Constructions of Hadamard Di erence Sets - CiteSeerX
Constructions of Hadamard Di erence Sets Richard M. Wilson and Qing Xiang Dept. of Mathematics, California Institute of Technology, Pasadena, CA 91125 June 24, 1998 Abstract Using a spread of PG(3; p) and certain projective two-weight codes, we give a general construction of Hadamard di erence sets in groups H (Zp )4 , where H is either the Klein 4-group or the cyclic group of order 4, and p is an odd prime. In the case p 3 (mod 4), we use an ovoidal bration of PG(3; p) to construct Hadamard di erence sets, this construction includes Xia's construction of Hadamard di erence sets as a special case. In the case p 1 (mod 4), we construct new reversible Hadamard di erence sets by explicitly constructing the two-weight codes needed in our general construction method. Using a well-known composition theorem, we conclude that there exist Hadamard di erence sets with parameters (4m2 ; 2m2 ? m; m2 ? m), where m = 2a 3b 52c1 132c2 172c3 p21 p22 p2t with a; b; c1 ; c2 ; c3 positive integers, and where each pj is a prime congruent to 3 modulo 4, 1 j t. E-mail: [email protected], [email protected]
0
1 Introduction Let G be a nite group of order v. A k-element subset D of G is called a (v; k; ) di erence set in G if the list of \di erences" d d? , d ; d 2 D, d 6= d , represents each nonidentity element in G exactly times. Using multiplicative notation for the group operation, D is a (v; k; ) di erence set in G if and only if it satis es the following equation in Z [G]: 1
2
1
1
2
1
2
DD ? = (k ? )1G + G; (
1)
where D = Pd2D d, D ? = Pd2D d? , and 1G is the identity element of G. D is called reversible if D ? = D. In the case G is an abelian group, using the Fourier inversion formula, we have the following standard lemma in the theory of di erence sets. Lemma A. Let G be an abelian p group of order v. A k-subset D is a (v; k; ) di erence set in G if and only if j (D)j = k ? for every nontrivial character of G. Furthermore, D = D ? if and only if (D) = (D) for every character of G. The di erence sets considered in this paper have parameters (
(
(
1)
1
1)
1)
(v; k; ) = (4m ; 2m ? m; m ? m): 2
2
2
These di erence sets are called Hadamard di erence sets (HDS), since their 1 incidence matrices are Hadamard matrices. Alternative names used by other authors are Menon di erence sets and H-sets. The central problem in the study of HDS is for each integer m, which groups of order 4m contain a Hadamard di erence set. This problem remains open, for abelian groups and nonabelian groups as well. However considerable progress has been made on the construction of Hadamard di erence sets in recent years. For example, in 1992, Xia [10] constructed Hadamard di erence sets in groups H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. Smith [8] constructed a nonabelian reversible Hadamard di erence set in the group ha; b; cja = b = c = [a; b] = cac? a? = cbc? b? = 1i. In October, 1995, Van Eupen and Tonchev [5] constructed a reversible Hadamard di erence set in Z Z (Z ) , which is the rst example of an abelian Hadamard di erence set with order divisible by a prime congruent to 1 modulo 4. In this paper, we rst give a general construction method for Hadamard di erence sets in groups H (Zp) , where H is either group of order 4, p is an odd prime, by assuming the existence of certain projective two-weight codes. This method applies to both cases that p 3 (mod 4) and p 1 (mod 4). In section 3, we explain Xia's construction by using our general construction method. This was actually done by Xiang and Chen in [11], we include this section here for the convenience of the reader. In section 4, we use an 2
4
4
4
t
5
1
2
1
2
2
2
4
1
5
4
5
4
ovoidal bration of PG(3; p) (see [1], [4], [6]), and spreads associated with it to construct Hadamard di erence sets in H (Zp) , where H is either group of order 4, and p is a prime congruent to 3 modulo 4. This construction includes Xia's construction of Hadamard di erence sets as a special case. In section 5, we explicitly construct those projective twoweight codes needed in our general construction for HDS when p = 5; 13; 17. Using a well-known composition theorem of Hadamard di erence sets (for example, see [7], [9]), we conclude that there exist Hadamard di erence sets with parameters (4m ; 2m ? m; m ? m), where m = 2a3b 5 c1 13 c2 17 c3 p p pt with a; b; c ; c ; c positive integers, and where each pj is a prime congruent to 3 modulo 4, 1 j t. 4
2
2
2
2
2 1
2 2
2
1
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2
2
3
2 The Construction We begin with the de nition of a projective (n; k; h ; h ) set in PG(k ? 1; q), where q is a power of prime p. De nition. A projective (n; k; h ; h ) set O is a proper, non-empty set of n points of the projective space PG(k ? 1; q) with the property that every hyperplane meets O in h points or h points. Let O = fhy i; hy i; ; hynig be a set of n points in PG(k ? 1; q). Associated with PG(k ? 1; q) is the k-dimensional vector space W = Vk (q). Let = fv 2 W jhvi 2 Og be the set of vectors in W corresponding to O. For w 2 GF (q)k , de ne an additive character of GF (q)k as follows: w : x 7! T r wx ; x 2 GF (q)k ; 1
1
2
2
1
2
1
2
(
)
where is a primitive p-th root of unity and Tr is the trace from GF (q) to GF (p). It is easy to see that w , w 2 GF (q)k , are all the additive characters of GF (q)k . For any nontrivial additive character w of GF (q)k , we have
w ( ) = (q ? 1)jw? \ fy ; y ; ; yngj + (?1)(n ? jw? \ fy ; y ; ; yngj) = qjw? \ fy ; y ; ; yngj ? n; 1
1
2
1
2
2
where w? = fy 2 GF (q)k jy w = 0g, and y w is the usual dot product. Hence we have the following lemma. Lemma 2.1. O is a projective (n; k; h ; h ) set if and only if w ( ) = qh ? n or qh ? n, for every nontrivial additive character w , w 2 GF (q)k . Also we mention that projective (n; k; h ; h ) sets are equivalent to projective two-weight codes, and certain strongly regular Cayley graphs. We refer the reader to the survey papers [3], [7] for more detailed discussion of these three objects. 1
2
1
1
2
2
2
Let = PG(3; p) denote projective 3-space over GF (p), where p is an odd prime. A spread of is any collection of p + 1 pairwise disjoint lines of , necessarily partitioning the points of . A partial spread in is a set of lines no two of which 4intersect. Also, for p? 2 p 2 p ? convenience, we will call a subset C of type Q if C is a projective ( p? ; 4; ; ) set. Theorem 2.2. Assume that S = fL ; L ; ; Lp2 g is a spread of . If there exist two subsets C , C of type Q in such that jC \ L2i j = (p + 1)=2, 1 i s, and jC \ Lj j = (p + 1)=2, (s + 1) j 2s, where s = p , then there exists a Hadamard di erence set in H (Zp ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set obtained is reversible. Proof: Let C = (L [ L [ [ Ls ) n C , C = (Ls [ Ls [ [ L s ) n C . We rst prove that C , C are also two subsets of type Q in . Associated with is the 4-dimensional vector space W = V (p) over GF (p). Let C = fw 2 W jhwi 2 C g, C = fw 2 W jhwi 2 C g. Since C is a set of type Q in , by Lemma 2.1 we have (C ) = p2 ? ? p or p2? , for every nontrivial additive character of W . We will use W to denote the additive character group 2 of W , and de ne U = f 2 W j (C ) = p ? ? p g, V = f 2 W j (C ) = p2? g. Let L = fw 2 W jhwi 2 [si Lig. Since fL ; L ; ; Lsg is a partial spread, we have 3
2
3
3
3
3
3
4(
1
0
1
2
2
1
+1 2
2
0
3
+1
3
+2
2
3
4
0
0
3
2
1
2
1
(
1
2
4
4
0
4
=1
2
1
0
0
1
3
0
1
( +1) 4
0
4
2
1) 4
3
+1
3
1
(
1 1)
1
4
2
p2 ; if 2 N ; ? (L ) = p2 ? ; if 2 T , where N = f 2 W n f gj is nontrivial on every Li; 1 i sg, and T = f 2 W n f gj is trivial on exactly one Li , for some i, 1 i sg. We contend that T \ U = ;. For each Lj , 1 j 2s, which is now viewed as a 2-dimensional subspace of W , let L?j = f 2 W j is trivial on Lj g. Then jL?j j = jW=Lj j = p . For 1 j s, we de ne j = j(L?j n f g) \ U j, j = j(L?j n f g) \ V j. Then j + j = p ? 1. For every 2 L?j , we have (C ) = (C n (C \ Lj )) + jC \ Lj j. Therefore +1 2 1
1
1
1
2
1
0
1
0
1
2
0
2
0
0
X 2L? j
0
(C ) = 0
0
X
0
X
(w) + p jC \ Lj j: 2
w2C0 n(C0 \Lj ) 2L? j
0
Noting that P 2L? (w) = 0 if there is a 2 L?j such that (w) 6= 1, we have P 2L? (C ) = p jC \ Lj j. That is p4? + j ( p2? ? p ) + j p2? = p jC \ Lj j. Simplifying this we get j
2
0
j
1
1
4
4
1
2
2
4
0
1 ? p + = jC \ L j: j j 2 2
0
3
0
Since jC \ Lj j = p2? for every j , 1 j s, we have j = p ? 1; j = 0; 1 j s. Hence T \ U = ;. For any nontrivial 2 W , (C ) = (L ) ? (C ). Since T \ U = ;, we have 1
0
2
2
1
2
(C ) =
1
( p2 ?1
0
? p ; if 2 N \ V ; if 2 (N \ U ) [ (T \ V ). 2
1
4 p2 ?1 ;
2
1
4
1
1
This shows that C is a set of type Q in . Similarly, de ne C = fw 2 W jhwi 2 C g, C = fw 2 W jhwi 2 C g. Let L = fw 2 W jhwi 2 [i s s Li g. Since C is a set of type Q in and fLs ; Ls ; ; L s g is a partial spread, we have 2
3
1
2 = +1
1
3
3
1
3
(C ) =
( p2 ?1
+1
+2
2
2
? p ; if 2 X ; if 2 Y . 2
4 p2 ?1 ;
1
4
and (
p2 ; if 2 N ; ? (L ) = p2 ? ; if 2 T , +1 2 1
2
2
2
2
where N = T , and T = N . By the same argument as above, we can show that T \ X = ;, hence 2
1
2
1
2
(C ) = 3
( p2 ?1 4 p2 ?1 4
? p ; if 2 N \ Y ; ; if 2 (N \ X ) [ (T \ Y ). 2
2
2
2
Assume that A is the union of any p2? lines from Ls ; Ls ; ; L s , B is the union of any p2 ? lines from L ; L ; ; L , and we view A, B as subsets in the vector space W (we make s the convention that A, B , when viewed as subsets in W , do not contain the zero vector). De ne 1
1
4
+1
4
1
+2
2
D = C [ A;
D = C [ B;
D = C [ A;
D = C [ B:
0
2
0
1
2
3
For any nontrivial 2 W , we distinguish two cases. 4
1
3
2
(1). Ker Lj , for some j , 1 j s. In2 this case, 2 N = T . Since T \ U = ;, we have 2 V . Therefore, (D ) = p ? + (? p2 ? ) = 0, and (D ) = p2 ? + (? p2 ? ) = 0. 2
4
1
1
1
1
2
4
0
1
1
4
4
(
p2 ? ; if L 2 B ; p ? j (B ) = p2 ? ? ; if Lj 62 B . 1
2
4
1
4
Hence
Lj 62 B , or, 2 X and Lj 2 B ; (D ) = 0;p ; ifif 22 YY and and Lj 2 B , or, 2 X and Lj 62 B ; 1
2
and
Lj 2 B , or, 2 X and Lj 62 B ; (D ) = 0;p ; ifif 22 YY and and Lj 62 B , or, 2 X and Lj 2 B . 3
2
This shows that (D ) = (D ) = 0, and only one of (D ), (D ) vanishes, the other is p . (2). Ker Lj , for some j , (s + 1) j 2s. In this case, 2 N = T . Since T \ X = ;, we have 2 Y . In a manner similar to that of case (1), we can show that (D ) = (D ) = 0, and only one of (D ), (D ) vanishes, the other is p . We rst consider the group Z Z (Zp) . Let us denote the elements of Z Z by f1; a; b; abg. De ne D = D [ aD [ bD [ ab(W n D ). We contend that D is a reversible Hadamard di erence set in Z Z (W; +). Let be an arbitrary nontrivial character of Z Z W . If is trivial, is nontrivial, then 0
2
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2
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4
2
1
2
0
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2
2
2
(D) = jD j + (a)jD j + (b)jD j + (ab)jW n D j = p (ab); 0
1
2
3
so j (D)j = p . If is nontrivial, by the discussion in the two cases above, we have 2
(D) = p ; 2
hence j (D)j = p . 2
5
2
2
By Lemma A, D is a Hadamard di erence set. Since (D) = (D) for every nontrivial character of Z Z W , D is reversible. In the case the group is Z (Zp) , let the elements of Z be f1; c; c ; c g, and D = D [ cD [ c D [ c (W n D ). Then it is easy to show that D is a Hadamard di erence set in Z (W; +). This completes the proof of the theorem. 2 2
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3
4
3 On Xia's Construction In 1992, Xia [10] constructed Hadamard di erence sets in groups H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. Xia's construction depends on very complicated calculations involving cyclotomic classes of high order. Xiang and Chen [11] have given a simpler proof for Xia's construction by using additive characters of nite elds. In view of Theorem 2.2, in order to construct Hadamard di erence sets in H (Zp) , where H is either group of order 4 and p is a prime congruent to 3 modulo 4, all we need are a spread in , and sets C , C of type Q in satisfying the conditions in Theorem 2.2. Let p be a prime congruent to 3 modulo 4 and let be a primitive element of GF (p ). We model by viewing GF (p ) as 4-dimensional vector space over GF (2p). Thus the points of 2 2 p p ? i p i p p i ig, are represented by h1i; h i; ; h i. Let Li = fh i; h i; ; h 0 i p . Then it is easy to see that S = fL ; L ; ; Lp2 g is a spread in . Let C = fh1i; h i; h i; ; h p2 p ? ig, C = fh i; h i; h i; ; h p2 p ? ig. Since p 3 (mod 4), by uniform cyclotomy (see [2], [7]),2 C , C are two sets of type Q in . Also it is easy to see that jC \ L i j = p , 0 i p ? , and jC \ L i j = p , 0 i p2? . Therefore by Theorem 2.2, we have Corollary 3.1. There exists a Hadamard di erence set in H (Zp) , where p is a prime congruent to 3 modulo 4, and H is either the Klein 4-group or the cyclic group of order 4. In the rst case, the Hadamard di erence set constructed by Theorem 2.2 is reversible. Using a composition theorem of Turyn [9], it is routine to construct Hadamard di erence sets in H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. 4
4
4
t
4
3
0
1
3
4
4
3
(
+1)( +1)
1
(
2
0
0
4
8
(
+1)( +1)
4
1
0
0
+1 2
2
1
2
+1)+
(
3
+1)+ 3
1 5
9
(
+1)( +1)
3
1
3
1
+1 2
2 +1
1
2
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4
4
4
t
4 General Construction in the case p 3 (mod 4) In this section, we give a general construction of Hadamard di erence sets in H (Zp) , where H is either group of order 4, and p is a prime congruent to 3 modulo 4, by using an ovoidal bration of PG(3; p) in [1], [4] and [6] (page 253). We introduce the following 4
6
notation as in [1], [4]. Let p be a prime congruent to 3 modulo 4. We view GF (p ) as a 4-dimesional vector space over GF (p), and hence the 1-dimensional subspaces of this vector space can be thought of as the projective points of = PG(3; p). Similarly, we identify the lines of with the 2-dimensional vector subspaces of GF (p ) over GF (p). We also let hAi denote the vector subspace generated by the set A over GF (p). If is a2 primitive element of GF (p ), then o( ) = p ? 1 = (p + 1)(p ? 1)(p + 1) and hence p p is a primitive element of GF (p). We therefore identify the points of with fh tijt = 0; 1; 2; ; (p + 1)(p + 1) ? 1g. If we now let i = fh tijt i (mod p + 1) g, each
i is an ovoid (Theorem 3 of [4]) and the points of are thus partitioned into p + 1 disjoint ovoids: = [ [ [ p : () Moreover, each line of is tangent to precisely 0 or 2 of these ovoids (Lemma 1 and Theorem 4 of [4]). Lines of the former type will be called \secant-type" and those of the latter \tangent-type". If Ls = h ; s p i denotes the line of joining the points h i and h s p i of for any integer s with 1 s p , then Ls is a secant-type line if and only if s is odd (Theorem 4 of [4]). If L is any line of , let [L] denote the line orbit of L under the collineation of corresponding to multiplication by p . Also let Lp denote the image of the line L under the collineation corresponding to the Frobenius automorphism, and let L d denote the image of L under multiplication by d. We quote the following theorem and corollary from [1]. Theorem B. Using the above notation, let s be an odd integer with 1 s p . Consider p s p the secant-type line Ls = h ; i, necessarily secant to ovoids in the bration (). Then there exists a positive integer d such that (Ls)p d is a secant-type line meeting the p ovoids of () missed by Ls. Moreover, if s 6= p2 , d is unique modulo p + 1. Corollary C. [Ls] [ [(Ls )p d] is a spread of . This spread is regular precisely when 2 s= p . Now we use the spread in Corollary C to construct Hadamard di erence sets. By Theorem 2.2, we need to come up with two sets C , C in of type Q satisfying the conditions in Theorem 2.2. Let Ls and (Ls)p d be the secant-type lines in Theorem B. We assume that Ls is secant to
t1 ; t2 ; ; t , and (Ls)p d is secant to t +1 ; t +2 ; ; t2 , where r = p . By Theorem B, ft ; t ; ; t r g = f0; 1; 2; ; pg. Since p 3 (mod 4), r is even. Let C be the union of any r=2 ovoids from f t1 ; t2 ; ; t g, and let C be the union of any r=2 ovoids from f t +1 ; t +2 ; ; t2 g. Then we have the following lemma. 4
3
3
4
4
( +1)(
4
2
+1)
3
2
3
3
0
1
3
0
( +1)
( +1)
0
3
0
3
2
3
2( +1)
2
0
+1 2
( +1)
+1 2
+1 2
3
+1 2
0
r
1
2
1
r
r
r
2
r
+1 2
0
1
r
r
3
r
7
Lemma 4.1. C meets every line in [Ls ] in p points, and C meets every line in [(Ls)p d] p +1 2
0
1
points. C0 , C1 are two sets of type Q in 3 . Proof. The rst assertion is clear by the de nition of C0 and C1 . For the proof of the second part, we observe that every plane of 3 must meet each of the p + 1 ovoids in () in a point or an oval. A simple counting argument then shows that each plane of 3 is tangent to 1 of the ovoids in () and meets the other p ovoids in disjoint ovals. Let be an arbitrary (p?1)2 r plane of 3 . Then j \ C0j = 1 + ( 2 ? 1)(p + 1) = 4 if C0 contains some t such that 2 j \ t j = 1, 1 j r, and j \ C0j = 2r (p + 1) = (p+1) if j \ t j = p + 1, for every t 4 contained in C0 . This shows that C0 is a set of type Q in 3 . Similarly, we can show that C1 is also a set of type Q in 3 . This completes the proof of the lemma. 2 Corollary 4.2. Let [Ls] [ [(Ls)p d], C0, C1 be de ned as above. Then there exists a Hadamard di erence set in H (Zp )4 , where H is either group of order 4, and p is a prime congruent to 3 modulo 4, by using the spread [Ls ] [ [(Ls )p d ] and the sets C0 and C1 of type Q in 3 . Proof. This is clear from Lemma 4.1 and Theorem 2.2. 2 Remarks. (1) If we let s = p22+1 , then Ls = GF (p2), Lps = Ls, and [Ls] [ [ Ls] is the regular spread in Section 3. Also we note that Ls meets the ovoids 0 ; 2 ; 4; ; p?1 (Lemma 1 of [4]), and Ls meets the ovoids 1 ; 3; 5 ; ; p. If we choose the union of
0 ; 4; 8 ; ; p?3 as C0 and the union of 1 ; 5; 9 ; ; p?2 as C1, then the construction in this section will give rise to Xia's construction. (2) Let G = K4 P , where K4 is the Klein 4-group and P = Zp4 , p is a prime congruent to 3 modulo 4. Two di erence sets D and D0 in G are said to be equivalent if D0 = gD for some automorphism of G and some element g of G. Since K4 and P have relatively prime orders, they must be invariant under every automorphism of G. Therefore the automorphism group of G has size jGL(2; 2)jjGL(4; p)j = 6p6(p4 ? 1)(p3 ? 1)(p2 ? 1)(p ? 1). From Theorem 2.2 and the construction in this section, we see that there are at least in
+1 2
j
j
i
2 p2 +1 2 2 +1 p+1 p 2 4! 2 p+1 p22?1 4 4
i
p+1 2 p2 +1 2 2 2 + 1) p+1 p2 ?1
(p 4 4 = 4p 6p (p ? 1)(p ? 1)(p ? 1)(p ? 1) 2p (p ? 1)(p ? 1)(p ? 1)(p ? 1) 4
6
4
3
2
2
10
4
3
2
pairwise inequivalent Hadamard di erence sets in G.
5 The case p 1 (mod 4) In this section, we construct sets of type Q in = PG(3; p) with p 1 (mod 4). Again let W be the 4-dimensional vector space over GF (p) associated with . We may consider W 3
3
8
as a direct product GF (p ) GF (p ). Let g be a primitive element of GF (p ). L1 will denote the line f0g GF (p ), and for any d in GF (p ), Ld will denote the line f(x; dxp)jx 2 GF (p )g. It is easy to verify that S = fLd jd 2 GF (p )g [ fL1g is a spread in . We now consider the action of T = g0 g0? on the points of which are now viewed as 1-dimensional subspaces over GF (p) of the 4-dimensional vector space GF (p ) GF (p ) over GF (p). The orbits of the action of T on the points of are as follows. (1). Four \short" orbits, each of length p . We choose (0; 1), (0; g), (1; 0), and (g; 0) as the representatives of these four orbits. (2). 4(p + 1) \long" orbits, each of length p2 ? . The representatives of these 4(p + 1) orbits can be chosen as (1; 1); (1; g); (1; g ); ; (1; g p ); (g; 1); (g; g); (g; g ); ; (g; g p ): 2
2
2
2
2
2
2
3
2
2
3
2
2
3
+1 2
1
4
2
2 +1
2
2 +1
It is clear that each short orbit consists of p points of L or L1. Next we show that each long orbit consists of p points of p? lines from the set of lines fLd jd 6= 0; d 2 GF (p )g. For example, take the long orbit represented by (1; gi), 0 i 2p + 1. Let a = gi. The points in this orbit are represented by +1 2
0
1
+1 2
2
2
(1; a) ! (g ; ag? ) ! : : : ! (gp? ; ag ?p) ! (gp? ; ag ?p) ! (gp ; ag? ?p) ! : : : ! (g p? ; ag ? p) ! 2
1
1
2
+1
3
1
2
4
3
4
2
(g`; ag?`) ! (g` ; ag?`? ) ! : : : ! (g` p? ; ag?`?p ) +2
2
+
2
3
+3
where ` = p? . Each column in the above diagram consists of p points of some line Ld , d 6= 0, d 2 GF (p ), hence the orbit represented by (1; gi) consists of p points of p? lines from the set of lines Ld ; d 6= 0; d 2 GF (p ). This argument applies to any orbit represented by (g; gi), 0 i 2p + 1. For p = 5 let g be a root of x + x + 2 2 GF (5)[x]. With the help of a computer (we will give more details about the computer search at the end of this section), we found that the union of the following orbits (
1) 2
+1 2
2
2
2
(1; g); (1; g ); (1; g ); (1; g ); (g; 1); (g; g ); (1; 0) 2
9
11
9
8
+1 2
1
2
forms a set of type Q in PG(3; 5), which we will call C , also the union of the following orbits (1; g ); (1; g ); (1; g ); (g; g ); (g; g ); (g; g ); (0; 1) forms another set of type Q in PG(3; 5), which we denote by C . Let S = fLd jd 2 GF (5 )g[ fL1g. We have seen that each orbit of T intersects the lines in S in 0 or 3 points, also no two orbits of T in C , C intersect the same line, hence C , C satisfy the conditions of Theorem 2.2, therefore there exists a Hadamard di erence set in H (Z ) , where H is either group of order 4. We state this as a corollary. Corollary 5.1. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set is reversible. Remark. Van Eupen and Tonchev ([5]) were the rst to construct a reversible Hadamard di erence set in Z Z (Z ) . We remark that the structure of the Hadamard di erence set in Z Z (Z ) constructed in Corollary 5.1 is di erent from that of Van Eupen and Tonchev's Hadamard di erence set. For example, in Theorem 2.2 (hence in Corollary 5.1), we choose A, B both as union of lines from a spread in , while in Van Eupen and Tonchev's example, one projective (36; 4; 6; 11) set in PG(3; 5) comes from the union of 6 lines, the other does not. In the case p = 13, let g be a root of x + x + 2 2 GF (13)[x]. With the help of a computer, we found the following two sets of type Q in PG(3; 13). The union of the following orbits (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (1; 0) forms a set of type Q in PG(3; 13), which we will denote by C . And the union of the following orbits (1; 1), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (g; g), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (0; 1) forms another set of type Q, which we will denote by C . Let S = fLd jd 2 GF (13 )g[fL1g. It is easy to see that S , C , C satisfy the conditions in Theorem 2.2. By Theorem 2.2, we have Corollary 5.2. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case the Hadamard di erence set is reversible. When p = 17, let g be a root of x + x + 3 2 GF (17)[x]. With the help of a computer, we found the following two sets of type Q in PG(3; 17). The union of the following orbits 0
6
8
10
5
9
10
2
1
0
1
0
1
5
4
5
2
2
2
5
2 4
5
4
4
3
2
5
6
16
9
13
15
17
18
24
4
5
8
14
11
12
23
0
2
24
4
7
8
12
25
6
7
27
2
1
0
1
13
2
10
4
(1; g ), (g; g), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (1; g ), (1; 0) forms a set of type Q in PG(3; 17), which we will denote by C . And the union of the following orbits (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (0; 1) forms another set of type Q in PG(3; 17), which we will denote by C . Let S = fLd jd 2 GF (17 )g[fL1g. Then it is easy to check that S , C , C satisfy the conditions in Theorem 2.2. By Theorem 2.2, we have Corollary 5.3. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set is reversible. Remark. We give more details about our computer search in what follows. In order to search for sets of type Q in by computer, we rst noted that T also permutes the planes in four \short" orbits and 4(p + 1) \long" orbits. We formed a square nonnegative integral matrix M whose rows were indexed by the orbits of T on the points, whose columns were indexed by the orbits of T on the planes, and where the entry in row i and column j was the the cardinality of the intersection of a (any) plane in plane-orbit j with the i-th orbit of points. A union of point-orbits is a set of type Q if and only if the sum of the corresponding rows of M has entries (p 1) =4 only. For example, when p = 17, the matrix M is square of order 76. The sum of all rows was a constant vector of 307's. We searched for 19 rows (one row corresponding to a short orbit and the others to long orbits) so that the sum of the 19 rows had entries 64 and 81 only. Our search was not exhaustive but simply moved from one set of 19 rows to another set by deleting one row|with a large entry in a column where the sum exceeded 81|and randomly adding another one. This was done with Mathematica on a PC. Finally, using a composition theorem of Turyn [9], it is routine to construct (4m ; 2m ? m; m ?m) Hadamard di erence sets with m = 2a 3b5 c1 13 c2 17 c3 p p pt , where a; b; c ; c ; c are positive integers and each pj is a prime congruent to 3 modulo 4, 1 j t. 2
7
25
32
6
31
12
34
11
33
15
21
14
19
18
26
22
0
5
4
18
6
17
5
31
9
30
8
4
10
9
11
10
17
16
20
1
2
0
1
17
4
3
2
2
2
2
2
2
2 1
2 2
2
2
1
2
Acknowledgment: We thank G. Ebert, W. Kantor and V. Tonchev for their helpful dis-
cussions.
11
3
References [1] R. D. Baker and G. L. Ebert \Construction of two-dimensional ag-transitive planes", Geometriae Dedicata 27 (1988), 9{14. [2] L. D. Baumert, W. H. Mills, and R. L. Ward \Uniform cyclotomy", J. Number Theory 14 (1982), 67{82. [3] A. R. Calderbank and W. M. Kantor \The geometry of two-weight codes", Bull. London Math. Soc. 18 (1986), 97{122 [4] G. L. Ebert \Partitioning projective geometries into caps", Can. J. Math. Vol. 37, No. 6, (1988), 1163{1175. [5] M. van Eupen and V. D. Tonchev \The existence of a reversible Hadamard di erence set in Z Z (Z ) and an in nite class of Williamson matrices", Preprint, Oct. 1995. [6] J. H. van Lint and R. M. Wilson \A Course in Combinatorics", Cambridge University Press, 1992. [7] S. L. Ma \A survey of partial di erence sets", Designs, Codes and Cryptography,4 (1994) 221{261. [8] K. W. Smith \Non-abelian Hadamard di erence sets", J. Combin. Theory Ser. A 70 (1995), 144{156. [9] R. J. Turyn \A special class of Williamson matrices and di erence sets", J. Combin. Theory Ser. A 36 (1984), 111{115. [10] M. Y. Xia \Some in nite classes of special Williamson matrices and di erence sets", J. Combin. Theory Ser. A 61 (1992), 230{242. [11] Q. Xiang and Y. Q. Chen \On Xia's construction of Hadamard di erence sets", Finite Fields and Their Applications 2, (1996), 87{95. 2
2
5
4
12
0
1 Introduction Let G be a nite group of order v. A k-element subset D of G is called a (v; k; ) di erence set in G if the list of \di erences" d d? , d ; d 2 D, d 6= d , represents each nonidentity element in G exactly times. Using multiplicative notation for the group operation, D is a (v; k; ) di erence set in G if and only if it satis es the following equation in Z [G]: 1
2
1
1
2
1
2
DD ? = (k ? )1G + G; (
1)
where D = Pd2D d, D ? = Pd2D d? , and 1G is the identity element of G. D is called reversible if D ? = D. In the case G is an abelian group, using the Fourier inversion formula, we have the following standard lemma in the theory of di erence sets. Lemma A. Let G be an abelian p group of order v. A k-subset D is a (v; k; ) di erence set in G if and only if j (D)j = k ? for every nontrivial character of G. Furthermore, D = D ? if and only if (D) = (D) for every character of G. The di erence sets considered in this paper have parameters (
(
(
1)
1
1)
1)
(v; k; ) = (4m ; 2m ? m; m ? m): 2
2
2
These di erence sets are called Hadamard di erence sets (HDS), since their 1 incidence matrices are Hadamard matrices. Alternative names used by other authors are Menon di erence sets and H-sets. The central problem in the study of HDS is for each integer m, which groups of order 4m contain a Hadamard di erence set. This problem remains open, for abelian groups and nonabelian groups as well. However considerable progress has been made on the construction of Hadamard di erence sets in recent years. For example, in 1992, Xia [10] constructed Hadamard di erence sets in groups H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. Smith [8] constructed a nonabelian reversible Hadamard di erence set in the group ha; b; cja = b = c = [a; b] = cac? a? = cbc? b? = 1i. In October, 1995, Van Eupen and Tonchev [5] constructed a reversible Hadamard di erence set in Z Z (Z ) , which is the rst example of an abelian Hadamard di erence set with order divisible by a prime congruent to 1 modulo 4. In this paper, we rst give a general construction method for Hadamard di erence sets in groups H (Zp) , where H is either group of order 4, p is an odd prime, by assuming the existence of certain projective two-weight codes. This method applies to both cases that p 3 (mod 4) and p 1 (mod 4). In section 3, we explain Xia's construction by using our general construction method. This was actually done by Xiang and Chen in [11], we include this section here for the convenience of the reader. In section 4, we use an 2
4
4
4
t
5
1
2
1
2
2
2
4
1
5
4
5
4
ovoidal bration of PG(3; p) (see [1], [4], [6]), and spreads associated with it to construct Hadamard di erence sets in H (Zp) , where H is either group of order 4, and p is a prime congruent to 3 modulo 4. This construction includes Xia's construction of Hadamard di erence sets as a special case. In section 5, we explicitly construct those projective twoweight codes needed in our general construction for HDS when p = 5; 13; 17. Using a well-known composition theorem of Hadamard di erence sets (for example, see [7], [9]), we conclude that there exist Hadamard di erence sets with parameters (4m ; 2m ? m; m ? m), where m = 2a3b 5 c1 13 c2 17 c3 p p pt with a; b; c ; c ; c positive integers, and where each pj is a prime congruent to 3 modulo 4, 1 j t. 4
2
2
2
2
2 1
2 2
2
1
2
2
2
3
2 The Construction We begin with the de nition of a projective (n; k; h ; h ) set in PG(k ? 1; q), where q is a power of prime p. De nition. A projective (n; k; h ; h ) set O is a proper, non-empty set of n points of the projective space PG(k ? 1; q) with the property that every hyperplane meets O in h points or h points. Let O = fhy i; hy i; ; hynig be a set of n points in PG(k ? 1; q). Associated with PG(k ? 1; q) is the k-dimensional vector space W = Vk (q). Let = fv 2 W jhvi 2 Og be the set of vectors in W corresponding to O. For w 2 GF (q)k , de ne an additive character of GF (q)k as follows: w : x 7! T r wx ; x 2 GF (q)k ; 1
1
2
2
1
2
1
2
(
)
where is a primitive p-th root of unity and Tr is the trace from GF (q) to GF (p). It is easy to see that w , w 2 GF (q)k , are all the additive characters of GF (q)k . For any nontrivial additive character w of GF (q)k , we have
w ( ) = (q ? 1)jw? \ fy ; y ; ; yngj + (?1)(n ? jw? \ fy ; y ; ; yngj) = qjw? \ fy ; y ; ; yngj ? n; 1
1
2
1
2
2
where w? = fy 2 GF (q)k jy w = 0g, and y w is the usual dot product. Hence we have the following lemma. Lemma 2.1. O is a projective (n; k; h ; h ) set if and only if w ( ) = qh ? n or qh ? n, for every nontrivial additive character w , w 2 GF (q)k . Also we mention that projective (n; k; h ; h ) sets are equivalent to projective two-weight codes, and certain strongly regular Cayley graphs. We refer the reader to the survey papers [3], [7] for more detailed discussion of these three objects. 1
2
1
1
2
2
2
Let = PG(3; p) denote projective 3-space over GF (p), where p is an odd prime. A spread of is any collection of p + 1 pairwise disjoint lines of , necessarily partitioning the points of . A partial spread in is a set of lines no two of which 4intersect. Also, for p? 2 p 2 p ? convenience, we will call a subset C of type Q if C is a projective ( p? ; 4; ; ) set. Theorem 2.2. Assume that S = fL ; L ; ; Lp2 g is a spread of . If there exist two subsets C , C of type Q in such that jC \ L2i j = (p + 1)=2, 1 i s, and jC \ Lj j = (p + 1)=2, (s + 1) j 2s, where s = p , then there exists a Hadamard di erence set in H (Zp ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set obtained is reversible. Proof: Let C = (L [ L [ [ Ls ) n C , C = (Ls [ Ls [ [ L s ) n C . We rst prove that C , C are also two subsets of type Q in . Associated with is the 4-dimensional vector space W = V (p) over GF (p). Let C = fw 2 W jhwi 2 C g, C = fw 2 W jhwi 2 C g. Since C is a set of type Q in , by Lemma 2.1 we have (C ) = p2 ? ? p or p2? , for every nontrivial additive character of W . We will use W to denote the additive character group 2 of W , and de ne U = f 2 W j (C ) = p ? ? p g, V = f 2 W j (C ) = p2? g. Let L = fw 2 W jhwi 2 [si Lig. Since fL ; L ; ; Lsg is a partial spread, we have 3
2
3
3
3
3
3
4(
1
0
1
2
2
1
+1 2
2
0
3
+1
3
+2
2
3
4
0
0
3
2
1
2
1
(
1
2
4
4
0
4
=1
2
1
0
0
1
3
0
1
( +1) 4
0
4
2
1) 4
3
+1
3
1
(
1 1)
1
4
2
p2 ; if 2 N ; ? (L ) = p2 ? ; if 2 T , where N = f 2 W n f gj is nontrivial on every Li; 1 i sg, and T = f 2 W n f gj is trivial on exactly one Li , for some i, 1 i sg. We contend that T \ U = ;. For each Lj , 1 j 2s, which is now viewed as a 2-dimensional subspace of W , let L?j = f 2 W j is trivial on Lj g. Then jL?j j = jW=Lj j = p . For 1 j s, we de ne j = j(L?j n f g) \ U j, j = j(L?j n f g) \ V j. Then j + j = p ? 1. For every 2 L?j , we have (C ) = (C n (C \ Lj )) + jC \ Lj j. Therefore +1 2 1
1
1
1
2
1
0
1
0
1
2
0
2
0
0
X 2L? j
0
(C ) = 0
0
X
0
X
(w) + p jC \ Lj j: 2
w2C0 n(C0 \Lj ) 2L? j
0
Noting that P 2L? (w) = 0 if there is a 2 L?j such that (w) 6= 1, we have P 2L? (C ) = p jC \ Lj j. That is p4? + j ( p2? ? p ) + j p2? = p jC \ Lj j. Simplifying this we get j
2
0
j
1
1
4
4
1
2
2
4
0
1 ? p + = jC \ L j: j j 2 2
0
3
0
Since jC \ Lj j = p2? for every j , 1 j s, we have j = p ? 1; j = 0; 1 j s. Hence T \ U = ;. For any nontrivial 2 W , (C ) = (L ) ? (C ). Since T \ U = ;, we have 1
0
2
2
1
2
(C ) =
1
( p2 ?1
0
? p ; if 2 N \ V ; if 2 (N \ U ) [ (T \ V ). 2
1
4 p2 ?1 ;
2
1
4
1
1
This shows that C is a set of type Q in . Similarly, de ne C = fw 2 W jhwi 2 C g, C = fw 2 W jhwi 2 C g. Let L = fw 2 W jhwi 2 [i s s Li g. Since C is a set of type Q in and fLs ; Ls ; ; L s g is a partial spread, we have 2
3
1
2 = +1
1
3
3
1
3
(C ) =
( p2 ?1
+1
+2
2
2
? p ; if 2 X ; if 2 Y . 2
4 p2 ?1 ;
1
4
and (
p2 ; if 2 N ; ? (L ) = p2 ? ; if 2 T , +1 2 1
2
2
2
2
where N = T , and T = N . By the same argument as above, we can show that T \ X = ;, hence 2
1
2
1
2
(C ) = 3
( p2 ?1 4 p2 ?1 4
? p ; if 2 N \ Y ; ; if 2 (N \ X ) [ (T \ Y ). 2
2
2
2
Assume that A is the union of any p2? lines from Ls ; Ls ; ; L s , B is the union of any p2 ? lines from L ; L ; ; L , and we view A, B as subsets in the vector space W (we make s the convention that A, B , when viewed as subsets in W , do not contain the zero vector). De ne 1
1
4
+1
4
1
+2
2
D = C [ A;
D = C [ B;
D = C [ A;
D = C [ B:
0
2
0
1
2
3
For any nontrivial 2 W , we distinguish two cases. 4
1
3
2
(1). Ker Lj , for some j , 1 j s. In2 this case, 2 N = T . Since T \ U = ;, we have 2 V . Therefore, (D ) = p ? + (? p2 ? ) = 0, and (D ) = p2 ? + (? p2 ? ) = 0. 2
4
1
1
1
1
2
4
0
1
1
4
4
(
p2 ? ; if L 2 B ; p ? j (B ) = p2 ? ? ; if Lj 62 B . 1
2
4
1
4
Hence
Lj 62 B , or, 2 X and Lj 2 B ; (D ) = 0;p ; ifif 22 YY and and Lj 2 B , or, 2 X and Lj 62 B ; 1
2
and
Lj 2 B , or, 2 X and Lj 62 B ; (D ) = 0;p ; ifif 22 YY and and Lj 62 B , or, 2 X and Lj 2 B . 3
2
This shows that (D ) = (D ) = 0, and only one of (D ), (D ) vanishes, the other is p . (2). Ker Lj , for some j , (s + 1) j 2s. In this case, 2 N = T . Since T \ X = ;, we have 2 Y . In a manner similar to that of case (1), we can show that (D ) = (D ) = 0, and only one of (D ), (D ) vanishes, the other is p . We rst consider the group Z Z (Zp) . Let us denote the elements of Z Z by f1; a; b; abg. De ne D = D [ aD [ bD [ ab(W n D ). We contend that D is a reversible Hadamard di erence set in Z Z (W; +). Let be an arbitrary nontrivial character of Z Z W . If is trivial, is nontrivial, then 0
2
1
2
2
1
3
2
1
2
3
2
0
2
4
2
1
2
0
2
2
3
2
2
2
(D) = jD j + (a)jD j + (b)jD j + (ab)jW n D j = p (ab); 0
1
2
3
so j (D)j = p . If is nontrivial, by the discussion in the two cases above, we have 2
(D) = p ; 2
hence j (D)j = p . 2
5
2
2
By Lemma A, D is a Hadamard di erence set. Since (D) = (D) for every nontrivial character of Z Z W , D is reversible. In the case the group is Z (Zp) , let the elements of Z be f1; c; c ; c g, and D = D [ cD [ c D [ c (W n D ). Then it is easy to show that D is a Hadamard di erence set in Z (W; +). This completes the proof of the theorem. 2 2
2
2
4
3
0
2
1
3
2
4
4
3
4
3 On Xia's Construction In 1992, Xia [10] constructed Hadamard di erence sets in groups H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. Xia's construction depends on very complicated calculations involving cyclotomic classes of high order. Xiang and Chen [11] have given a simpler proof for Xia's construction by using additive characters of nite elds. In view of Theorem 2.2, in order to construct Hadamard di erence sets in H (Zp) , where H is either group of order 4 and p is a prime congruent to 3 modulo 4, all we need are a spread in , and sets C , C of type Q in satisfying the conditions in Theorem 2.2. Let p be a prime congruent to 3 modulo 4 and let be a primitive element of GF (p ). We model by viewing GF (p ) as 4-dimensional vector space over GF (2p). Thus the points of 2 2 p p ? i p i p p i ig, are represented by h1i; h i; ; h i. Let Li = fh i; h i; ; h 0 i p . Then it is easy to see that S = fL ; L ; ; Lp2 g is a spread in . Let C = fh1i; h i; h i; ; h p2 p ? ig, C = fh i; h i; h i; ; h p2 p ? ig. Since p 3 (mod 4), by uniform cyclotomy (see [2], [7]),2 C , C are two sets of type Q in . Also it is easy to see that jC \ L i j = p , 0 i p ? , and jC \ L i j = p , 0 i p2? . Therefore by Theorem 2.2, we have Corollary 3.1. There exists a Hadamard di erence set in H (Zp) , where p is a prime congruent to 3 modulo 4, and H is either the Klein 4-group or the cyclic group of order 4. In the rst case, the Hadamard di erence set constructed by Theorem 2.2 is reversible. Using a composition theorem of Turyn [9], it is routine to construct Hadamard di erence sets in H Zp1 Zp2 Zp , where H is either group of order 4, and each pj is a prime congruent to 3 modulo 4, 1 j t. 4
4
4
t
4
3
0
1
3
4
4
3
(
+1)( +1)
1
(
2
0
0
4
8
(
+1)( +1)
4
1
0
0
+1 2
2
1
2
+1)+
(
3
+1)+ 3
1 5
9
(
+1)( +1)
3
1
3
1
+1 2
2 +1
1
2
4
4
4
4
t
4 General Construction in the case p 3 (mod 4) In this section, we give a general construction of Hadamard di erence sets in H (Zp) , where H is either group of order 4, and p is a prime congruent to 3 modulo 4, by using an ovoidal bration of PG(3; p) in [1], [4] and [6] (page 253). We introduce the following 4
6
notation as in [1], [4]. Let p be a prime congruent to 3 modulo 4. We view GF (p ) as a 4-dimesional vector space over GF (p), and hence the 1-dimensional subspaces of this vector space can be thought of as the projective points of = PG(3; p). Similarly, we identify the lines of with the 2-dimensional vector subspaces of GF (p ) over GF (p). We also let hAi denote the vector subspace generated by the set A over GF (p). If is a2 primitive element of GF (p ), then o( ) = p ? 1 = (p + 1)(p ? 1)(p + 1) and hence p p is a primitive element of GF (p). We therefore identify the points of with fh tijt = 0; 1; 2; ; (p + 1)(p + 1) ? 1g. If we now let i = fh tijt i (mod p + 1) g, each
i is an ovoid (Theorem 3 of [4]) and the points of are thus partitioned into p + 1 disjoint ovoids: = [ [ [ p : () Moreover, each line of is tangent to precisely 0 or 2 of these ovoids (Lemma 1 and Theorem 4 of [4]). Lines of the former type will be called \secant-type" and those of the latter \tangent-type". If Ls = h ; s p i denotes the line of joining the points h i and h s p i of for any integer s with 1 s p , then Ls is a secant-type line if and only if s is odd (Theorem 4 of [4]). If L is any line of , let [L] denote the line orbit of L under the collineation of corresponding to multiplication by p . Also let Lp denote the image of the line L under the collineation corresponding to the Frobenius automorphism, and let L d denote the image of L under multiplication by d. We quote the following theorem and corollary from [1]. Theorem B. Using the above notation, let s be an odd integer with 1 s p . Consider p s p the secant-type line Ls = h ; i, necessarily secant to ovoids in the bration (). Then there exists a positive integer d such that (Ls)p d is a secant-type line meeting the p ovoids of () missed by Ls. Moreover, if s 6= p2 , d is unique modulo p + 1. Corollary C. [Ls] [ [(Ls )p d] is a spread of . This spread is regular precisely when 2 s= p . Now we use the spread in Corollary C to construct Hadamard di erence sets. By Theorem 2.2, we need to come up with two sets C , C in of type Q satisfying the conditions in Theorem 2.2. Let Ls and (Ls)p d be the secant-type lines in Theorem B. We assume that Ls is secant to
t1 ; t2 ; ; t , and (Ls)p d is secant to t +1 ; t +2 ; ; t2 , where r = p . By Theorem B, ft ; t ; ; t r g = f0; 1; 2; ; pg. Since p 3 (mod 4), r is even. Let C be the union of any r=2 ovoids from f t1 ; t2 ; ; t g, and let C be the union of any r=2 ovoids from f t +1 ; t +2 ; ; t2 g. Then we have the following lemma. 4
3
3
4
4
( +1)(
4
2
+1)
3
2
3
3
0
1
3
0
( +1)
( +1)
0
3
0
3
2
3
2( +1)
2
0
+1 2
( +1)
+1 2
+1 2
3
+1 2
0
r
1
2
1
r
r
r
2
r
+1 2
0
1
r
r
3
r
7
Lemma 4.1. C meets every line in [Ls ] in p points, and C meets every line in [(Ls)p d] p +1 2
0
1
points. C0 , C1 are two sets of type Q in 3 . Proof. The rst assertion is clear by the de nition of C0 and C1 . For the proof of the second part, we observe that every plane of 3 must meet each of the p + 1 ovoids in () in a point or an oval. A simple counting argument then shows that each plane of 3 is tangent to 1 of the ovoids in () and meets the other p ovoids in disjoint ovals. Let be an arbitrary (p?1)2 r plane of 3 . Then j \ C0j = 1 + ( 2 ? 1)(p + 1) = 4 if C0 contains some t such that 2 j \ t j = 1, 1 j r, and j \ C0j = 2r (p + 1) = (p+1) if j \ t j = p + 1, for every t 4 contained in C0 . This shows that C0 is a set of type Q in 3 . Similarly, we can show that C1 is also a set of type Q in 3 . This completes the proof of the lemma. 2 Corollary 4.2. Let [Ls] [ [(Ls)p d], C0, C1 be de ned as above. Then there exists a Hadamard di erence set in H (Zp )4 , where H is either group of order 4, and p is a prime congruent to 3 modulo 4, by using the spread [Ls ] [ [(Ls )p d ] and the sets C0 and C1 of type Q in 3 . Proof. This is clear from Lemma 4.1 and Theorem 2.2. 2 Remarks. (1) If we let s = p22+1 , then Ls = GF (p2), Lps = Ls, and [Ls] [ [ Ls] is the regular spread in Section 3. Also we note that Ls meets the ovoids 0 ; 2 ; 4; ; p?1 (Lemma 1 of [4]), and Ls meets the ovoids 1 ; 3; 5 ; ; p. If we choose the union of
0 ; 4; 8 ; ; p?3 as C0 and the union of 1 ; 5; 9 ; ; p?2 as C1, then the construction in this section will give rise to Xia's construction. (2) Let G = K4 P , where K4 is the Klein 4-group and P = Zp4 , p is a prime congruent to 3 modulo 4. Two di erence sets D and D0 in G are said to be equivalent if D0 = gD for some automorphism of G and some element g of G. Since K4 and P have relatively prime orders, they must be invariant under every automorphism of G. Therefore the automorphism group of G has size jGL(2; 2)jjGL(4; p)j = 6p6(p4 ? 1)(p3 ? 1)(p2 ? 1)(p ? 1). From Theorem 2.2 and the construction in this section, we see that there are at least in
+1 2
j
j
i
2 p2 +1 2 2 +1 p+1 p 2 4! 2 p+1 p22?1 4 4
i
p+1 2 p2 +1 2 2 2 + 1) p+1 p2 ?1
(p 4 4 = 4p 6p (p ? 1)(p ? 1)(p ? 1)(p ? 1) 2p (p ? 1)(p ? 1)(p ? 1)(p ? 1) 4
6
4
3
2
2
10
4
3
2
pairwise inequivalent Hadamard di erence sets in G.
5 The case p 1 (mod 4) In this section, we construct sets of type Q in = PG(3; p) with p 1 (mod 4). Again let W be the 4-dimensional vector space over GF (p) associated with . We may consider W 3
3
8
as a direct product GF (p ) GF (p ). Let g be a primitive element of GF (p ). L1 will denote the line f0g GF (p ), and for any d in GF (p ), Ld will denote the line f(x; dxp)jx 2 GF (p )g. It is easy to verify that S = fLd jd 2 GF (p )g [ fL1g is a spread in . We now consider the action of T = g0 g0? on the points of which are now viewed as 1-dimensional subspaces over GF (p) of the 4-dimensional vector space GF (p ) GF (p ) over GF (p). The orbits of the action of T on the points of are as follows. (1). Four \short" orbits, each of length p . We choose (0; 1), (0; g), (1; 0), and (g; 0) as the representatives of these four orbits. (2). 4(p + 1) \long" orbits, each of length p2 ? . The representatives of these 4(p + 1) orbits can be chosen as (1; 1); (1; g); (1; g ); ; (1; g p ); (g; 1); (g; g); (g; g ); ; (g; g p ): 2
2
2
2
2
2
2
3
2
2
3
2
2
3
+1 2
1
4
2
2 +1
2
2 +1
It is clear that each short orbit consists of p points of L or L1. Next we show that each long orbit consists of p points of p? lines from the set of lines fLd jd 6= 0; d 2 GF (p )g. For example, take the long orbit represented by (1; gi), 0 i 2p + 1. Let a = gi. The points in this orbit are represented by +1 2
0
1
+1 2
2
2
(1; a) ! (g ; ag? ) ! : : : ! (gp? ; ag ?p) ! (gp? ; ag ?p) ! (gp ; ag? ?p) ! : : : ! (g p? ; ag ? p) ! 2
1
1
2
+1
3
1
2
4
3
4
2
(g`; ag?`) ! (g` ; ag?`? ) ! : : : ! (g` p? ; ag?`?p ) +2
2
+
2
3
+3
where ` = p? . Each column in the above diagram consists of p points of some line Ld , d 6= 0, d 2 GF (p ), hence the orbit represented by (1; gi) consists of p points of p? lines from the set of lines Ld ; d 6= 0; d 2 GF (p ). This argument applies to any orbit represented by (g; gi), 0 i 2p + 1. For p = 5 let g be a root of x + x + 2 2 GF (5)[x]. With the help of a computer (we will give more details about the computer search at the end of this section), we found that the union of the following orbits (
1) 2
+1 2
2
2
2
(1; g); (1; g ); (1; g ); (1; g ); (g; 1); (g; g ); (1; 0) 2
9
11
9
8
+1 2
1
2
forms a set of type Q in PG(3; 5), which we will call C , also the union of the following orbits (1; g ); (1; g ); (1; g ); (g; g ); (g; g ); (g; g ); (0; 1) forms another set of type Q in PG(3; 5), which we denote by C . Let S = fLd jd 2 GF (5 )g[ fL1g. We have seen that each orbit of T intersects the lines in S in 0 or 3 points, also no two orbits of T in C , C intersect the same line, hence C , C satisfy the conditions of Theorem 2.2, therefore there exists a Hadamard di erence set in H (Z ) , where H is either group of order 4. We state this as a corollary. Corollary 5.1. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set is reversible. Remark. Van Eupen and Tonchev ([5]) were the rst to construct a reversible Hadamard di erence set in Z Z (Z ) . We remark that the structure of the Hadamard di erence set in Z Z (Z ) constructed in Corollary 5.1 is di erent from that of Van Eupen and Tonchev's Hadamard di erence set. For example, in Theorem 2.2 (hence in Corollary 5.1), we choose A, B both as union of lines from a spread in , while in Van Eupen and Tonchev's example, one projective (36; 4; 6; 11) set in PG(3; 5) comes from the union of 6 lines, the other does not. In the case p = 13, let g be a root of x + x + 2 2 GF (13)[x]. With the help of a computer, we found the following two sets of type Q in PG(3; 13). The union of the following orbits (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (1; 0) forms a set of type Q in PG(3; 13), which we will denote by C . And the union of the following orbits (1; 1), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (1; g ), (g; g), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (g; g ), (0; 1) forms another set of type Q, which we will denote by C . Let S = fLd jd 2 GF (13 )g[fL1g. It is easy to see that S , C , C satisfy the conditions in Theorem 2.2. By Theorem 2.2, we have Corollary 5.2. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case the Hadamard di erence set is reversible. When p = 17, let g be a root of x + x + 3 2 GF (17)[x]. With the help of a computer, we found the following two sets of type Q in PG(3; 17). The union of the following orbits 0
6
8
10
5
9
10
2
1
0
1
0
1
5
4
5
2
2
2
5
2 4
5
4
4
3
2
5
6
16
9
13
15
17
18
24
4
5
8
14
11
12
23
0
2
24
4
7
8
12
25
6
7
27
2
1
0
1
13
2
10
4
(1; g ), (g; g), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (1; g ), (1; 0) forms a set of type Q in PG(3; 17), which we will denote by C . And the union of the following orbits (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (1; g ), (g; g ), (0; 1) forms another set of type Q in PG(3; 17), which we will denote by C . Let S = fLd jd 2 GF (17 )g[fL1g. Then it is easy to check that S , C , C satisfy the conditions in Theorem 2.2. By Theorem 2.2, we have Corollary 5.3. There exists a Hadamard di erence set in H (Z ) , where H is either the Klein 4-group or the cyclic group of order 4; in the rst case, the Hadamard di erence set is reversible. Remark. We give more details about our computer search in what follows. In order to search for sets of type Q in by computer, we rst noted that T also permutes the planes in four \short" orbits and 4(p + 1) \long" orbits. We formed a square nonnegative integral matrix M whose rows were indexed by the orbits of T on the points, whose columns were indexed by the orbits of T on the planes, and where the entry in row i and column j was the the cardinality of the intersection of a (any) plane in plane-orbit j with the i-th orbit of points. A union of point-orbits is a set of type Q if and only if the sum of the corresponding rows of M has entries (p 1) =4 only. For example, when p = 17, the matrix M is square of order 76. The sum of all rows was a constant vector of 307's. We searched for 19 rows (one row corresponding to a short orbit and the others to long orbits) so that the sum of the 19 rows had entries 64 and 81 only. Our search was not exhaustive but simply moved from one set of 19 rows to another set by deleting one row|with a large entry in a column where the sum exceeded 81|and randomly adding another one. This was done with Mathematica on a PC. Finally, using a composition theorem of Turyn [9], it is routine to construct (4m ; 2m ? m; m ?m) Hadamard di erence sets with m = 2a 3b5 c1 13 c2 17 c3 p p pt , where a; b; c ; c ; c are positive integers and each pj is a prime congruent to 3 modulo 4, 1 j t. 2
7
25
32
6
31
12
34
11
33
15
21
14
19
18
26
22
0
5
4
18
6
17
5
31
9
30
8
4
10
9
11
10
17
16
20
1
2
0
1
17
4
3
2
2
2
2
2
2
2 1
2 2
2
2
1
2
Acknowledgment: We thank G. Ebert, W. Kantor and V. Tonchev for their helpful dis-
cussions.
11
3
References [1] R. D. Baker and G. L. Ebert \Construction of two-dimensional ag-transitive planes", Geometriae Dedicata 27 (1988), 9{14. [2] L. D. Baumert, W. H. Mills, and R. L. Ward \Uniform cyclotomy", J. Number Theory 14 (1982), 67{82. [3] A. R. Calderbank and W. M. Kantor \The geometry of two-weight codes", Bull. London Math. Soc. 18 (1986), 97{122 [4] G. L. Ebert \Partitioning projective geometries into caps", Can. J. Math. Vol. 37, No. 6, (1988), 1163{1175. [5] M. van Eupen and V. D. Tonchev \The existence of a reversible Hadamard di erence set in Z Z (Z ) and an in nite class of Williamson matrices", Preprint, Oct. 1995. [6] J. H. van Lint and R. M. Wilson \A Course in Combinatorics", Cambridge University Press, 1992. [7] S. L. Ma \A survey of partial di erence sets", Designs, Codes and Cryptography,4 (1994) 221{261. [8] K. W. Smith \Non-abelian Hadamard di erence sets", J. Combin. Theory Ser. A 70 (1995), 144{156. [9] R. J. Turyn \A special class of Williamson matrices and di erence sets", J. Combin. Theory Ser. A 36 (1984), 111{115. [10] M. Y. Xia \Some in nite classes of special Williamson matrices and di erence sets", J. Combin. Theory Ser. A 61 (1992), 230{242. [11] Q. Xiang and Y. Q. Chen \On Xia's construction of Hadamard di erence sets", Finite Fields and Their Applications 2, (1996), 87{95. 2
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