dual diameter t if Ej C = 0 for j > t. Then our axioms amount to assuming, for each 0 t d, the existence of a collection Et of antidesigns such that the members of Et are equicardinal and their characteristic vectors span a space containing Vt. See Meyerowitz [14] for a case study of the connection between antidesigns and Q-posets. 0
1
Lemma 2.1 Let (L; ) be a regular semilattice [5] and let (X; A) be the association scheme carried by the top bre. Then there exists a subposet (P ; ) of (L; ) which is a Q-poset for (X; A). Proof: See [5] for de nitions of these objects. There, Delsarte proves that there exist bres L ; Lj ; . . . ; Ljd of (L; ) which, together with the incidence matrices between them, satisfy conditions (1)-(3) above. 2 0
1
For example, the Hamming lattice is a Q-poset for the Hamming scheme H (n; q) and, for v 2k, the truncated Boolean lattice is a Q-poset for the Johnson scheme J (v; k). For any Q-poset derived from a regular semilattice in this way, there are constants
i;j = jfy 2 P j : z y xgj; i;j = jfy 2 P j : z ygj which are independent of z 2 P i and x 2 X provided z x. If such constants exist for a Q-poset (P ; ), let us say that (P ; ) is a regular Q-poset (cf. [5]). In such cases, a set D for which t is well-de ned is automatically a t-design in (P ; ). The following lemma was proved in [5] for regular semilattices.
Lemma 2.2 Let (X; A) be a symmetric association scheme. Let (P ; ) be a Q-poset for (X; A) with incidence matrices Wi . Then for D X with characteristic vector D , the following are equivalent: (i) D is a poset t-design in (P ; );
(ii) there exist constants i, 0 i t such that Wi D = i1; (iii) for 1 j t, Ej D = 0; (iv) for 1 j t, (aQ)j = 0. 6
Proof: The equivalence of (i) and (ii) is immediate from the de nitions. The equivalence of (iii) and (iv) is entirely standard [4, p27]. (ii))(iii): Let e be any row of Ej (1 j t). By (3), we may write e = rWj for some vector r of length jP j j. Let be the row sum of Wj (using (2)); then we have Wj 1T = 1T . Since j = 6 0, 0 = he; 1i = rWj 1T = r1T : This allows us to compute he; D i. We have he; Di = rWj D = j r1T = 0: (iii))(ii): Let w be any row of Wi (0 i t). With ej = wEj , we have w = Pij ej using (3). Now if = jX j hw; 1i, then e = 1 and is independent of the choice of w since =0
1
0
Wi has constant row sum. Consequently,
hw; Di =
Xi j =0
hej ; Di = he ; D i = jDj 0
is independent of the choice of w and we have Wi D = i 1T where i jX j = jDj with being the row sum of Wi . 2 While we use the term Q-poset, the existence of such a poset attached to an association scheme (X; A) does not seem to imply that the scheme is Q-polynomial. It remains open to decide whether there exists a non-Q-polynomial association scheme with an attached Q-poset. Example 2.1: The Shrikhande graph (see [2, p104]) is a distance-regular graph of diameter two. This is a Cayley graph for Z Z with (0; 0) adjacent to (0; 1), (1; 0), and (1; 3). The adjacency matrix of this graph has eigenvalues = 6, = 2, and = ?2. Song [18] found several Q-posets for this association scheme. One of them is described as follows. Let P = fX g and let P = X . Let Y = f(a; b) : b = 0; 1g. There are twelve subsets of X in the orbit of Y under the automorphsim group of this graph. Call these Y ; . . . ; Y and set P = fY ; . . . ; Y g. Now order P = P [ P [ P by reverse inclusion. This is a Q-poset for the Shrikhande graph. For example, the set D = f(a; a) : a 2 Z g is both a poset 1-design and a Delsarte 1-design for this situation. 4
4
0
0
1
1
2
2
1
12
0
1
1
2
12
4
2.2 A combinatorial characterisation of designs in product schemes For 1 i m, let (Yi ; Ai ) be a di -class association scheme with adjacency matrices Ai . The direct product of these schemes is the association scheme (X; A) = (Y ; A ) (Ym ; Am) de ned by X = Y Y Ym 1
1
1
2
7
and
A = f mi Mi : Mi 2 Ai; 1 i mg =1
where
mi Mi = M M Mm 1
=1
2
is the m-fold Kronecker product of matrices. The fact that this always gives an association scheme follows from the properties ( mi Mi )( mi Ni ) = mi (Mi Ni ); =1
=1
=1
( mi Mi) ( mi Ni ) = mi (Mi Ni ) of the Kronecker product. From the rst identity, it follows that the P -matrix for this product scheme is given by the Kronecker product of the P -matrices for the component schemes (Yi ; Ai ). Similarly, the second change-of-basis matrix Q for the product scheme is given by mi Qi where Qi is the Q-matrix for the i component scheme. =1
=1
=1
th
=1
Assume that each component scheme (Yi ; Ai ) has an attached Q-poset (Pi ; i ). Consider the product poset (P ; ) given by
P = fp = (p ; p ; . . . ; pm) : pi 2 Pig 1
2
with p q if pi i qi for each i. On this poset, we have a natural vector-valued height function. If p = (p ; p ; . . . ; pm) and `(pj ) = ij for 1 j m, then we will write `(p) = (i ; i ; . . . ; im) and we will say p has height (i ; i ; . . . ; im). Let Ck denote the totally ordered chain on [k]. If, for each i, the i component scheme has di classes, then the vector-valued height function ` is an order-preserving map from the product poset P onto the product of chains C = Cd1 Cdm . (We will also use to denote the partial order on C .) For j = (j ; . . . ; jm ) 2 C , Wj will denote the incidence matrix of objects in P of height j versus objects of maximum height, which are just the elements of X . Similarly, the adjacency matrices and primitive idempotents of the product association scheme are indexed by elements of C . Finally, if D X has inner distribution a, the entries of both a and its MacWilliams transform aQ are indexed by vectors in C . Recall that T C is a downset in (C ; ) if, whenever j 2 T and i j , we necessarily have i 2 T . For a xed downset T C , we say that D X is a poset T -design if there exist constants j (j 2 T ) such that, for every p in P , `(p) = j 2 T implies 1
2
1
1
2
th
1
jfx 2 D : p xgj = j : In the special case where t is a positive integer and
T = fj 2 C : j + + jm tg; 1
we will simply say that D is a poset t-design. 8
2
We now present a few examples, limiting attention to products of Hamming lattices and truncated Boolean lattices. Since all Q-posets involved are regular, it suces to specify j only for the maximal elements j 2 T . All remaining constants j (j 2 T ) can be computed from these. In such cases, our de nition of T -design is highly redundant. Example 2.2: A Room square of side 2v ? 1 (see [7]) is an arrangement of all unordered pairs from a set V of size 2v inside a (2v ? 1) (2v ? 1) array in such a way as to satisfy the following conditions: each cell of the array contains at most one unordered pair; each unordered pair of elements from V appears exactly once in the array; in each row and in each column, the unordered pairs which occur form a one-factor (i.e., a partition of V into 2-sets). Such an object can be viewed as a poset T -design in (the product poset naturally associated to) H (2; 2v ? 1) J (2v; 2) where T = f(0; 0); (1; 0); (0; 1); (1; 1); (0; 2)g and ; = 1. (An extension of this concept is the Room d-cube which can be encoded as a T -design in H (d; 2v ? 1) J (2v; 2) with restrictions on its inner distribution.) More generally, let B denote a (possibly trivial) 2-(v; k; ) block design. An -resolution of B is a partition of the blocks of B into 1-(v; k; ) designs, called resolution classes. Two resolutions are orthogonal if any resolution class from one shares at most one block in common with any resolution class from the second. It is not dicult to see that a set of n pairwise orthogonal resolutions of a 2-(v; k; ) block design is equivalent to a T -design in H (n; q)
J (v; k) where q = (v ? 1)=( (k ? 1)) is the number of resolution classes in any resolution, T = f(0; 0); (1; 0); (0; 1); (1; 1); (0; 2)g; ; = , and the inner distribution a satis es a i;j = 0 unless i = j = 0 or i d ? 1. (This last condition ensures orthogonality.) Example 2.3: Let B be the set of blocks of a t-(v; k; ) design. A resolution of strength s is a partition of B into s-(v; k; ) designs for some . Such a structure is equivalent to a T -design (of speci ed cardinality) in J (w; 1) J (v; k) where ? ? vt ks w = ?v?k s t and T = f(0; 0); (0; 1); . . . ; (0; t); (1; 0); (1; 1); . . . ; (1; s)g: The most familiar case is t = 2, s = 1. These are called -resolvable designs. These designs are well-studied (see also [11]). 2 As is standard, let T = T ? f0g. We are now prepared to prove the main result of the paper. (1 1)
(1 1)
(
9
)
Theorem 2.3 For 1 i m, let (Yi; Ai) be a symmetric di-class association scheme with primitive idempotents Ei ; Ei . . . ; Eidi (in that order). Let (Pi ; i) be a Q-poset for (Yi; Ai ) with incidence matrices Wij (0 j di). Let (X; A) be the product association scheme with primitive idempotents Ej (j 2 C ). For 0
each such j , de ne
1
Wj = mi Wiji : =1
Let (P ; ) denote the product poset constructed as above from the (Pi ; i). Let T be any downset in C . Then for D X with characteristic vector D , the following are equivalent: (i) D is a poset T -design in (P ; ); (ii) there exist constants j , j 2 T such that Wj D = j 1; (iii) for j 2 T , Ej D = 0 (i.e., D is a Delsarte T -design); (iv) for j 2 T , (aQ)j = 0.
Proof: The equivalence of (i) and (ii) follows from the fact that Wj is the incidence matrix of objects of height j versus objects in X in the product poset (P ; ). The equivalence of (iii) and (iv) is again entirely standard. (ii))(iii): Let j 2 T and let e = mi ei be any row of Ej = mi Eiji where each ei is a row of Eiji . Then e is orthogonal to the all-ones vector 1 since j = 6 0. Since each (Pi; i) is a Q-poset for (Yi ; Ai ), there exist vectors r ; . . . ; rm such that ei = ri Wiji . So we have 0 = e1T = ( i riWiji )( i 1T ) = ( iri )( i Wiji 1T ) = m( iri )1T where i is the (constant) row sum of Wiji . This shows that ( i ri)1T = 0. Now we are able to compute the inner product he; D i: ( i riWiji ) TD = ( i ri)Wj TD = ( i ri)j 1T where j is as purported in (ii). Putting this together with the above, we get he; D i = 0. Since e was an arbitrary row of Ej , we obtain Ej D = 0 as desired. (iii))(ii): Let j 2 T and assume Ek D = 0 for all k 2 T ? f0g with k j. Take any row w = iwi of Wj . By axiom (3) for the Q-poset (Pi ; i), we can write =1
=1
1
1
wi =
ji X k=0
2
eik
for each 1 i m where eik = wi Eik 2 rowsp Eik . Note that ei = i 1 where i is independent of the choice of row wi . (This follows from the observation that 0
rowsum Wiji = hwi ; 1i =
10
ji X k=0
heik ; 1i = ijYij:)
Hence
hw; Di =
X kj
h ieiki ; Di = h iei ; D i 0
since, for k 6= 0, we have Ek D = 0 and ieiki is in the rowspace of Ek . So we have
hw; Di = mh1; D i = mjDj; which is independent of the row w chosen. Hence this nal quantity is our j and we have proven Wj D = j 1. 2 1
2
1
2
An immediate consequence of this result is a linear programming bound for any family of combinatorial objects which canPbe characterized by condition (i) of the theorem. Specifically, one may minimize jDj = ai subject to the inequalities of Theorem 1.1 and the equations given by condition (iv), yielding a lower bound on the size of D.
Example 2.4: In [10], Levenshtein introduces the concept of a split orthogonal array. Given q; n ; n ; t ; t , we wish to nd an M (n + n ) array with entries in Q = f0; 1; . . . ; q ? 1g 1
2
1
2
1
2
such that, upon choosing any t columns from among the rst n columns and any t columns from among the last n columns, all (t + t )-tuples over the alphabet Q occur equally often. This is equivalent to a T -design in the product scheme H (n ; q) H (n ; q) where 1
1
2
1
2
2
1
2
T = f(i ; i ) : 0 i t ; 0 i t g: 1
2
1
1
2
2
For such objects, Delsarte's linear programming bound (Thm. 1.1) specialises to [10, Thm. 6.3] and our main theorem above appears (in part) as [10, Thm. 5.9]. 2
Example 2.5: Mixed-level orthogonal arrays are studied by Sloane and Stufken in [17]. A mixed-level orthogonal array OA(M; qn1 qn2 qmnm ; t) of strength t is an M n matrix P (n = ni ) whose columns are broken into m blocks | the i block contains ni columns and has entries from f0; 1; . . . ; qi ? 1g | having the property that, upon choosing any t columns, any two t-tuples which could possibly occur (given the alphabets) appear equally often. These designs are of particular interest in industrial engineering. Observe that the above de nition is equivalent to a poset t-design in a product of Hamming lattices. Hence, by the result above, it is also equivalent to a Delsarte T -design in P the scheme H (n ; q ) H (nm; qm) where T = fj : ji tg. In this case, Delsarte's linear programming bound appears as [17, Theorem 1] and the theorem above specialises to imply [17, Theorem 2]. 2 1
1
2
th
1
Example 2.6: In the product of Johnson schemes J (v ; k ) J (v ; k ), consider Delsarte T -designs for T = f(i ; i ) : i + i tg. These are called mixed t-designs and are studied 1
1
2
1
1
2
2
2
in [11] where the results established here are applied to that case. In combinatorial terms, a mixed t-(v ; k ; v ; k ; ) design is a collection D of ordered pairs (b ; b ) where bi is a ki-subset 1
1
2
2
1
11
2
of a xed set Vi of vi points having the property that, for any t ; t with t + t t, there is a constant t1 ;t2 such that the number of pairs (b ; b ) in D with Si bi is independent of the choice of S and S for any subsets Si Vi satisfying jSi j = ti . The equivalence is established using the above theorem via the obvious characterisation in terms of poset t-designs in the product of two truncated Boolean lattices. See [11] for a construction of such designs from ordinary t-designs using the (generalised) Assmus-Mattson Theorem [6]. 2 Example 2.7: Consider a product scheme of the form H (n; q) J (v; k). The associated poset is a product of a Hamming lattice and a truncated Boolean lattice. Let us call a poset t-design in this poset a fused orthogonal array design of strength t. The elements of D are ordered pairs with rst coordinates forming an orthogonal array and second coordinates forming a combinatorial t-design. If we specify any i positions in the rst coordinate, the value of the n-tuples in these positions determines a partition of D and this must be a resolution of the design formed by the second coordinates into qi designs of equal size and strength (at least) t ? i. Similarly, if we specify any i symbols and isolate those pairs whose second coordinate contains these points, the corresponding rst coordinates form an orthogonal array of strength at least t ? i. A simple example is the following 2-design in H (4; 2) J (4; 3): 1
1
1
(0000; f1; 2; 3g), (0011; f1; 2; 4g), (0101; f1; 3; 4g), (1001; f2; 3; 4g),
2
1
2
2
2
(1111; f1; 2; 3g), (1100; f1; 2; 4g), (1010; f1; 3; 4g), (0110; f2; 3; 4g). 2
3 Bounds for designs in products of Q-polynomial schemes
3.1 The Delsarte bound
One well-known theorem [4, Theorem 5.19] of Delsarte gives a lower bound on the size of a t-design in a Q-polynomial association scheme. This bound has both the Rao bound for orthogonal arrays and the bound of Ray-Chaudhuri and Wilson for block designs as special cases. In this section, we generalise this bound further to the case of T -designs in products of Q-polynomial association schemes. Let (X; A) be an association scheme with v = jX j. Consider the Krein parameters qij (k) de ned by d X 1 Ei Ej = v qij (k)Ek : k =0
12
The well-known Krein condition states that each qij (k) 0. We say (X; A) is Q-polynomial if the following conditions are satis ed for all i; j; k: (k > i + j ) qij (k) = 0) and (i + j d ) qij (i + j ) > 0): (See [1, Sec. III.1] or [2, Sec. 2.7] for much more information.) Let C be the poset formed by the product of chains Cd1 Cdm . For E C , de ne E + E = f(i + j ; . . . ; im + jm) : i; j 2 Eg: For j 2 C , let fj = rank Ej . 1
1
Lemma 3.1 Let (X; A) be the product of association schemes (Yh; Ah ) (1 h m). Let the Krein parameters of (Yh ; Ah ) be denoted by qihjh (kh ). Then for i; j; k 2 C , the corresponding Krein parameter of (X; A) is given by qij (k) =
m Y
h=1
qih jh (kh ):
Furthermore, if each component scheme is Q-polynomial, then we have qij (k ) = 0 whenever there exists an h for which kh > ih + jh. 2
Theorem 3.2 (Delsarte Bound) Let (X; A) be the product of Q-polynomial association schemes (Yi ; Ai ) (1 i m). Let T be a downset in C and let D X be a Delsarte T -design. If E C satis es (E + E ) \ C T , then X jDj fj : j2E
Moreover, if equality holds, then, for ` 6= 0,
X j2E
Q`j = 0
whenever D contains a pair of `-related elements.
Proof: We begin with a derivation of the dual of Delsarte's linear program for T -designs. Each matrix M 2 A may be expanded in the form M =v If we change bases, we have
M=
X j 2C
X i2C
13
j Ej : i Ai
where
i =
X j 2C
(i 2 C ):
Qij j
Consider only matrices M satisfying the conditions (a) M is a non-negative matrix; (b) j 0 for j 62 T ; (c) = 1. Let D be any Delsarte T -design in (X; A). Since M is non-negative, we have 0
TD M D =
X i2C
i TD Ai D TD A D = jDj: 0
On the other hand, we have X TD M D = v j TD Ej D j 2C
= v TD E D + v 0
0
X j2T
0
0
j TD Ej D + v
X j 62T
j TD Ej D :
Since = 1, E = v J and Ej D = 0 for j 2 T , this reduces to 0
0
1
TD M D = jDj + v 2
X j 62T
j TD Ej D ;
which is bounded above by jDj since, for j 62 T , j 0 and TD Ej D 0. Putting the two inequalities together gives the bound jDj: In this way, each matrix M 2 A satisfying conditions (a)-(c) gives a lower bound on the size of a T -design. (Notice that is the value on the diagonal of M .) Now let E T be as hypothesized. Consider the matrix X N = Ej : 2
0
0
j 2E
If we square each entry of this matrix, we obtain the non-negative matrix
N N =
XX h2E i2E
Eh Ei =
0 X @1 X X j2C
14
v h2E
i2E
1 qhi (j )A Ej :
Notice that, by Lemma 3.1, qhi (j ) = 0 whenever h; i 2 E and j 62 T . Thus the coecient of P Ej vanishes whenever j 62 T . On the other hand, since qhi (0) = h;i fh , we have v E fh as the coecient of E in this expansion. Let
= Pv f ; h2E h and take M = (N N ). Then M is non-negative and, if we expand X M = v j Ej ; 1
0
2
j 2C
we obtain = 1 and j = 0 for j 62 T . So M satis es conditions (a), (b), and (c). Consequently, for any T -design D, the size of D is bounded below by the common value of the diagonal entries of M . P The diagonal entries of Eh are all equal to fh =v; the diagonal entries of N are therefore v E fh . So each diagonal entry of M is equal to 0
1
P f! X h2E h
= fh : v 2
h2E
This proves the inequality
jDj
X h2E
fh :
Now if equality holds in this last inequality, then we must have equality throughout. In particular, for each ` 6= 0, we must have ` ( TD A` D ) = 0. In other words, if ` 6= 0, and D contains two `-related elements, then X j Q`j = 0: 2 j 2C
Example 3.1: For split orthogonal arrays, we obtain the bound M
bX t1 =2c bX t2 =2c i=0 j =0
n i
1
n
i j (q ? 1) 2
j
+
on the number of rows, M , in the array. Obviously, the right-hand side factors into the product of the ordinary Rao bounds for OA(ti; ni ; q), i = 1; 2. This result is implied by Theorem 6.3 in [10]. 2
Example 3.2: For mixed-level orthogonal arrays, the theorem above gives
X
m Y nh
i1 ++im bt=2c h=1
15
ih ( q ? 1) h ih
as a lower bound on the number of rows of the array. This bound was already derived by elementary means in [9]. Now if equality holds in the bound, the second statement of the theorem forces many entries of the inner distribution to equal zero. For example, in the case t = 2, if ` 6= 0 and D contains a pair of `-related elements, then ` q + `mqm = jDj. For t = 2 and 4, these restrictions were already found by Mukerjee and Wu [15]. 2 1 1
Example 3.3: For mixed t-designs in J (v ; k ) J (v ; k ), we may take E = f(i ; i ) : i + i ug where u = bt=2c. This gives the bound 1
1
2
1
2
2
1
2
X v v v v jDj ? i ?1 ? i ?1 ; i i i i u 1
0
which simpli es to
1
1
1+ 2
2
1
2
2
2
v + v v + v jDj u ? u ? 1 : 1
2
1
2
For example, for t = 2, we nd jDj v + v ? 1 and for t = 4, jDj (v + v ? 3)(v + v ). 1
1 2
2
1
2
1
2
2 Example 3.4: For a fused orthogonal array design of strength t in H (n; q) J (v; k), we
obtain
v v X n j jDj (q ? 1) j ? j ?1 j j u j 1
1
1+ 2
2
2
where u := bt=2c. This can be written in terms of a hypergeometric series:
v
jDj u
2
?u; ?n F ; q?1 : 2 1
v?u+1
3.2 The degree bound and association schemes induced by designs
Continuing the treatment in parallel to that of designs in Q-polynomial association schemes, we prove in this section a relationship between the size of T and the degree s of a T -design, focusing on the case where this bound is tight. Let D be a design in an association scheme (X; A). Let A denote the Bose-Mesner algebra of (X; A). For M 2 A , denote by M the submatrix of M whose rows and columns are indexed by the elements of D. In the case where (X; A) is Q-polynomial and D has degree s and strength t, a well-known result [4, Theorem 5.25] of Delsarte states that bt=2c s and, if t 2s ? 2, then the vector space A of submatrices fM : M 2 A g is the Bose-Mesner algebra of a (Q-polynomial) association scheme. We express this by saying that D induces an association scheme in (X; A). 16
Now let D be a Delsarte T -design in a product (X; A) of Q-polynomial association schemes. Recall that the degree of D is the number of indices i 6= 0 for which Ai is a nonzero matrix. Call D a tight design in (X; A) if, for some E , equality holds in the bound of Theorem 3.2.
Theorem 3.3 Let D be a Delsarte T -design of degree s in the product (X; A) of Qpolynomial association schemes. If (E + E ) \ C T , then jEj s + 1: Moreover, if equality holds, then D is a tight design and D induces an association scheme in (X; A). Secondly, if jEj = s, then either D is a tight design or D induces an association scheme in (X; A).
Proof: The proof is modelled after the proof of Theorem 5.25 in [4]. Let S be a v v matrix diagonalising the Bose-Mesner algebra A of the product scheme. Then S may be partitioned into v fj submatrices Sj where the columns of Sj form an orthonormal basis for colsp Ej (j 2 C ). Clearly Sj SjT = Ej . Let Hj be the submatrix of Sj
obtained by restriction to the rows indexed by elements of D. Then Hj HjT = Ej . For i; j 2 C , we have (Delsarte [4, Theorem 3.15]):
qij (k)(aQ)k = 0 for all k 2 C ? f0g if and only if
0;
if i 6= j ; = jDjI; if i = j . Consider the vector space A = fM : M 2 A g. This is spanned by the matrices fAi : i 2 Cg and so has dimension s + 1 where s is the degree of D. For i; j 2 E , we apply Lemma 3.1 to conclude that qij (k) = 0 whenever k 62 T . As Ei Ej = HiHiT Hj HjT , we obtain (6) Ei Ei = jDj Ei and (assuming i 6= j ) (7) Ei Ej = 0: So these jEj members of A are linearly independent. This proves that jEj s + 1. If equality holds, then (by (6) and (7)) A has a basis of mutually orthogonal idempotents and hence is closed under matrix multiplication. It is easy to see that this vector space is closed under Schur multiplication and contains I and J . Therefore it is the Bose-Mesner algebra of an association scheme (see [2, Theorem 2.6.1]).
HiT Hj
17
Write E = fi ; . . . ; ir g. De ne 1
GE = [Hi1 Hi2 Hir ]: P GE has jDj rows and rj fij columns. From the preceding observations, we see that GTE GE = jDjI . (This gives an alternative proof of the bound in Theorem 3.2.) p If D is not a tight design, we may add columns to form an orthogonal matrix (1= jDj)G with G = [GE K ]: Then GGT = jDjI , yielding r X T KK = jDjI ? Hij HiTj ; =1
j =1
Now HiHiT = Ei . Thus KK T 2 A and, by Equations (6) and (7), ( 1 KK T )Ei = 0 jDj for i 2 E . Since K T K = I , KK T is not the zero matrix. So we have jEj + 1 linearly independent matrices in A . Thus, if jEj = s + 1, D must be a tight design. Finally, assume jEj = s and that D is not a tight design. If we take c = 1=jDj, the matrices fcEj : j 2 Eg [ fcKK T g form a basis of primitive idempotents for A . So A is closed under both ordinary and Schur multiplication and contains I and J . Consequently, A is the Bose-Mesner algebra of an association scheme. 2 For example, if D is a t-design in a product of m Q-polynomial schemes, then the degree of D is at least the size of
E = f(i ; . . . ; im) : 1 i + im t=2; all ij 0g: ? This gives the bound s + 1 mm u where u = bt=2c. For example, if D is a mixed-level orthogonal array of strength two in H (n ; q ) H (nm ; qm) then ai is non-zero for at 1
1
+
least m non-zero index vectors i.
1
1
Remarks: (i) If D is a tight design, the bound jEj s may fail. For example, consider a projective
plane 2-(n + n + 1; n + 1; 1). Delete a block x and let D denote the set of n + n remaining blocks, viewed as a 2-design in J (n +1; 1) J (n ; n). Then D has degree two in this product scheme yet the set E = f(0; 0); (0; 1), (1; 0)g satis es (E + E ) \C T . See [11] for an investigation of tight 2-designs in J (v ; k )
J (v ; k ). 2
2
2
1
2
2
18
1
(ii) In contrast to the situation where (X; A) is Q-polynomial, the induced association
scheme found above is not necessarily Q-polynomial, even though we have assumed that each component scheme is Q-polynomial.
4 Summary Two decades ago, a theory was established for the study of t-designs in Q-polynomial association schemes. In certain cases, these objects have useful combinatorial interpretations. Motivated by several applications, the present paper begins to establish a theory of T designs in products of Q-polynomial association schemes. In the language of Q-posets, a combinatorial characterisation (Theorem 2.3) of such objects is given. Independent of this characterisation, two results are proved for designs in products of Q-polynomial association schemes which are analogues of Delsarte's results on t-designs. In addition, the linear programming bound for such designs comes \for free". A variety of questions arise from this investigation. Which association schemes admit a Qposet? Must such a scheme be Q-polynomial? Is the bound in Theorem 3.2 the best possible? What are the tight designs? Do any new association schemes arise from Theorem 3.3? The theory in the latter part of the paper can be extended to arbitrary association schemes, but will only be non-trivial in cases where a large number of Krein parameters vanish. See [13] for an application of such techniques to a problem in numerical integration. It also seems useful to focus on particular classes of product schemes, such as products of Hamming schemes (split orthogonal arrays and mixed orthogonal arrays), products of Johnson schemes (mixed block designs [11] and split designs [12]), or H (n; q) J (v; k) (Room d-cubes and their generalisations). Acknowledgements: The author is grateful to S. Song and C. Godsil for helpful comments. In addition, the referee o ered several suggestions which improved the manuscript.
References [1] E. Bannai and T. Ito, \Algebraic Combinatorics I: Association Schemes," BenjaminCummings Lecture Note Ser. 58, The Benjamin-Cummings Publishing Company, Inc., London, 1984. [2] A.E. Brouwer, A.M. Cohen, and A. Neumaier, \Distance-Regular Graphs," Springer-Verlag, Berlin, 1989. [3] C.J. Colbourn and J.H. Dinitz, \Handbook of Combinatorial Designs," CRC Press, Boca Raton, 1996. 19
[4] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl. 10 (1973). [5] P. Delsarte, Association schemes and t-designs in regular semilattices, J. Combin. Th. (A) 20 (1976), 230-243. [6] P. Delsarte, Pairs of vectors in the space of an association scheme, Philips Res. Reports 32 (1977), 373-411. [7] J.H. Dinitz and D.R. Stinson (eds.), \Contemporary Design Theory: A Collection of Surveys," Wiley, New York, 1992. [8] C.D. Godsil, \Algebraic Combinatorics," Chapman and Hall, London, 1993. [9] A.S. Hedayat, N.J.A. Sloane and J. Stufken, \Orthogonal Arrays," SpringerVerlag, New York, 1998. [10] V. Levenshtein, Split orthogonal arrays and maximum resilient systems of functions, Designs, Codes and Crypt. 12 (1997), 131-160. [11] W.J. Martin, Mixed block designs, J. Combin. Designs 6 #2 (1998), 151-163. [12] W.J. Martin, Completely regular designs, to appear in: J. Combin. Designs. [13] W.J. Martin and D.R. Stinson, Association schemes for ordered orthogonal arrays and (T; M; S )-nets, (1997), submitted for publication. [14] A.D. Meyerowitz, Cycle-balanced partitions in distance-regular graphs, preprint. [15] R. Mukerjee and C.F.J. Wu, On the existence of saturated and nearly saturated asymmetrical orthogonal arrays, Ann. Stat. 23 (1995), 2102-2115. [16] C. Roos, On antidesigns and designs in an association scheme, Delft Progress Report 7 (1982), 98-109. [17] N.J.A. Sloane and J. Stufken, A linear programming bound for orthogonal arrays with mixed levels, J. Stat. Plan. Inf. 56 (1996), 295-306. [18] S.Y. Song, Posets related to some association schemes, Journal of the Korean Society of Mathematical Education 25 (1987), 57-69. [19] D. Stanton, t-Designs in classical association schemes, Graphs Combin. 2 (1986), 283-286. [20] P. Terwilliger, Incidence algebras of uniform posets, in: \Coding Theory and Design Theory, Proceedings of the IMA" (D. K. Ray-Chaudhuri, ed.) Springer-Verlag, New York, 1991. 20
Let (Y; A) be an association scheme with primitive idempotents E ; E ; . . . ; Ed . For T f1; . . . ; dg, a Delsarte T -design in (Y; A) is a subset D of Y whose characteristic vector is annihilated by the idempotents Ej (j 2 T ). The case most studied is that in which (Y; A) is Q-polynomial and T = f1; . . . ; tg. For many such examples, a combinatorial characterization is known, giving an equivalence between Delsarte T -designs and 0
1
poset t-designs in what we call here \Q-posets". For example, combinatorial t-designs (i.e., block designs) can be described via the truncated Boolean lattice while orthogonal arrays can be described via the Hamming lattice. For 1 i m, let (Yi; Ai ) be a Q-polynomial association scheme. Assume that Delsarte t-designs in each (Yi; Ai ) are characterised as poset t-designs in a Q-poset Pi attached to that scheme. With these assumptions, we consider the product association scheme (Yi ; A). The primitive idempotents for this scheme naturally inherit the partial order structure of a product of chains. Our main result, Theorem 2.3, characterises Delsarte T -designs in (Yi; A) as poset designs in the product poset Pi where T is any downset in the product of chains. Using this characterisation, we immediately obtain linear programming bounds for a wide variety of combinatorial objects. On the other hand, if we assume each component scheme is Q-polynomial, we obtain bounds on the size and degree of such a T -design analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.
1 Overview and Background We are interested in applications of the theory of association schemes to problems in coding theory and design theory. This investigation is motivated by two applications. In [10], The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through grant OGP0155422 1
1
Levenshtein introduces split orthogonal arrays (see Example 2.4 below) and applies the theory of Krawtchouk polynomials to obtain bounds on the size of such objects. It was clear to the author early on that similar results hold for mixed-level orthogonal arrays. Independently, Sloane and Stufken [17] derived a linear programming bound for these objects. Our approach shows how the two studies can be carried out simultaneously and elegantly if one considers the product of Hamming association schemes and the associated product of Hamming lattices. In fact, much more can be accomplished if one abstracts the essential features of the relationship between the Hamming association scheme and (incidence matrices of) the Hamming lattice. With this motivation, we introduce in Section 2.1 the concept of a Q-poset attached to an association scheme. (While the de nition can be applied to arbitrary association schemes, the case where the scheme is Q-polynomial is our focus.) The main result of the paper is Theorem 2.3 in Section 2.2 which uses these partially ordered sets to characterise Delsarte T -designs in a product association scheme where each component scheme is assumed to have an attached Q-poset and where T is a downset in a product of chains. The balance of the paper explores bounds on T -designs in product schemes where each component is Q-polynomial. The general results include: a linear programming bound (which falls out immediately from Theorem 2.3 using the standard theory); a Delsarte bound (Theorem 3.2) analogous to the Rao bound for orthogonal arrays and the RayChaudhuri/Wilson bound for combinatorial t-designs; a degree bound (Theorem 3.3). Numerous examples are included to demonstrate the power and wide applicability of these bounds.
1.1 Association Schemes
All material in this section is standard (see [4, 1, 2, 8]). Let X be a nite set of cardinality v > 0 and let A = fA ; A ; . . . ; Ad g be a set of v v symmetric 01-matrices with rows and columns indexed by X and with linear span A Rvv. We say that (X; A) is a (symmetric) association scheme if A satis es (i) A = I ; (ii) Pdi Ai = J (the all-ones matrix); (iii) for 0 i; j d, AiAj 2 A . To each matrix Ai (i 6= 0) is associated its graph Gi with vertex set X : Ai is the adjacency matrix of Gi . The above de nition is customarily given in the language of relations. We will use graph and matrix terminology interchangeably. The vector space A spanned by the matrices in A is, by condition (iii), an algebra; this is the Bose-Mesner algebra of the association scheme. Extensive background material on association schemes can be found in the references. 0
0
=0
2
1
A. The Hamming Scheme
In terms of applications, two important association schemes are the Hamming scheme and the Johnson scheme. The Hamming scheme H (n; q) has as its vertices the set X of all words of length n over an alphabet Q having q elements. The graphs G ; . . . ; Gn are de ned by Hamming distance: if x; y 2 X , then (x; y) 2 Gk where k is the number of positions in which the words x and y di er. It is well-known that the mapping (x; y) 7! k is a metric. So the entire association scheme is determined by the Hamming graph G . Thus the Hamming scheme is a metric (hence, P -polynomial) association scheme. (Metric association schemes are the same as distance-regular graphs.) As examples, for q = 2, G is the n-cube and for n = 2, G is the grid graph Kq Kq . Several authors have associated to this scheme a partially ordered set (P ; ), commonly called the Hamming lattice (see [5, 19, 14, 8]). Adjoin to the alphabet Q a new symbol \". The elements of P are the words of length n over Q [ fg. For x; y 2 P , we have x y if and only if, for all i, yi = implies xi = . This poset has rank function ` given by `(x) = jfi : xi 6= gj. So the elements of X are precisely the members of P having maximum rank n. It is immediately evident that a subset D X is an orthogonal array of strength t in H (n; q) if and only if, in (P ; ), there exists such that every element of rank t is dominated by exactly members of D. 1
1
1
1
B. The Johnson Scheme Let V be a set of n points and let k satisfy 1 k n=2. The Johnson scheme J (n; k) has vertex set X consisting of all k-element subsets of V and (x; y) 2 Gi (1 i k) if jx \ yj = k ? i. Again, we have a metric association scheme; G is called the Johnson graph. To this association scheme, we attach the truncated boolean lattice (P ; ) where P = fS V : jS j kg with rank function `(S ) = jS j. A combinatorial t-design is simply 1
any subset D of X having the property that every element S of rank t is dominated by exactly members of D for some xed . The Johnson graphs and Hamming graphs are examples of distance-regular graphs and these two families are well-studied. In particular, Delsarte [5] analysed the relationship between these schemes and the above-mentioned posets (as well as their q-analogues) in the language of regular semilattices.
1.2 Linear Programming
Let (X; A) be an association scheme with adjacency matrices A = fA ; . . . ; Ad g and BoseMesner algebra A . Since the Ai are 01-matrices with pairwise disjoint supports, this algebra is also closed under Schur (entrywise) multiplication, denoted . The matrices Ai form a basis of mutually orthogonal idempotents for A with respect to this multiplication (Ai Aj = i;j Ai). There is also a unique basis fE ; E ; . . . ; Ed g of mutually orthogonal idempotents with respect to ordinary matrix multiplication (which we call the \primitive idempotents"). 0
0
1
3
The change-of-basis matrices P and Q, de ned by
Ai =
d X j =0
PjiEj
and
d X 1 Ej = v Qij Ai i
(1)
=0
satisfy the orthogonality relations PQ = vI and MP = QT K (2) where K is a diagonal matrix with Kii equal to the valency of (the regular graph) Gi and M is a diagonal matrix with Mjj = fj := rank Ej . We refer to these simply as the \P -matrix" and \Q-matrix" of the scheme. For the convenience of the reader mainly interested in applications, we record here the change-of-basis matrices for the two examples described above. For the Hamming scheme H (n; q), we have Pji = Ki (j ) where
Ki (x) =
Xi
(?1)` (q ? 1)i?`
xn ? x
(3) ` i ? ` ` is the Krawtchouk polynomial familiar to coding theorists and we have Q = P . (This is a self-dual association scheme.) For the Johnson scheme J (n; k), we have Pji = Ei (j ) where =0
Ei (x) =
Xi
(?1)`
xk ? xn ? k ? x
(4) i?` i?` Hahn polynomial of degree i. The rank of Ej in this case is given by fj = ?dual ?isnthe n , the valency of G is given by k = ?k?n?k, and Q is obtained from P using the ? i i i i j? j orthogonality relations above. Given any subset D of the vertices of an association scheme (X; A) on v points, we build the characteristic vector of D, denoted D , which is a 01-vector of length v with y-entry equal to 1 if and only if y 2 D. We are also interested in the inner distribution vector of D: a = [a ; a ; . . . ; ad] is de ned by (5) ai = jD1 j TD Ai D which, in the case of a distance-regular graph, gives the average number of elements of D at distance i from any element of D. For linear codes, a is simply the list of coecients of the weight enumerator. For combinatorial t-designs, ai measures the average number of blocks meeting a given block in k ? i points. Consider the product aQ, which is called the MacWilliams transform of a. By Equation (1b), we have (aQ)j = v TD Ej D jDj and, since Ej = Ej , this quantity is always non-negative (see also [2, Sec. 2.5]): `=0
`
1
0
1
2
4
Theorem 1.1 (Delsarte's Linear Programming Bound [4, Thm. 3.3]) For any sub-
set D of the vertices X of an association scheme, the MacWilliams transform of the inner distribution vector is non-negative: aQ 0. 2
For T f1; . . . dg, a Delsarte T -design is a subset D X satisfying Ej D = 0 for j 2 T . This is clearly equivalent to the condition (aQ)j = 0 for all j 2 T . In the case where T = f1; . . . ; tg, we will simply say D is a Delsarte t-design.
2 Combinatorial characterizations of Delsarte T -designs 2.1 Q-posets
Let (P ; ) be a partially ordered set (poset). If x y, we say y dominates x. If x y and there is no z with x z y, we say y covers x. For a non-negative integer d, write [d] = f0; 1; . . . ; dg. A rank function on (P ; ) is a surjection ` : P ! [d] satisfying the following condition: `(x) = i y x `(y) = i + 1. A ranked poset is then simply a poset (P ; ) with a speci ed rank function. If (P ; ) is a ranked poset with rank function `, then P is partitioned into bres P ; P ; . . . ; P d where P i = fx 2 P : `(x) = ig consists of those objects of rank i. For i 2 [d], let Wi be the incidence matrix of the i bre P i versus the top bre P d. The rows of Wi are indexed by elements of P i and the columns are indexed by elements of P d . As indicated above, a number of design-theoretic objects have a common de nition in the language of posets. Let (P ; ) be a ranked poset as above. Extending Delsarte's de nition [5], let us call D P d a poset t-design in (P ; ) if there exist constants ; ; . . . ; t such that, for 0 i t and for all x 2 P i , if
0
and
covers
, then
1
th
0
1
jfy 2 D : x ygj = i: For example, a combinatorial t-design is equivalent to a poset t-design in the truncated Boolean lattice and an orthogonal array of strength t is a poset t-design in the Hamming lattice. Let (X; A) be a symmetric association scheme having d classes and a xed ordering E ; . . . ; Ed on its primitive idempotents. Denote by Vj the j eigenspace, rowsp Ej , of the scheme. We say a ranked poset (P ; ) is a Q-poset for (X; A) (with respect to the ordering 0; 1; . . . ; d) if the following axioms are satis ed: (1) the top bre P d is the set X ; (2) for 0 i d, Wi has constant row sum; th
0
5
(3) for 0 i d, Vi rowsp Wi V V . . . Vi. Note: Unless explicitly stated to the contrary, the only ordering assumed on the primitive 0
1
idempotents of an association scheme is the natural ordering E ; E ; . . . ; Ed . This ordering, usually dictated by the application, is to be assumed in any statement regarding attached Q-posets or the Q-polynomial property. Remark: Viewing the rows of the incidence matrices Wi as subsets of X , our axioms (2) and (3) can be stated as conditions on certain collections Et (0 t d) of \antidesigns". Following Roos [16], we call a subset C of X an antidesign with dual diameter t if Ej C = 0 for j > t. Then our axioms amount to assuming, for each 0 t d, the existence of a collection Et of antidesigns such that the members of Et are equicardinal and their characteristic vectors span a space containing Vt. See Meyerowitz [14] for a case study of the connection between antidesigns and Q-posets. 0
1
Lemma 2.1 Let (L; ) be a regular semilattice [5] and let (X; A) be the association scheme carried by the top bre. Then there exists a subposet (P ; ) of (L; ) which is a Q-poset for (X; A). Proof: See [5] for de nitions of these objects. There, Delsarte proves that there exist bres L ; Lj ; . . . ; Ljd of (L; ) which, together with the incidence matrices between them, satisfy conditions (1)-(3) above. 2 0
1
For example, the Hamming lattice is a Q-poset for the Hamming scheme H (n; q) and, for v 2k, the truncated Boolean lattice is a Q-poset for the Johnson scheme J (v; k). For any Q-poset derived from a regular semilattice in this way, there are constants
i;j = jfy 2 P j : z y xgj; i;j = jfy 2 P j : z ygj which are independent of z 2 P i and x 2 X provided z x. If such constants exist for a Q-poset (P ; ), let us say that (P ; ) is a regular Q-poset (cf. [5]). In such cases, a set D for which t is well-de ned is automatically a t-design in (P ; ). The following lemma was proved in [5] for regular semilattices.
Lemma 2.2 Let (X; A) be a symmetric association scheme. Let (P ; ) be a Q-poset for (X; A) with incidence matrices Wi . Then for D X with characteristic vector D , the following are equivalent: (i) D is a poset t-design in (P ; );
(ii) there exist constants i, 0 i t such that Wi D = i1; (iii) for 1 j t, Ej D = 0; (iv) for 1 j t, (aQ)j = 0. 6
Proof: The equivalence of (i) and (ii) is immediate from the de nitions. The equivalence of (iii) and (iv) is entirely standard [4, p27]. (ii))(iii): Let e be any row of Ej (1 j t). By (3), we may write e = rWj for some vector r of length jP j j. Let be the row sum of Wj (using (2)); then we have Wj 1T = 1T . Since j = 6 0, 0 = he; 1i = rWj 1T = r1T : This allows us to compute he; D i. We have he; Di = rWj D = j r1T = 0: (iii))(ii): Let w be any row of Wi (0 i t). With ej = wEj , we have w = Pij ej using (3). Now if = jX j hw; 1i, then e = 1 and is independent of the choice of w since =0
1
0
Wi has constant row sum. Consequently,
hw; Di =
Xi j =0
hej ; Di = he ; D i = jDj 0
is independent of the choice of w and we have Wi D = i 1T where i jX j = jDj with being the row sum of Wi . 2 While we use the term Q-poset, the existence of such a poset attached to an association scheme (X; A) does not seem to imply that the scheme is Q-polynomial. It remains open to decide whether there exists a non-Q-polynomial association scheme with an attached Q-poset. Example 2.1: The Shrikhande graph (see [2, p104]) is a distance-regular graph of diameter two. This is a Cayley graph for Z Z with (0; 0) adjacent to (0; 1), (1; 0), and (1; 3). The adjacency matrix of this graph has eigenvalues = 6, = 2, and = ?2. Song [18] found several Q-posets for this association scheme. One of them is described as follows. Let P = fX g and let P = X . Let Y = f(a; b) : b = 0; 1g. There are twelve subsets of X in the orbit of Y under the automorphsim group of this graph. Call these Y ; . . . ; Y and set P = fY ; . . . ; Y g. Now order P = P [ P [ P by reverse inclusion. This is a Q-poset for the Shrikhande graph. For example, the set D = f(a; a) : a 2 Z g is both a poset 1-design and a Delsarte 1-design for this situation. 4
4
0
0
1
1
2
2
1
12
0
1
1
2
12
4
2.2 A combinatorial characterisation of designs in product schemes For 1 i m, let (Yi ; Ai ) be a di -class association scheme with adjacency matrices Ai . The direct product of these schemes is the association scheme (X; A) = (Y ; A ) (Ym ; Am) de ned by X = Y Y Ym 1
1
1
2
7
and
A = f mi Mi : Mi 2 Ai; 1 i mg =1
where
mi Mi = M M Mm 1
=1
2
is the m-fold Kronecker product of matrices. The fact that this always gives an association scheme follows from the properties ( mi Mi )( mi Ni ) = mi (Mi Ni ); =1
=1
=1
( mi Mi) ( mi Ni ) = mi (Mi Ni ) of the Kronecker product. From the rst identity, it follows that the P -matrix for this product scheme is given by the Kronecker product of the P -matrices for the component schemes (Yi ; Ai ). Similarly, the second change-of-basis matrix Q for the product scheme is given by mi Qi where Qi is the Q-matrix for the i component scheme. =1
=1
=1
th
=1
Assume that each component scheme (Yi ; Ai ) has an attached Q-poset (Pi ; i ). Consider the product poset (P ; ) given by
P = fp = (p ; p ; . . . ; pm) : pi 2 Pig 1
2
with p q if pi i qi for each i. On this poset, we have a natural vector-valued height function. If p = (p ; p ; . . . ; pm) and `(pj ) = ij for 1 j m, then we will write `(p) = (i ; i ; . . . ; im) and we will say p has height (i ; i ; . . . ; im). Let Ck denote the totally ordered chain on [k]. If, for each i, the i component scheme has di classes, then the vector-valued height function ` is an order-preserving map from the product poset P onto the product of chains C = Cd1 Cdm . (We will also use to denote the partial order on C .) For j = (j ; . . . ; jm ) 2 C , Wj will denote the incidence matrix of objects in P of height j versus objects of maximum height, which are just the elements of X . Similarly, the adjacency matrices and primitive idempotents of the product association scheme are indexed by elements of C . Finally, if D X has inner distribution a, the entries of both a and its MacWilliams transform aQ are indexed by vectors in C . Recall that T C is a downset in (C ; ) if, whenever j 2 T and i j , we necessarily have i 2 T . For a xed downset T C , we say that D X is a poset T -design if there exist constants j (j 2 T ) such that, for every p in P , `(p) = j 2 T implies 1
2
1
1
2
th
1
jfx 2 D : p xgj = j : In the special case where t is a positive integer and
T = fj 2 C : j + + jm tg; 1
we will simply say that D is a poset t-design. 8
2
We now present a few examples, limiting attention to products of Hamming lattices and truncated Boolean lattices. Since all Q-posets involved are regular, it suces to specify j only for the maximal elements j 2 T . All remaining constants j (j 2 T ) can be computed from these. In such cases, our de nition of T -design is highly redundant. Example 2.2: A Room square of side 2v ? 1 (see [7]) is an arrangement of all unordered pairs from a set V of size 2v inside a (2v ? 1) (2v ? 1) array in such a way as to satisfy the following conditions: each cell of the array contains at most one unordered pair; each unordered pair of elements from V appears exactly once in the array; in each row and in each column, the unordered pairs which occur form a one-factor (i.e., a partition of V into 2-sets). Such an object can be viewed as a poset T -design in (the product poset naturally associated to) H (2; 2v ? 1) J (2v; 2) where T = f(0; 0); (1; 0); (0; 1); (1; 1); (0; 2)g and ; = 1. (An extension of this concept is the Room d-cube which can be encoded as a T -design in H (d; 2v ? 1) J (2v; 2) with restrictions on its inner distribution.) More generally, let B denote a (possibly trivial) 2-(v; k; ) block design. An -resolution of B is a partition of the blocks of B into 1-(v; k; ) designs, called resolution classes. Two resolutions are orthogonal if any resolution class from one shares at most one block in common with any resolution class from the second. It is not dicult to see that a set of n pairwise orthogonal resolutions of a 2-(v; k; ) block design is equivalent to a T -design in H (n; q)
J (v; k) where q = (v ? 1)=( (k ? 1)) is the number of resolution classes in any resolution, T = f(0; 0); (1; 0); (0; 1); (1; 1); (0; 2)g; ; = , and the inner distribution a satis es a i;j = 0 unless i = j = 0 or i d ? 1. (This last condition ensures orthogonality.) Example 2.3: Let B be the set of blocks of a t-(v; k; ) design. A resolution of strength s is a partition of B into s-(v; k; ) designs for some . Such a structure is equivalent to a T -design (of speci ed cardinality) in J (w; 1) J (v; k) where ? ? vt ks w = ?v?k s t and T = f(0; 0); (0; 1); . . . ; (0; t); (1; 0); (1; 1); . . . ; (1; s)g: The most familiar case is t = 2, s = 1. These are called -resolvable designs. These designs are well-studied (see also [11]). 2 As is standard, let T = T ? f0g. We are now prepared to prove the main result of the paper. (1 1)
(1 1)
(
9
)
Theorem 2.3 For 1 i m, let (Yi; Ai) be a symmetric di-class association scheme with primitive idempotents Ei ; Ei . . . ; Eidi (in that order). Let (Pi ; i) be a Q-poset for (Yi; Ai ) with incidence matrices Wij (0 j di). Let (X; A) be the product association scheme with primitive idempotents Ej (j 2 C ). For 0
each such j , de ne
1
Wj = mi Wiji : =1
Let (P ; ) denote the product poset constructed as above from the (Pi ; i). Let T be any downset in C . Then for D X with characteristic vector D , the following are equivalent: (i) D is a poset T -design in (P ; ); (ii) there exist constants j , j 2 T such that Wj D = j 1; (iii) for j 2 T , Ej D = 0 (i.e., D is a Delsarte T -design); (iv) for j 2 T , (aQ)j = 0.
Proof: The equivalence of (i) and (ii) follows from the fact that Wj is the incidence matrix of objects of height j versus objects in X in the product poset (P ; ). The equivalence of (iii) and (iv) is again entirely standard. (ii))(iii): Let j 2 T and let e = mi ei be any row of Ej = mi Eiji where each ei is a row of Eiji . Then e is orthogonal to the all-ones vector 1 since j = 6 0. Since each (Pi; i) is a Q-poset for (Yi ; Ai ), there exist vectors r ; . . . ; rm such that ei = ri Wiji . So we have 0 = e1T = ( i riWiji )( i 1T ) = ( iri )( i Wiji 1T ) = m( iri )1T where i is the (constant) row sum of Wiji . This shows that ( i ri)1T = 0. Now we are able to compute the inner product he; D i: ( i riWiji ) TD = ( i ri)Wj TD = ( i ri)j 1T where j is as purported in (ii). Putting this together with the above, we get he; D i = 0. Since e was an arbitrary row of Ej , we obtain Ej D = 0 as desired. (iii))(ii): Let j 2 T and assume Ek D = 0 for all k 2 T ? f0g with k j. Take any row w = iwi of Wj . By axiom (3) for the Q-poset (Pi ; i), we can write =1
=1
1
1
wi =
ji X k=0
2
eik
for each 1 i m where eik = wi Eik 2 rowsp Eik . Note that ei = i 1 where i is independent of the choice of row wi . (This follows from the observation that 0
rowsum Wiji = hwi ; 1i =
10
ji X k=0
heik ; 1i = ijYij:)
Hence
hw; Di =
X kj
h ieiki ; Di = h iei ; D i 0
since, for k 6= 0, we have Ek D = 0 and ieiki is in the rowspace of Ek . So we have
hw; Di = mh1; D i = mjDj; which is independent of the row w chosen. Hence this nal quantity is our j and we have proven Wj D = j 1. 2 1
2
1
2
An immediate consequence of this result is a linear programming bound for any family of combinatorial objects which canPbe characterized by condition (i) of the theorem. Specifically, one may minimize jDj = ai subject to the inequalities of Theorem 1.1 and the equations given by condition (iv), yielding a lower bound on the size of D.
Example 2.4: In [10], Levenshtein introduces the concept of a split orthogonal array. Given q; n ; n ; t ; t , we wish to nd an M (n + n ) array with entries in Q = f0; 1; . . . ; q ? 1g 1
2
1
2
1
2
such that, upon choosing any t columns from among the rst n columns and any t columns from among the last n columns, all (t + t )-tuples over the alphabet Q occur equally often. This is equivalent to a T -design in the product scheme H (n ; q) H (n ; q) where 1
1
2
1
2
2
1
2
T = f(i ; i ) : 0 i t ; 0 i t g: 1
2
1
1
2
2
For such objects, Delsarte's linear programming bound (Thm. 1.1) specialises to [10, Thm. 6.3] and our main theorem above appears (in part) as [10, Thm. 5.9]. 2
Example 2.5: Mixed-level orthogonal arrays are studied by Sloane and Stufken in [17]. A mixed-level orthogonal array OA(M; qn1 qn2 qmnm ; t) of strength t is an M n matrix P (n = ni ) whose columns are broken into m blocks | the i block contains ni columns and has entries from f0; 1; . . . ; qi ? 1g | having the property that, upon choosing any t columns, any two t-tuples which could possibly occur (given the alphabets) appear equally often. These designs are of particular interest in industrial engineering. Observe that the above de nition is equivalent to a poset t-design in a product of Hamming lattices. Hence, by the result above, it is also equivalent to a Delsarte T -design in P the scheme H (n ; q ) H (nm; qm) where T = fj : ji tg. In this case, Delsarte's linear programming bound appears as [17, Theorem 1] and the theorem above specialises to imply [17, Theorem 2]. 2 1
1
2
th
1
Example 2.6: In the product of Johnson schemes J (v ; k ) J (v ; k ), consider Delsarte T -designs for T = f(i ; i ) : i + i tg. These are called mixed t-designs and are studied 1
1
2
1
1
2
2
2
in [11] where the results established here are applied to that case. In combinatorial terms, a mixed t-(v ; k ; v ; k ; ) design is a collection D of ordered pairs (b ; b ) where bi is a ki-subset 1
1
2
2
1
11
2
of a xed set Vi of vi points having the property that, for any t ; t with t + t t, there is a constant t1 ;t2 such that the number of pairs (b ; b ) in D with Si bi is independent of the choice of S and S for any subsets Si Vi satisfying jSi j = ti . The equivalence is established using the above theorem via the obvious characterisation in terms of poset t-designs in the product of two truncated Boolean lattices. See [11] for a construction of such designs from ordinary t-designs using the (generalised) Assmus-Mattson Theorem [6]. 2 Example 2.7: Consider a product scheme of the form H (n; q) J (v; k). The associated poset is a product of a Hamming lattice and a truncated Boolean lattice. Let us call a poset t-design in this poset a fused orthogonal array design of strength t. The elements of D are ordered pairs with rst coordinates forming an orthogonal array and second coordinates forming a combinatorial t-design. If we specify any i positions in the rst coordinate, the value of the n-tuples in these positions determines a partition of D and this must be a resolution of the design formed by the second coordinates into qi designs of equal size and strength (at least) t ? i. Similarly, if we specify any i symbols and isolate those pairs whose second coordinate contains these points, the corresponding rst coordinates form an orthogonal array of strength at least t ? i. A simple example is the following 2-design in H (4; 2) J (4; 3): 1
1
1
(0000; f1; 2; 3g), (0011; f1; 2; 4g), (0101; f1; 3; 4g), (1001; f2; 3; 4g),
2
1
2
2
2
(1111; f1; 2; 3g), (1100; f1; 2; 4g), (1010; f1; 3; 4g), (0110; f2; 3; 4g). 2
3 Bounds for designs in products of Q-polynomial schemes
3.1 The Delsarte bound
One well-known theorem [4, Theorem 5.19] of Delsarte gives a lower bound on the size of a t-design in a Q-polynomial association scheme. This bound has both the Rao bound for orthogonal arrays and the bound of Ray-Chaudhuri and Wilson for block designs as special cases. In this section, we generalise this bound further to the case of T -designs in products of Q-polynomial association schemes. Let (X; A) be an association scheme with v = jX j. Consider the Krein parameters qij (k) de ned by d X 1 Ei Ej = v qij (k)Ek : k =0
12
The well-known Krein condition states that each qij (k) 0. We say (X; A) is Q-polynomial if the following conditions are satis ed for all i; j; k: (k > i + j ) qij (k) = 0) and (i + j d ) qij (i + j ) > 0): (See [1, Sec. III.1] or [2, Sec. 2.7] for much more information.) Let C be the poset formed by the product of chains Cd1 Cdm . For E C , de ne E + E = f(i + j ; . . . ; im + jm) : i; j 2 Eg: For j 2 C , let fj = rank Ej . 1
1
Lemma 3.1 Let (X; A) be the product of association schemes (Yh; Ah ) (1 h m). Let the Krein parameters of (Yh ; Ah ) be denoted by qihjh (kh ). Then for i; j; k 2 C , the corresponding Krein parameter of (X; A) is given by qij (k) =
m Y
h=1
qih jh (kh ):
Furthermore, if each component scheme is Q-polynomial, then we have qij (k ) = 0 whenever there exists an h for which kh > ih + jh. 2
Theorem 3.2 (Delsarte Bound) Let (X; A) be the product of Q-polynomial association schemes (Yi ; Ai ) (1 i m). Let T be a downset in C and let D X be a Delsarte T -design. If E C satis es (E + E ) \ C T , then X jDj fj : j2E
Moreover, if equality holds, then, for ` 6= 0,
X j2E
Q`j = 0
whenever D contains a pair of `-related elements.
Proof: We begin with a derivation of the dual of Delsarte's linear program for T -designs. Each matrix M 2 A may be expanded in the form M =v If we change bases, we have
M=
X j 2C
X i2C
13
j Ej : i Ai
where
i =
X j 2C
(i 2 C ):
Qij j
Consider only matrices M satisfying the conditions (a) M is a non-negative matrix; (b) j 0 for j 62 T ; (c) = 1. Let D be any Delsarte T -design in (X; A). Since M is non-negative, we have 0
TD M D =
X i2C
i TD Ai D TD A D = jDj: 0
On the other hand, we have X TD M D = v j TD Ej D j 2C
= v TD E D + v 0
0
X j2T
0
0
j TD Ej D + v
X j 62T
j TD Ej D :
Since = 1, E = v J and Ej D = 0 for j 2 T , this reduces to 0
0
1
TD M D = jDj + v 2
X j 62T
j TD Ej D ;
which is bounded above by jDj since, for j 62 T , j 0 and TD Ej D 0. Putting the two inequalities together gives the bound jDj: In this way, each matrix M 2 A satisfying conditions (a)-(c) gives a lower bound on the size of a T -design. (Notice that is the value on the diagonal of M .) Now let E T be as hypothesized. Consider the matrix X N = Ej : 2
0
0
j 2E
If we square each entry of this matrix, we obtain the non-negative matrix
N N =
XX h2E i2E
Eh Ei =
0 X @1 X X j2C
14
v h2E
i2E
1 qhi (j )A Ej :
Notice that, by Lemma 3.1, qhi (j ) = 0 whenever h; i 2 E and j 62 T . Thus the coecient of P Ej vanishes whenever j 62 T . On the other hand, since qhi (0) = h;i fh , we have v E fh as the coecient of E in this expansion. Let
= Pv f ; h2E h and take M = (N N ). Then M is non-negative and, if we expand X M = v j Ej ; 1
0
2
j 2C
we obtain = 1 and j = 0 for j 62 T . So M satis es conditions (a), (b), and (c). Consequently, for any T -design D, the size of D is bounded below by the common value of the diagonal entries of M . P The diagonal entries of Eh are all equal to fh =v; the diagonal entries of N are therefore v E fh . So each diagonal entry of M is equal to 0
1
P f! X h2E h
= fh : v 2
h2E
This proves the inequality
jDj
X h2E
fh :
Now if equality holds in this last inequality, then we must have equality throughout. In particular, for each ` 6= 0, we must have ` ( TD A` D ) = 0. In other words, if ` 6= 0, and D contains two `-related elements, then X j Q`j = 0: 2 j 2C
Example 3.1: For split orthogonal arrays, we obtain the bound M
bX t1 =2c bX t2 =2c i=0 j =0
n i
1
n
i j (q ? 1) 2
j
+
on the number of rows, M , in the array. Obviously, the right-hand side factors into the product of the ordinary Rao bounds for OA(ti; ni ; q), i = 1; 2. This result is implied by Theorem 6.3 in [10]. 2
Example 3.2: For mixed-level orthogonal arrays, the theorem above gives
X
m Y nh
i1 ++im bt=2c h=1
15
ih ( q ? 1) h ih
as a lower bound on the number of rows of the array. This bound was already derived by elementary means in [9]. Now if equality holds in the bound, the second statement of the theorem forces many entries of the inner distribution to equal zero. For example, in the case t = 2, if ` 6= 0 and D contains a pair of `-related elements, then ` q + `mqm = jDj. For t = 2 and 4, these restrictions were already found by Mukerjee and Wu [15]. 2 1 1
Example 3.3: For mixed t-designs in J (v ; k ) J (v ; k ), we may take E = f(i ; i ) : i + i ug where u = bt=2c. This gives the bound 1
1
2
1
2
2
1
2
X v v v v jDj ? i ?1 ? i ?1 ; i i i i u 1
0
which simpli es to
1
1
1+ 2
2
1
2
2
2
v + v v + v jDj u ? u ? 1 : 1
2
1
2
For example, for t = 2, we nd jDj v + v ? 1 and for t = 4, jDj (v + v ? 3)(v + v ). 1
1 2
2
1
2
1
2
2 Example 3.4: For a fused orthogonal array design of strength t in H (n; q) J (v; k), we
obtain
v v X n j jDj (q ? 1) j ? j ?1 j j u j 1
1
1+ 2
2
2
where u := bt=2c. This can be written in terms of a hypergeometric series:
v
jDj u
2
?u; ?n F ; q?1 : 2 1
v?u+1
3.2 The degree bound and association schemes induced by designs
Continuing the treatment in parallel to that of designs in Q-polynomial association schemes, we prove in this section a relationship between the size of T and the degree s of a T -design, focusing on the case where this bound is tight. Let D be a design in an association scheme (X; A). Let A denote the Bose-Mesner algebra of (X; A). For M 2 A , denote by M the submatrix of M whose rows and columns are indexed by the elements of D. In the case where (X; A) is Q-polynomial and D has degree s and strength t, a well-known result [4, Theorem 5.25] of Delsarte states that bt=2c s and, if t 2s ? 2, then the vector space A of submatrices fM : M 2 A g is the Bose-Mesner algebra of a (Q-polynomial) association scheme. We express this by saying that D induces an association scheme in (X; A). 16
Now let D be a Delsarte T -design in a product (X; A) of Q-polynomial association schemes. Recall that the degree of D is the number of indices i 6= 0 for which Ai is a nonzero matrix. Call D a tight design in (X; A) if, for some E , equality holds in the bound of Theorem 3.2.
Theorem 3.3 Let D be a Delsarte T -design of degree s in the product (X; A) of Qpolynomial association schemes. If (E + E ) \ C T , then jEj s + 1: Moreover, if equality holds, then D is a tight design and D induces an association scheme in (X; A). Secondly, if jEj = s, then either D is a tight design or D induces an association scheme in (X; A).
Proof: The proof is modelled after the proof of Theorem 5.25 in [4]. Let S be a v v matrix diagonalising the Bose-Mesner algebra A of the product scheme. Then S may be partitioned into v fj submatrices Sj where the columns of Sj form an orthonormal basis for colsp Ej (j 2 C ). Clearly Sj SjT = Ej . Let Hj be the submatrix of Sj
obtained by restriction to the rows indexed by elements of D. Then Hj HjT = Ej . For i; j 2 C , we have (Delsarte [4, Theorem 3.15]):
qij (k)(aQ)k = 0 for all k 2 C ? f0g if and only if
0;
if i 6= j ; = jDjI; if i = j . Consider the vector space A = fM : M 2 A g. This is spanned by the matrices fAi : i 2 Cg and so has dimension s + 1 where s is the degree of D. For i; j 2 E , we apply Lemma 3.1 to conclude that qij (k) = 0 whenever k 62 T . As Ei Ej = HiHiT Hj HjT , we obtain (6) Ei Ei = jDj Ei and (assuming i 6= j ) (7) Ei Ej = 0: So these jEj members of A are linearly independent. This proves that jEj s + 1. If equality holds, then (by (6) and (7)) A has a basis of mutually orthogonal idempotents and hence is closed under matrix multiplication. It is easy to see that this vector space is closed under Schur multiplication and contains I and J . Therefore it is the Bose-Mesner algebra of an association scheme (see [2, Theorem 2.6.1]).
HiT Hj
17
Write E = fi ; . . . ; ir g. De ne 1
GE = [Hi1 Hi2 Hir ]: P GE has jDj rows and rj fij columns. From the preceding observations, we see that GTE GE = jDjI . (This gives an alternative proof of the bound in Theorem 3.2.) p If D is not a tight design, we may add columns to form an orthogonal matrix (1= jDj)G with G = [GE K ]: Then GGT = jDjI , yielding r X T KK = jDjI ? Hij HiTj ; =1
j =1
Now HiHiT = Ei . Thus KK T 2 A and, by Equations (6) and (7), ( 1 KK T )Ei = 0 jDj for i 2 E . Since K T K = I , KK T is not the zero matrix. So we have jEj + 1 linearly independent matrices in A . Thus, if jEj = s + 1, D must be a tight design. Finally, assume jEj = s and that D is not a tight design. If we take c = 1=jDj, the matrices fcEj : j 2 Eg [ fcKK T g form a basis of primitive idempotents for A . So A is closed under both ordinary and Schur multiplication and contains I and J . Consequently, A is the Bose-Mesner algebra of an association scheme. 2 For example, if D is a t-design in a product of m Q-polynomial schemes, then the degree of D is at least the size of
E = f(i ; . . . ; im) : 1 i + im t=2; all ij 0g: ? This gives the bound s + 1 mm u where u = bt=2c. For example, if D is a mixed-level orthogonal array of strength two in H (n ; q ) H (nm ; qm) then ai is non-zero for at 1
1
+
least m non-zero index vectors i.
1
1
Remarks: (i) If D is a tight design, the bound jEj s may fail. For example, consider a projective
plane 2-(n + n + 1; n + 1; 1). Delete a block x and let D denote the set of n + n remaining blocks, viewed as a 2-design in J (n +1; 1) J (n ; n). Then D has degree two in this product scheme yet the set E = f(0; 0); (0; 1), (1; 0)g satis es (E + E ) \C T . See [11] for an investigation of tight 2-designs in J (v ; k )
J (v ; k ). 2
2
2
1
2
2
18
1
(ii) In contrast to the situation where (X; A) is Q-polynomial, the induced association
scheme found above is not necessarily Q-polynomial, even though we have assumed that each component scheme is Q-polynomial.
4 Summary Two decades ago, a theory was established for the study of t-designs in Q-polynomial association schemes. In certain cases, these objects have useful combinatorial interpretations. Motivated by several applications, the present paper begins to establish a theory of T designs in products of Q-polynomial association schemes. In the language of Q-posets, a combinatorial characterisation (Theorem 2.3) of such objects is given. Independent of this characterisation, two results are proved for designs in products of Q-polynomial association schemes which are analogues of Delsarte's results on t-designs. In addition, the linear programming bound for such designs comes \for free". A variety of questions arise from this investigation. Which association schemes admit a Qposet? Must such a scheme be Q-polynomial? Is the bound in Theorem 3.2 the best possible? What are the tight designs? Do any new association schemes arise from Theorem 3.3? The theory in the latter part of the paper can be extended to arbitrary association schemes, but will only be non-trivial in cases where a large number of Krein parameters vanish. See [13] for an application of such techniques to a problem in numerical integration. It also seems useful to focus on particular classes of product schemes, such as products of Hamming schemes (split orthogonal arrays and mixed orthogonal arrays), products of Johnson schemes (mixed block designs [11] and split designs [12]), or H (n; q) J (v; k) (Room d-cubes and their generalisations). Acknowledgements: The author is grateful to S. Song and C. Godsil for helpful comments. In addition, the referee o ered several suggestions which improved the manuscript.
References [1] E. Bannai and T. Ito, \Algebraic Combinatorics I: Association Schemes," BenjaminCummings Lecture Note Ser. 58, The Benjamin-Cummings Publishing Company, Inc., London, 1984. [2] A.E. Brouwer, A.M. Cohen, and A. Neumaier, \Distance-Regular Graphs," Springer-Verlag, Berlin, 1989. [3] C.J. Colbourn and J.H. Dinitz, \Handbook of Combinatorial Designs," CRC Press, Boca Raton, 1996. 19
[4] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Reports Suppl. 10 (1973). [5] P. Delsarte, Association schemes and t-designs in regular semilattices, J. Combin. Th. (A) 20 (1976), 230-243. [6] P. Delsarte, Pairs of vectors in the space of an association scheme, Philips Res. Reports 32 (1977), 373-411. [7] J.H. Dinitz and D.R. Stinson (eds.), \Contemporary Design Theory: A Collection of Surveys," Wiley, New York, 1992. [8] C.D. Godsil, \Algebraic Combinatorics," Chapman and Hall, London, 1993. [9] A.S. Hedayat, N.J.A. Sloane and J. Stufken, \Orthogonal Arrays," SpringerVerlag, New York, 1998. [10] V. Levenshtein, Split orthogonal arrays and maximum resilient systems of functions, Designs, Codes and Crypt. 12 (1997), 131-160. [11] W.J. Martin, Mixed block designs, J. Combin. Designs 6 #2 (1998), 151-163. [12] W.J. Martin, Completely regular designs, to appear in: J. Combin. Designs. [13] W.J. Martin and D.R. Stinson, Association schemes for ordered orthogonal arrays and (T; M; S )-nets, (1997), submitted for publication. [14] A.D. Meyerowitz, Cycle-balanced partitions in distance-regular graphs, preprint. [15] R. Mukerjee and C.F.J. Wu, On the existence of saturated and nearly saturated asymmetrical orthogonal arrays, Ann. Stat. 23 (1995), 2102-2115. [16] C. Roos, On antidesigns and designs in an association scheme, Delft Progress Report 7 (1982), 98-109. [17] N.J.A. Sloane and J. Stufken, A linear programming bound for orthogonal arrays with mixed levels, J. Stat. Plan. Inf. 56 (1996), 295-306. [18] S.Y. Song, Posets related to some association schemes, Journal of the Korean Society of Mathematical Education 25 (1987), 57-69. [19] D. Stanton, t-Designs in classical association schemes, Graphs Combin. 2 (1986), 283-286. [20] P. Terwilliger, Incidence algebras of uniform posets, in: \Coding Theory and Design Theory, Proceedings of the IMA" (D. K. Ray-Chaudhuri, ed.) Springer-Verlag, New York, 1991. 20
