Constructions of Partial Di erence Sets and Relative Di ... - CiteSeerX
Constructions of Partial Di erence Sets and Relative Di erence Sets Using Galois Rings D. K. Ray-Chaudhuri Department of Mathematics, Ohio State University, Columbus, OH 43210 Qing Xiang Dept. of Mathematics, California Institute of Technology, Pasadena, CA 91125 Dedicated to Professor Dr. Hanfried Lenz
Abstract We use Galois rings to construct partial di erence sets and relative di erence sets in non-elementary abelian -groups. As an example, we also use Galois ring (4 2) to construct a (96,20,4) di erence set in 4 4 6 . p
GR
Z
Z
E-mail: [email protected]
0
Z
;
1 Introduction Let G be a nite group of order v. A k-element subset D of G is called a (v; k; ; )partial di erence set (PDS) in G if the di erences d1 d?2 1 , d1 ; d2 2 D, d1 6= d2 , represent each nonidentity element in D exactly times and each nonidentity element not contained in D exactly times. D is called abelian if G is abelian. It is well known that a PDS D with e 62 D and fd?1 : d 2 Dg = D is equivalent to a strongly regular Cayley graph, such a PDS is called regular. For a survey on partial di erence sets, we refer the reader to Ma [8]. Assume that v = mn and that G contains a normal subgroup N of order n. A k-element subset R of G is called an (m; n; k; )-relative di erence set (or,in short an (m; n; k; )-RDS) in G relative to N if the di erences d1d?2 1, d1; d2 2 R, d1 6= d2, represent each element in G n N exactly times and each nonidentity element in N zero time. If G = H N , where H is some subgroup of G, then R is called a splitting RDS. For more details about relative di erence sets, we refer the reader to the survey of Pott [11]. Finite eld theory , in particular, the cyclotomy of nite elds, is very useful in the construction of di erence sets, partial di erence sets and relative di erence sets in elementary abelian p-groups or in groups with elementary abelian Sylow p-subgroups. In the case the group is not elementary abelian, we can no longer use nite elds for the purpose of construction. In this paper, we use Galois rings to construct new families of partial di erence sets and relative di erence sets in non-elementary abelian p-groups. We also use this approach to construct a (96,20,4) di erence set in Z4 Z4 Z6. The following characterizations of abelian partial di erence sets and relative di erence sets will be used in our constructions. Lemma A. Let G be an abelian group of order v and D be a subset of G such that fd?1 : d 2 Dg = D. Suppose k, and are positive integers satisfying k2 = v +( ? )k + where = k ? if e 2 D and = k ? if e 62 D. Then D is a (v; k; ; )-partial di erence set in G if and only if, for any character of G, 8 < X j (r)j = >: r2R
kp if is principal on G. pk ? n if is nonprincipal on G but principal on N . k if is nonprincipal on N .
2 Galois Rings over Z=p2Z Let Fp be the nite eld with p elements, where p is a prime and (x) be a primitive polynomial of degree t over Fp. Let (x) be a polynomial over Z=p Z obtained from (x) by taking a preimage of (x) underPthe homomorphism : Z=p Z [x] ! Fp[x] which is de ned P n by f (x) = i ai xi 7! f (x) = ni ai xi, where ai is the reduction of ai mod p. There is a unique monic (x) whose root g satis es gp ? = 1. The ring Z=p Z [g] is an algebraic extension of Z=p Z , it is the Galois extension of Z=p Z of degree t. This extension Z=p Z [g] is called a Galois ring, and is denoted by GR(p ; t). GR(p ; t) is a nite local ring, it has the unique maximal ideal B = f0; p; pg; ; pgq? g, where q = pt, the residue class eld GR(p ; t)=B = K = f0; 1; g; ; gq? g is isomorphic to Fq (the nite eld with pt elements). We can take the Teichmuller system T = f0; 1; g; ; gq? g as a set of representatives of GR(p ; t)=B . Therefore an arbitrary element of GR(p ; t) is uniquely represented as = + p , , 2 T . We denote the set of invertible elements of GR(p ; t) by GR(p ; t) = GR(p ; t) n B . Every element of GR(p ; t) has a unique representation in the form gi(1 + p ), 0 i q ? 2, 2 T . GR(p ; t) is a multiplicative group of order (pt ? 1)pt which is a direct product H U , where H is the cyclic group of order pt ? 1 generated by g, and U is the group of principal units of GR(p ; t), that is elements of the form 1 + p ; 2 T . U has the structure of an elementary abelian group of order pt and is isomorphic to the additive group of K via the map 1 + p 7! , 2 T . For the proof of the above assertions on the structure of GR(p ; t) and more detailed description of Galois rings, we refer the reader to MacDonald [9]. 2
2
=0
=0
t
1
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
1 2
0
1
2
2
2
2
3 Frobenius and Trace Maps We will use the following lemma about the additive characters of K = f0; 1; g; ; gq? g in the future. Lemma C. All additive characters y of K , for y 2 K , are given as follows 2
y (x) = ptr xy ; x 2 K (
2
)
where p is a primitive p-th root of unity, and tr is the relative trace from K to Fp. The Frobenius map f from GR(p ; t) to GR(p ; t) is the ring automorphism f : + p 7! p + p p, , 2 T . The Galois group of GR(p ; t) over Z=p Z is a cyclic group of order t which is generated by f . The set of elements of GR(p ; t) invariant under f is identical with Z=p Z . The relative trace from GR(p ; t) to Z=p Z is de ned by 2
0
0
1
2
2
1
0
2
1
2
2
2
2
T ( ) = + f + + f ? t
1
It is easy to see that the following diagram is commutative. GR(?p ; t) ?! GR(p?; t)=B ? ? 2
2
yT
?!
Z=p Z 2
Hence
ytr
Fp
T = tr
(3:1)
where is the natural homomorphis. About the additive characters of GR(p ; t), we have the following lemma. Lemma D. All additive characters of GR(p ; t), for 2 GR(p ; t), are given as follows 2
2
2
( ) = pT ; 2 GR(p ; t) (
2
)
2
where p is a primitive p -th root of unity, and T is the relative trace from GR(p ; t) to Z=p Z . For the proofs of Lemma C and Lemma D, please see Yamamoto and Yamada [12]. 2
2
2
2
4 Partial Di erence Sets We will construct two families of regular PDS in the additive group of GR(p ; t). Our strategy is to come up with a \candidate" subset of the group, then use characters to check that all the character sums are correct. We will follow the notation used in Section 2 and 3. Considering the subgroup H of GR(p ; t) , we enumerate the cosets of H in GR(p ; t) as follows, E = H; E = (1 + p)H; Eg = (1 + pg)H; ; Eg ? = (1 + pgq? )H , where q = pt. We note that E , E , Eg , ,Eg ? , B n f0g, f0g are the orbits of the multiplication action of H on GR(p ; t), and jE j = jE j = jEg j = = jEg ? j = q ? 1. 2
2
2
q
q
0
1
2
2
0
1
2
2
q
2
3
0
1
Let A = fx 2 K : tr(x) = 0g, and v (p) = pq?? . Then all nontrivial additive characters of K with kernel equals to A are 1 1
1
f : 2 GalQ(p)=Qg = f g 1
v1 (p)
; g
2v1 (p)
; ; g
?
(p 1)v1 (p)
g
(4:1)
De ne D = [a2AEa . Then we have the following theorem.
Theorem 4.1 If p is an odd prime, pjt, then D is an (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with n = pt , r = pt? ; D [ (B n f0g) is an (n ; r (n ? 1); n + r ? 3r ; r ? r )-PDS in the additive group of GR(p ; t) with r = pt? + 1. 2
2
2 1
1
2 1
2
2
1
2
2
1
1
1
1
Proof: Let be an arbitrary additive character of GR(p ; t). By Lemma D, we consider 2
the following three cases. (1) is principal, i.e. = . In this case, (D) = jDj = jAj(pt ? 1) = pt? (pt ? 1). (2) = pg , 0 u (q ? 2). In this case, has order p, and qX ? ((1 + pa)gi) (Ea ) = 1
0
u
2
i=0 qX ?2
=
i=0 qX ?2
=
i=0
pT
pa)gi pgu)
((1+
2
pT pg : (
2
i
)
By (3.1), we have
(Ea ) = =
qX ?2 i=0 qX ?2 i=0
pptr g 2
( i)
ptr g
( i)
= (K n f0g) = ?1: 1
Hence (D) = (?1)jAj = ?pt? . (3) = , = (1 + pb)gu, where 0 u (q ? 2), and b 2 T . In this case, has order p , and X (D) = (Ea ) 1
2
a2A
4
= = = = =
?2 X qX a2A i=0 ?2 X qX a2A i=0 ?2 X qX a2A i=0 ?2 X qX
((1 + pa)gi) pT
((1+
pa)gi (1+pb)gu )
pT
((1+
pT
((1+
2
pa+pb)gi )
2
pb)gi ) T (pagi ) p2
2
a2A i=0 qX ?2 i X i pT2((1+pb)g ) pT2(pag ): i=0 a2A
By (3.1), we have
(D) = = By (4.1), we know that
qX ?2 i=0 qX ?2 i=0
pb)gi ) X tr(gi a) p a2A
pT
((1+
pT
((1+
2
pb)gi )
2
gi (A):
(
v (p)j , 1 j p ? 1. g (A) = 0jAj ifif ii 6= = v (p)j , for some j , 1 j p ? 1. 1
i
1
Hence
(D) =
pX ?1 j =1
= jAj
jAjpT
pb)gv1 (p)j )
((1+
2
pX ?1
pT g
pT pbg
pT g
ptr bg
j =1
( v1 (p)j )
2
(
v1 (p)j
2
Again, by (3.1), we have
(D) = jAj
pX ?1 j =1
( v1 (p)j )
2
5
(
v1 (p)j
)
:
)
:
We note that gv p j 2 Fp and gv p j 2 Z=p Z because of (gv T (gv p j ) = tgv p j and tr(bgv p j ) = gv p j tr(b). So we have pX ? (D) = jAj ptg ptr b g : 1( )
1( )
1( )
1( )
1( )
2
1
v1 (p)j
2
j =1
If p jt, then (D) = jAj Ppj ? ptr b g
( ) v1 (p)j
1 =1
2
p j )p
1( )
1( )
= gv
p j.
1( )
Therefore
( ) v1 (p)j
. Therefore
( t?1 p (p ? 1) if tr(b) = 0. (D) = ? pt?1 if tr(b) 6= 0.
If pjjt, we assume that t = pw; (p; w) = 1, then pX ? (D) = jAj ppwg 1
= jAj therefore in this case,
j =1 pX ?1 j =1
v1 (p)j
2
(pw
ptr b g
( ) v1 (p)j
tr(b) )gv1 (p)j
+
( t?1 (b) ?w(mod p). (D) = p?pt?(1p ? 1) ifif tr tr(b) 6 ?w(mod p).
Summing up all these calculations, we have shown that, for any nonprincipal additive character of GR(p ; t), (D) = pt? (p ? 1) or ?pt? . By Lemma A, D is an (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with n = pt, r = pt? . For the proof of the second part of the theorem, we note that for any nonprincipal additive character of GR(p ; t), ( 1 if has order p . (B n f0g) = ? pt ? 1 if has order p . 2
2
1
1
2
2
2
1
2
2
Then by the above calculations of (D) and Lemma A, we see that D [ (B n f0g) is an (n ; r (n ? 1); n + r ? 3r ; r ? r )-PDS in the additive group of GR(p ; t) with r = pt? +1. This completes the proof of the theorem. 2 Remarks: (1). Yamamoto and Yamada [12] used a method similar to that in Theorem 4.1 to construct an Hadamard di erence set in the additive group of GR(4; t), where t is not necessarily even. 2
1
2 1
1
2 1
2
1
6
1
1
(2). The PDSs in Theorem 4.1 belong to the Latin square type (see Ma [8]). There are many known examples of Latin square type PDSs. One way to construct Latin square type PDS is to use a partial congruence partition of the group. Let G be a group of order n . A partial congruence partition of G with degree r (an (n; r)-PCP) is a set P of r subgroups of G of order n such that U \ V = feg for every pair of distinct elements U; V of P . Ma [6] showed that if P is an (n; r)-PCP of G, then D = [U 2P U n feg is a regular (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in G. We claim that the PCP construction for PDS can not give rise to the PDSs in Theorem 4.1. The reason is as follows. The additive group of GR(p ; t) is isomorphic to (Zp )t . If there is an (n; r)-PCP in the additive group of GR(p ; t), where n = pt, then r pbt= c + 1 (see Jungnickel [5]), so the PCP construction can only give (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with r pbt= c + 1, while in Theorem 4.1 r = pt? or pt? + 1. (3). Leung and Ma [6] used nite local ring to construct Latin square type PDS in R R, where R is a nite local ring. If R = GR(p ; t), Leung and Ma's construction can only give PDS in (Zp ) t while Theorem 4.1 can give PDSs in (Zp )t with t odd. This shows that Theorem 4.1 does give new PDSs. 2
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
5 Relative Di erence Sets In this section, we give two constructions of relative di erence sets. The rst is originally due to Brock [3], here we phrase the construction in terms of Galois rings, and give a character theoretic proof. The relative di erence sets in the second construction seem to be new although Davis [4] had constructed RDS with the same parameters in more general groups. The purpose for us to present these two constructions is to illustrate the use of Galois rings. We will follow the notation used in Section 2, 3 and 4. Let G = (GR(p ; t); +) (K; +), where (GR(p ; t); +) is the additive group of GR(p ; t), and (K; +) is the additive group of K . Let q = pt , and R = (E ; 0) [ (E ; 1) [ (Eg ; g) [ [ (Eg ? ; gq? ) [ (B; 0). Then we have the following theorem. 2
q
2
2
2
0
2
1
Theorem 5.1 R is a (q ; q; q ; q)-relative di erence set in G relative to (K; +). Proof: Let be an arbitrary character of G. Then = , where is an additive 2
2
character of GR(p ; t) and is an additive character of K . We consider two cases. Case 1. is a principal character. X (R) = (Ea ) + (B ) 2
a2K
If is principal, then (R) = jRj = q . 2
7
If is not principal, then we have two cases. (1) = pg . In this case, has order p, (B ) = jB j = q, and (Ea) = ?1, for every a 2 K by the calculations in the proof of Theorem 4.1. Hence (R) = (?1)q + q = 0. (2) = , = (1 + pb)gu, where 0 u (q ? 2), and b 2 T . In this case, has order p , (B ) = 0, and u
2
X a2K
?2 X qX
(Ea ) =
a2K i=0 ?2 X qX
=
a2K i=0 ?2 X qX
=
a2K i=0 ?2 X qX
=
((1 + pa)gi) pT
((1+
pT
((1+
pT
((1+
2
2
2
pa)gi (1+pb)gu ) pa+pb)gi ) pb)gi ) T (pagi ) p2
a2K i=0 qX ?2 X i i pT2((1+pb)g ) pT2(pag ): i=0 a2K
= By (3.1), we have X a2K
(Ea ) =
qX ?2 i=0 qX ?2
pT
= pT i = 0: Hence (R) = 0. Case 2. is nonprincipal on (K; +). (R) =
X a2K
X a2K
pb)gi )
((1+
2
=0
pb)gi )
((1+
2
ptr g a
( i )
gi (K )
(Ea) (a) + (B )
we also distinguish three cases. (1) is principal. (R) = jEa j(Pa2K (a)) + (B ) = q. (2) has order p, i.e.P = pg . In this case, (B ) = jB j = q, and (Ea ) = ?1, for every a 2 K . So (R) = ? a2K (a) + q = q. u
8
(3) has order p , i.e. = , = (1 + pb)gu. We assume that = g . In this case, (B ) = 0, and X (R) = (Ea ) (a) 2
v
a2K
= = = = =
?2 X qX
a2K i=0 ?2 X qX
pT
((1+
pT
((1+
2
2
pa)gi (1+pb)gu ) tr(gv a) p pa+pb)gi ) tr(gv a) p
a2K i=0 qX ?2 X i i pT2((1+pb)g ) pT2(pag ) ptr(gv a) i=0 a2K qX ?2 X i pT2((1+pb)g ) ptr((gi+gv )a) i=0 a2K qX ?2 i pT2((1+pb)g ) gv +gi (K ) i=0
If gv + gi = 0, then g g (K ) = jK j = q, If gv + gi 6= 0, then g g (K ) = 0. It is easy to see that there is a unique i, 0 i q ? 2, such that gv + gi = 0. Hence (R) = qpT pb g , and j (R)j = q. Hence we have shown that 8 > < q if is principal on G. j (R)j = >: 0 if is nonprincipal on G but principal on (K; +). q if is nonprincipal on (K; +). v
+ i
v
+ i
((1+
2
) i)
2
By Lemma B, R is a (q ; q; q ; q)-relative di erence set in G relative to (K; +). This completes the proof. 2 The RDS in Theorem 5.1 is splitting. In the following, we construct a non-splitting RDS in the same group. Again let G = (GR(p ; t); +) (K; +). De ne R = [a2K (Ea [f0g; a). We have the following theorem. 2
2
2
Theorem 5.2 R is a (q ; q; q ; q)-relative di erence set in G relative to B , where B is the 2
2
unique maximal ideal of GR(p2 ; t). Proof: Let be an arbitrary character of G. Then = , where is an additive character of GR(p2; t) and is an additive character of K .
9
If is principal, then (R) = jRj = q . If is principal, but is nonprincipal, then (R) = Pa2K q (a) = 0. IfP has order p, i.e. is principal on B but not principal on GR(p ; t), then (R) = a2K (1 + (Ea )) (a). Noting that (Ea ) = ?1, we have (R) = 0. If has order p , we consider two cases. Case 1. is principal. X (R) = ((Ea) + 1) a2K X X = (Ea ) + 1 2
2
2
a2K
a2K
From the proof of Theorem 5.1, we see that Pa2K (Ea) = 0 if has order p . Hence X (R) = 1 = q: 2
a2K
Case 2. is nonprincipal.
X
((Ea) + 1) (a) X X = (Ea ) (a) + (a) a2K a2K X (Ea ) (a) =
(R) =
a2K
a2K
From the proof of Theorem 5.1, we see that j (R)j = q. By Lemma B, we conclude that R is a (q ; q; q ; q)-relative di erence set in G relative to B . This completes the proof. 2 2
2
6 A (96,20,4) Di erence Set in Z4 Z4 Z6 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. A di erence set with parameters (v; k; ) = (qd (qd + qd? + + 2); qd(qd + qd? + + 1); qd(qd? + qd? + + 1)) where q = pf and p is a prime is called a McFarland di erence set. McFarland di erence sets were originally constructed in G = E L, where E denotes the elementary abelian group of order qd and L is an arbitrary 1
1
2
1
1
1
2
1
2
+1
2
+1
10
1
abelian group of order (qd + qd? + +2) by using hyperplanes of E (see [10]). When E is not elementary abelian, the problem of constructing di erence sets with McFarland's parameters is more dicult. For a long time, it was not known whether there exists a (96,20,4) di erence set in Z Z Z . In 1993, Arasu and Sehgal [1] constructed several (96,20,4) di erence sets in Z Z Z . But they did not give their method of construction. In the following, we show that by viewing Z Z as the additive group of GR(4; 2), a construction similar to that of McFarland [10] can give a (96,20,4) di erence Set in Z Z Z . Let (x) = x + x + 1 2 F [x]. (x) is a primitive polynomial of degree 2 over F . Let (x) = x + x + 1 2 Z=4Z [x]. (x) has a root g satisfying g = 1. Then GR(4; 2) = Z=4Z [g]. This ring has a unique maximal ideal B = f0; 2; 2g; 2g g, and GR(4; 2)=B = K = f0; 1; g; g g = F . We can take the Teichmuller system T = f0; 1; g; g g as a set of representatives of GR(4; 2)=B . Therefore an arbitrary element of GR(4; 2) is uniquely represented as = + 2 , , 2 T . The units of GR(4; 2) form a multiplicative group GR(4; 2) = H U = f1; g; g g f1 + 2 : 2 T g. We de ne E = H , E = 3H , Eg = (1 + 2g)H , Eg = (1 + 2g )H . Let T be the relative trace from GR(4; 2) to Z=4Z . The following are traces of the units of GR(4; 2). 1
4
4
4
6
4
6
4
4
4
2
2
2
4
6
2
3
2
2
2
4
0
1
0
2
1
0
2
2
1
T (1) = 2; T (g) = 3; T (g ) = 3 T (3) = 2; T (3g) = 1; T (3g ) = 1 T (1 + 2g) = 0; T (g + 2g ) = 1; T (g + 2) = 3 T (1 + 2g ) = 0; T (g + 2) = 3; T (g + 2g) = 1 Let G = (GR(4; 2); +) (Z=6Z; +). Assume that a is a generator for (Z=6Z; +). De ne D = (E [ f0g; 2a) [ (E [ f0g; 5a) [ (Eg [ f0g; a) [ (Eg [ f0g; 4a) [ (B; 0). Then we have the following theorem. 2
2
2
2
2
0
2
2
1
Theorem 6.1 D is a (96,20,4) di erence set in G.
be an arbitrary character of G. Then = , where is an additive character of GR(4; 2) and is a character of (Z=6Z; +). We consider two cases. Case 1. is a principal character. X (D) = (Ea [ f0g) + (B )
Proof: Let
a2K
If is principal, then (D) = jDj = 20. If has order 2, from the proof of Theorem 4.1, (Ea ) = ?1, for every a 2 K . Also (B ) = jB j = 4. Hence (D) = 4. IfP has order 4, we have (B ) = 0, and Pa2K (Ea [f0g) = Pa2K (Ea ) + jK j = 4, because a2K (Ea ) = 0 from the proof of Theorem 5.1. Therefore (D) = 4. 11
Case 2. is a nonprincipal character. If is principal, then (D) = ( (2a) + (5a) + (a) + (4a) + 1)4 = ?4 (3a): Hence j (D)j = 4. If has order 2, since (Ea ) = ?1, we have (D) = jB j = 4. If has order 4, then (B ) = 0, and X (D) = (Ea [ f0g) (ha): a2K
where h = 2a, h = 5a, hg = a, hg = 4a. Since has order 4, by Lemma D, = for some = (1 + 2b)gu, where 0 u 2, and b 2 T . X (Ea) = ((1 + 2a)gi) 0
2
1
2
i=0
2 X
=
i=0
2 X
=
i=0
T
((1+2 ) i (1+2 ) u )
ag
T
((1+2( + ) i ))
4
bg
a bg
4
= (Ea b) 1
(D) = = = Since K + b = K , we have (D) =
X a2K
X
+
( (Ea ) + 1) (ha )
( (Ea b) + 1) (ha) X ( (Ex) + 1) (hx?b) 1
a2K
1
x2K +b
X x2K
+
( (Ex) + 1) (hx?b) 1
From the trace calculations, we see that
(E ) + 1 = 1 + i + i + i = ?2i 1
2
0
12
3
3
(6:1)
(E ) + 1 = 1 + i + i + i = 2i (Eg ) + 1 = 1 + i + i + i = 2 (Eg ) + 1 = 1 + i + i + i = 2 1
2
1
0
1
1
0
2
3
3
p
where i = ?1 (we remark that there are more general character sum calculations than the above in [2]). By (6.1), we only need to consider 4 cases. If b = 0, then (D) = (2a)(?2i) + (5a)2i + (a)2 + (4a)2. Therefore ( 4 (a) if (3a) = 1. (D) = ? 4i (2a) if (3a) = ?1. If b = 1, then (D) = (5a)(?2i) + (2a)2i + (4a)2 + (a)2. Therefore ( a) if (3a) = 1. (D) = 44 i ((2 a) if (3a) = ?1. If b = g, then (D) = (a)(?2i) + (4a)2i + (2a)2 + (5a)2. Therefore ( a) if (3a) = 1. (D) = 4? 4(2 i (a) if (3a) = ?1. If b = g , then (D) = (4a)(?2i) + (a)2i + (5a)2 + (2a)2. Therefore ( (2a) if (3a) = 1. (D) = 44i (a) if (3a) = ?1. 2
Summing up, we have shown that for any nonprincipal character , j (D)j = 4, hence D is a (96,20,4) di erence set in G. This completes the proof. 2
13
References [1] K.T.Arasu, S.K.Sehgal \Some new di erence sets", Ohio State Mathematical Research Institute Preprints, Sept., 1993. [2] S.Boztas, R.Hammons, P.V.Kummar \4-phase sequences with near-optimum correlation properties" IEEE Trans. Inform. Theory 38 No.3 (1992) 1101-1113 [3] B.W.Brock \A new construction of circulant GH(p ; Zp)", Disc. Math. 112 (1993), 249252. [4] J.A.Davis \Constructions of relative di erence sets in p-groups", Disc. Math. 103 (1992), 7-15. [5] D.Jungnickel \Existence results for translation nets", In: Finite Geometries and designs. London Math. Soc. Lecture Notes 49, 1981, 172-196. [6] K.H.Leung, S.L.Ma \Constructions of partial di erence sets and relative di erence sets on p-groups", Bull. London Math. Soc. 22 (1990) 533-539. [7] S.L.Ma \Partial di erence sets", Disc. Math. 52 (1984),75-89 [8] S.L.Ma \A survey of partial di erence sets", Designs, Codes and Cryptography, 4 (1994) 221-261. [9] B.R.MacDonald \Finite rings with identity", New York, Marcel Dekker,1974. [10] R.L.McFarland \A family of di erence sets in non-cyclic groups", J. Combin. Theory Ser. A, 15 (1973) 1-10. [11] A.Pott \A survey on relative di erence sets" to appear in Di erence Set Conference Proceedings. [12] K.Yamamoto and M.Yamada \Hadamard di erence sets over an extension of Z=4Z ", Utilitas Mathematica 34 (1988) 169-178 2
14
Abstract We use Galois rings to construct partial di erence sets and relative di erence sets in non-elementary abelian -groups. As an example, we also use Galois ring (4 2) to construct a (96,20,4) di erence set in 4 4 6 . p
GR
Z
Z
E-mail: [email protected]
0
Z
;
1 Introduction Let G be a nite group of order v. A k-element subset D of G is called a (v; k; ; )partial di erence set (PDS) in G if the di erences d1 d?2 1 , d1 ; d2 2 D, d1 6= d2 , represent each nonidentity element in D exactly times and each nonidentity element not contained in D exactly times. D is called abelian if G is abelian. It is well known that a PDS D with e 62 D and fd?1 : d 2 Dg = D is equivalent to a strongly regular Cayley graph, such a PDS is called regular. For a survey on partial di erence sets, we refer the reader to Ma [8]. Assume that v = mn and that G contains a normal subgroup N of order n. A k-element subset R of G is called an (m; n; k; )-relative di erence set (or,in short an (m; n; k; )-RDS) in G relative to N if the di erences d1d?2 1, d1; d2 2 R, d1 6= d2, represent each element in G n N exactly times and each nonidentity element in N zero time. If G = H N , where H is some subgroup of G, then R is called a splitting RDS. For more details about relative di erence sets, we refer the reader to the survey of Pott [11]. Finite eld theory , in particular, the cyclotomy of nite elds, is very useful in the construction of di erence sets, partial di erence sets and relative di erence sets in elementary abelian p-groups or in groups with elementary abelian Sylow p-subgroups. In the case the group is not elementary abelian, we can no longer use nite elds for the purpose of construction. In this paper, we use Galois rings to construct new families of partial di erence sets and relative di erence sets in non-elementary abelian p-groups. We also use this approach to construct a (96,20,4) di erence set in Z4 Z4 Z6. The following characterizations of abelian partial di erence sets and relative di erence sets will be used in our constructions. Lemma A. Let G be an abelian group of order v and D be a subset of G such that fd?1 : d 2 Dg = D. Suppose k, and are positive integers satisfying k2 = v +( ? )k + where = k ? if e 2 D and = k ? if e 62 D. Then D is a (v; k; ; )-partial di erence set in G if and only if, for any character of G, 8 < X j (r)j = >: r2R
kp if is principal on G. pk ? n if is nonprincipal on G but principal on N . k if is nonprincipal on N .
2 Galois Rings over Z=p2Z Let Fp be the nite eld with p elements, where p is a prime and (x) be a primitive polynomial of degree t over Fp. Let (x) be a polynomial over Z=p Z obtained from (x) by taking a preimage of (x) underPthe homomorphism : Z=p Z [x] ! Fp[x] which is de ned P n by f (x) = i ai xi 7! f (x) = ni ai xi, where ai is the reduction of ai mod p. There is a unique monic (x) whose root g satis es gp ? = 1. The ring Z=p Z [g] is an algebraic extension of Z=p Z , it is the Galois extension of Z=p Z of degree t. This extension Z=p Z [g] is called a Galois ring, and is denoted by GR(p ; t). GR(p ; t) is a nite local ring, it has the unique maximal ideal B = f0; p; pg; ; pgq? g, where q = pt, the residue class eld GR(p ; t)=B = K = f0; 1; g; ; gq? g is isomorphic to Fq (the nite eld with pt elements). We can take the Teichmuller system T = f0; 1; g; ; gq? g as a set of representatives of GR(p ; t)=B . Therefore an arbitrary element of GR(p ; t) is uniquely represented as = + p , , 2 T . We denote the set of invertible elements of GR(p ; t) by GR(p ; t) = GR(p ; t) n B . Every element of GR(p ; t) has a unique representation in the form gi(1 + p ), 0 i q ? 2, 2 T . GR(p ; t) is a multiplicative group of order (pt ? 1)pt which is a direct product H U , where H is the cyclic group of order pt ? 1 generated by g, and U is the group of principal units of GR(p ; t), that is elements of the form 1 + p ; 2 T . U has the structure of an elementary abelian group of order pt and is isomorphic to the additive group of K via the map 1 + p 7! , 2 T . For the proof of the above assertions on the structure of GR(p ; t) and more detailed description of Galois rings, we refer the reader to MacDonald [9]. 2
2
=0
=0
t
1
2
2
2
2
2
2
2
2
2
2
2
2
2
0
2
1 2
0
1
2
2
2
2
3 Frobenius and Trace Maps We will use the following lemma about the additive characters of K = f0; 1; g; ; gq? g in the future. Lemma C. All additive characters y of K , for y 2 K , are given as follows 2
y (x) = ptr xy ; x 2 K (
2
)
where p is a primitive p-th root of unity, and tr is the relative trace from K to Fp. The Frobenius map f from GR(p ; t) to GR(p ; t) is the ring automorphism f : + p 7! p + p p, , 2 T . The Galois group of GR(p ; t) over Z=p Z is a cyclic group of order t which is generated by f . The set of elements of GR(p ; t) invariant under f is identical with Z=p Z . The relative trace from GR(p ; t) to Z=p Z is de ned by 2
0
0
1
2
2
1
0
2
1
2
2
2
2
T ( ) = + f + + f ? t
1
It is easy to see that the following diagram is commutative. GR(?p ; t) ?! GR(p?; t)=B ? ? 2
2
yT
?!
Z=p Z 2
Hence
ytr
Fp
T = tr
(3:1)
where is the natural homomorphis. About the additive characters of GR(p ; t), we have the following lemma. Lemma D. All additive characters of GR(p ; t), for 2 GR(p ; t), are given as follows 2
2
2
( ) = pT ; 2 GR(p ; t) (
2
)
2
where p is a primitive p -th root of unity, and T is the relative trace from GR(p ; t) to Z=p Z . For the proofs of Lemma C and Lemma D, please see Yamamoto and Yamada [12]. 2
2
2
2
4 Partial Di erence Sets We will construct two families of regular PDS in the additive group of GR(p ; t). Our strategy is to come up with a \candidate" subset of the group, then use characters to check that all the character sums are correct. We will follow the notation used in Section 2 and 3. Considering the subgroup H of GR(p ; t) , we enumerate the cosets of H in GR(p ; t) as follows, E = H; E = (1 + p)H; Eg = (1 + pg)H; ; Eg ? = (1 + pgq? )H , where q = pt. We note that E , E , Eg , ,Eg ? , B n f0g, f0g are the orbits of the multiplication action of H on GR(p ; t), and jE j = jE j = jEg j = = jEg ? j = q ? 1. 2
2
2
q
q
0
1
2
2
0
1
2
2
q
2
3
0
1
Let A = fx 2 K : tr(x) = 0g, and v (p) = pq?? . Then all nontrivial additive characters of K with kernel equals to A are 1 1
1
f : 2 GalQ(p)=Qg = f g 1
v1 (p)
; g
2v1 (p)
; ; g
?
(p 1)v1 (p)
g
(4:1)
De ne D = [a2AEa . Then we have the following theorem.
Theorem 4.1 If p is an odd prime, pjt, then D is an (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with n = pt , r = pt? ; D [ (B n f0g) is an (n ; r (n ? 1); n + r ? 3r ; r ? r )-PDS in the additive group of GR(p ; t) with r = pt? + 1. 2
2
2 1
1
2 1
2
2
1
2
2
1
1
1
1
Proof: Let be an arbitrary additive character of GR(p ; t). By Lemma D, we consider 2
the following three cases. (1) is principal, i.e. = . In this case, (D) = jDj = jAj(pt ? 1) = pt? (pt ? 1). (2) = pg , 0 u (q ? 2). In this case, has order p, and qX ? ((1 + pa)gi) (Ea ) = 1
0
u
2
i=0 qX ?2
=
i=0 qX ?2
=
i=0
pT
pa)gi pgu)
((1+
2
pT pg : (
2
i
)
By (3.1), we have
(Ea ) = =
qX ?2 i=0 qX ?2 i=0
pptr g 2
( i)
ptr g
( i)
= (K n f0g) = ?1: 1
Hence (D) = (?1)jAj = ?pt? . (3) = , = (1 + pb)gu, where 0 u (q ? 2), and b 2 T . In this case, has order p , and X (D) = (Ea ) 1
2
a2A
4
= = = = =
?2 X qX a2A i=0 ?2 X qX a2A i=0 ?2 X qX a2A i=0 ?2 X qX
((1 + pa)gi) pT
((1+
pa)gi (1+pb)gu )
pT
((1+
pT
((1+
2
pa+pb)gi )
2
pb)gi ) T (pagi ) p2
2
a2A i=0 qX ?2 i X i pT2((1+pb)g ) pT2(pag ): i=0 a2A
By (3.1), we have
(D) = = By (4.1), we know that
qX ?2 i=0 qX ?2 i=0
pb)gi ) X tr(gi a) p a2A
pT
((1+
pT
((1+
2
pb)gi )
2
gi (A):
(
v (p)j , 1 j p ? 1. g (A) = 0jAj ifif ii 6= = v (p)j , for some j , 1 j p ? 1. 1
i
1
Hence
(D) =
pX ?1 j =1
= jAj
jAjpT
pb)gv1 (p)j )
((1+
2
pX ?1
pT g
pT pbg
pT g
ptr bg
j =1
( v1 (p)j )
2
(
v1 (p)j
2
Again, by (3.1), we have
(D) = jAj
pX ?1 j =1
( v1 (p)j )
2
5
(
v1 (p)j
)
:
)
:
We note that gv p j 2 Fp and gv p j 2 Z=p Z because of (gv T (gv p j ) = tgv p j and tr(bgv p j ) = gv p j tr(b). So we have pX ? (D) = jAj ptg ptr b g : 1( )
1( )
1( )
1( )
1( )
2
1
v1 (p)j
2
j =1
If p jt, then (D) = jAj Ppj ? ptr b g
( ) v1 (p)j
1 =1
2
p j )p
1( )
1( )
= gv
p j.
1( )
Therefore
( ) v1 (p)j
. Therefore
( t?1 p (p ? 1) if tr(b) = 0. (D) = ? pt?1 if tr(b) 6= 0.
If pjjt, we assume that t = pw; (p; w) = 1, then pX ? (D) = jAj ppwg 1
= jAj therefore in this case,
j =1 pX ?1 j =1
v1 (p)j
2
(pw
ptr b g
( ) v1 (p)j
tr(b) )gv1 (p)j
+
( t?1 (b) ?w(mod p). (D) = p?pt?(1p ? 1) ifif tr tr(b) 6 ?w(mod p).
Summing up all these calculations, we have shown that, for any nonprincipal additive character of GR(p ; t), (D) = pt? (p ? 1) or ?pt? . By Lemma A, D is an (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with n = pt, r = pt? . For the proof of the second part of the theorem, we note that for any nonprincipal additive character of GR(p ; t), ( 1 if has order p . (B n f0g) = ? pt ? 1 if has order p . 2
2
1
1
2
2
2
1
2
2
Then by the above calculations of (D) and Lemma A, we see that D [ (B n f0g) is an (n ; r (n ? 1); n + r ? 3r ; r ? r )-PDS in the additive group of GR(p ; t) with r = pt? +1. This completes the proof of the theorem. 2 Remarks: (1). Yamamoto and Yamada [12] used a method similar to that in Theorem 4.1 to construct an Hadamard di erence set in the additive group of GR(4; t), where t is not necessarily even. 2
1
2 1
1
2 1
2
1
6
1
1
(2). The PDSs in Theorem 4.1 belong to the Latin square type (see Ma [8]). There are many known examples of Latin square type PDSs. One way to construct Latin square type PDS is to use a partial congruence partition of the group. Let G be a group of order n . A partial congruence partition of G with degree r (an (n; r)-PCP) is a set P of r subgroups of G of order n such that U \ V = feg for every pair of distinct elements U; V of P . Ma [6] showed that if P is an (n; r)-PCP of G, then D = [U 2P U n feg is a regular (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in G. We claim that the PCP construction for PDS can not give rise to the PDSs in Theorem 4.1. The reason is as follows. The additive group of GR(p ; t) is isomorphic to (Zp )t . If there is an (n; r)-PCP in the additive group of GR(p ; t), where n = pt, then r pbt= c + 1 (see Jungnickel [5]), so the PCP construction can only give (n ; r(n ? 1); n + r ? 3r; r ? r)-PDS in the additive group of GR(p ; t) with r pbt= c + 1, while in Theorem 4.1 r = pt? or pt? + 1. (3). Leung and Ma [6] used nite local ring to construct Latin square type PDS in R R, where R is a nite local ring. If R = GR(p ; t), Leung and Ma's construction can only give PDS in (Zp ) t while Theorem 4.1 can give PDSs in (Zp )t with t odd. This shows that Theorem 4.1 does give new PDSs. 2
2
2
2
2
2
2
2
2
2
2
2
1
2
1
2
2
2
2
5 Relative Di erence Sets In this section, we give two constructions of relative di erence sets. The rst is originally due to Brock [3], here we phrase the construction in terms of Galois rings, and give a character theoretic proof. The relative di erence sets in the second construction seem to be new although Davis [4] had constructed RDS with the same parameters in more general groups. The purpose for us to present these two constructions is to illustrate the use of Galois rings. We will follow the notation used in Section 2, 3 and 4. Let G = (GR(p ; t); +) (K; +), where (GR(p ; t); +) is the additive group of GR(p ; t), and (K; +) is the additive group of K . Let q = pt , and R = (E ; 0) [ (E ; 1) [ (Eg ; g) [ [ (Eg ? ; gq? ) [ (B; 0). Then we have the following theorem. 2
q
2
2
2
0
2
1
Theorem 5.1 R is a (q ; q; q ; q)-relative di erence set in G relative to (K; +). Proof: Let be an arbitrary character of G. Then = , where is an additive 2
2
character of GR(p ; t) and is an additive character of K . We consider two cases. Case 1. is a principal character. X (R) = (Ea ) + (B ) 2
a2K
If is principal, then (R) = jRj = q . 2
7
If is not principal, then we have two cases. (1) = pg . In this case, has order p, (B ) = jB j = q, and (Ea) = ?1, for every a 2 K by the calculations in the proof of Theorem 4.1. Hence (R) = (?1)q + q = 0. (2) = , = (1 + pb)gu, where 0 u (q ? 2), and b 2 T . In this case, has order p , (B ) = 0, and u
2
X a2K
?2 X qX
(Ea ) =
a2K i=0 ?2 X qX
=
a2K i=0 ?2 X qX
=
a2K i=0 ?2 X qX
=
((1 + pa)gi) pT
((1+
pT
((1+
pT
((1+
2
2
2
pa)gi (1+pb)gu ) pa+pb)gi ) pb)gi ) T (pagi ) p2
a2K i=0 qX ?2 X i i pT2((1+pb)g ) pT2(pag ): i=0 a2K
= By (3.1), we have X a2K
(Ea ) =
qX ?2 i=0 qX ?2
pT
= pT i = 0: Hence (R) = 0. Case 2. is nonprincipal on (K; +). (R) =
X a2K
X a2K
pb)gi )
((1+
2
=0
pb)gi )
((1+
2
ptr g a
( i )
gi (K )
(Ea) (a) + (B )
we also distinguish three cases. (1) is principal. (R) = jEa j(Pa2K (a)) + (B ) = q. (2) has order p, i.e.P = pg . In this case, (B ) = jB j = q, and (Ea ) = ?1, for every a 2 K . So (R) = ? a2K (a) + q = q. u
8
(3) has order p , i.e. = , = (1 + pb)gu. We assume that = g . In this case, (B ) = 0, and X (R) = (Ea ) (a) 2
v
a2K
= = = = =
?2 X qX
a2K i=0 ?2 X qX
pT
((1+
pT
((1+
2
2
pa)gi (1+pb)gu ) tr(gv a) p pa+pb)gi ) tr(gv a) p
a2K i=0 qX ?2 X i i pT2((1+pb)g ) pT2(pag ) ptr(gv a) i=0 a2K qX ?2 X i pT2((1+pb)g ) ptr((gi+gv )a) i=0 a2K qX ?2 i pT2((1+pb)g ) gv +gi (K ) i=0
If gv + gi = 0, then g g (K ) = jK j = q, If gv + gi 6= 0, then g g (K ) = 0. It is easy to see that there is a unique i, 0 i q ? 2, such that gv + gi = 0. Hence (R) = qpT pb g , and j (R)j = q. Hence we have shown that 8 > < q if is principal on G. j (R)j = >: 0 if is nonprincipal on G but principal on (K; +). q if is nonprincipal on (K; +). v
+ i
v
+ i
((1+
2
) i)
2
By Lemma B, R is a (q ; q; q ; q)-relative di erence set in G relative to (K; +). This completes the proof. 2 The RDS in Theorem 5.1 is splitting. In the following, we construct a non-splitting RDS in the same group. Again let G = (GR(p ; t); +) (K; +). De ne R = [a2K (Ea [f0g; a). We have the following theorem. 2
2
2
Theorem 5.2 R is a (q ; q; q ; q)-relative di erence set in G relative to B , where B is the 2
2
unique maximal ideal of GR(p2 ; t). Proof: Let be an arbitrary character of G. Then = , where is an additive character of GR(p2; t) and is an additive character of K .
9
If is principal, then (R) = jRj = q . If is principal, but is nonprincipal, then (R) = Pa2K q (a) = 0. IfP has order p, i.e. is principal on B but not principal on GR(p ; t), then (R) = a2K (1 + (Ea )) (a). Noting that (Ea ) = ?1, we have (R) = 0. If has order p , we consider two cases. Case 1. is principal. X (R) = ((Ea) + 1) a2K X X = (Ea ) + 1 2
2
2
a2K
a2K
From the proof of Theorem 5.1, we see that Pa2K (Ea) = 0 if has order p . Hence X (R) = 1 = q: 2
a2K
Case 2. is nonprincipal.
X
((Ea) + 1) (a) X X = (Ea ) (a) + (a) a2K a2K X (Ea ) (a) =
(R) =
a2K
a2K
From the proof of Theorem 5.1, we see that j (R)j = q. By Lemma B, we conclude that R is a (q ; q; q ; q)-relative di erence set in G relative to B . This completes the proof. 2 2
2
6 A (96,20,4) Di erence Set in Z4 Z4 Z6 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. A di erence set with parameters (v; k; ) = (qd (qd + qd? + + 2); qd(qd + qd? + + 1); qd(qd? + qd? + + 1)) where q = pf and p is a prime is called a McFarland di erence set. McFarland di erence sets were originally constructed in G = E L, where E denotes the elementary abelian group of order qd and L is an arbitrary 1
1
2
1
1
1
2
1
2
+1
2
+1
10
1
abelian group of order (qd + qd? + +2) by using hyperplanes of E (see [10]). When E is not elementary abelian, the problem of constructing di erence sets with McFarland's parameters is more dicult. For a long time, it was not known whether there exists a (96,20,4) di erence set in Z Z Z . In 1993, Arasu and Sehgal [1] constructed several (96,20,4) di erence sets in Z Z Z . But they did not give their method of construction. In the following, we show that by viewing Z Z as the additive group of GR(4; 2), a construction similar to that of McFarland [10] can give a (96,20,4) di erence Set in Z Z Z . Let (x) = x + x + 1 2 F [x]. (x) is a primitive polynomial of degree 2 over F . Let (x) = x + x + 1 2 Z=4Z [x]. (x) has a root g satisfying g = 1. Then GR(4; 2) = Z=4Z [g]. This ring has a unique maximal ideal B = f0; 2; 2g; 2g g, and GR(4; 2)=B = K = f0; 1; g; g g = F . We can take the Teichmuller system T = f0; 1; g; g g as a set of representatives of GR(4; 2)=B . Therefore an arbitrary element of GR(4; 2) is uniquely represented as = + 2 , , 2 T . The units of GR(4; 2) form a multiplicative group GR(4; 2) = H U = f1; g; g g f1 + 2 : 2 T g. We de ne E = H , E = 3H , Eg = (1 + 2g)H , Eg = (1 + 2g )H . Let T be the relative trace from GR(4; 2) to Z=4Z . The following are traces of the units of GR(4; 2). 1
4
4
4
6
4
6
4
4
4
2
2
2
4
6
2
3
2
2
2
4
0
1
0
2
1
0
2
2
1
T (1) = 2; T (g) = 3; T (g ) = 3 T (3) = 2; T (3g) = 1; T (3g ) = 1 T (1 + 2g) = 0; T (g + 2g ) = 1; T (g + 2) = 3 T (1 + 2g ) = 0; T (g + 2) = 3; T (g + 2g) = 1 Let G = (GR(4; 2); +) (Z=6Z; +). Assume that a is a generator for (Z=6Z; +). De ne D = (E [ f0g; 2a) [ (E [ f0g; 5a) [ (Eg [ f0g; a) [ (Eg [ f0g; 4a) [ (B; 0). Then we have the following theorem. 2
2
2
2
2
0
2
2
1
Theorem 6.1 D is a (96,20,4) di erence set in G.
be an arbitrary character of G. Then = , where is an additive character of GR(4; 2) and is a character of (Z=6Z; +). We consider two cases. Case 1. is a principal character. X (D) = (Ea [ f0g) + (B )
Proof: Let
a2K
If is principal, then (D) = jDj = 20. If has order 2, from the proof of Theorem 4.1, (Ea ) = ?1, for every a 2 K . Also (B ) = jB j = 4. Hence (D) = 4. IfP has order 4, we have (B ) = 0, and Pa2K (Ea [f0g) = Pa2K (Ea ) + jK j = 4, because a2K (Ea ) = 0 from the proof of Theorem 5.1. Therefore (D) = 4. 11
Case 2. is a nonprincipal character. If is principal, then (D) = ( (2a) + (5a) + (a) + (4a) + 1)4 = ?4 (3a): Hence j (D)j = 4. If has order 2, since (Ea ) = ?1, we have (D) = jB j = 4. If has order 4, then (B ) = 0, and X (D) = (Ea [ f0g) (ha): a2K
where h = 2a, h = 5a, hg = a, hg = 4a. Since has order 4, by Lemma D, = for some = (1 + 2b)gu, where 0 u 2, and b 2 T . X (Ea) = ((1 + 2a)gi) 0
2
1
2
i=0
2 X
=
i=0
2 X
=
i=0
T
((1+2 ) i (1+2 ) u )
ag
T
((1+2( + ) i ))
4
bg
a bg
4
= (Ea b) 1
(D) = = = Since K + b = K , we have (D) =
X a2K
X
+
( (Ea ) + 1) (ha )
( (Ea b) + 1) (ha) X ( (Ex) + 1) (hx?b) 1
a2K
1
x2K +b
X x2K
+
( (Ex) + 1) (hx?b) 1
From the trace calculations, we see that
(E ) + 1 = 1 + i + i + i = ?2i 1
2
0
12
3
3
(6:1)
(E ) + 1 = 1 + i + i + i = 2i (Eg ) + 1 = 1 + i + i + i = 2 (Eg ) + 1 = 1 + i + i + i = 2 1
2
1
0
1
1
0
2
3
3
p
where i = ?1 (we remark that there are more general character sum calculations than the above in [2]). By (6.1), we only need to consider 4 cases. If b = 0, then (D) = (2a)(?2i) + (5a)2i + (a)2 + (4a)2. Therefore ( 4 (a) if (3a) = 1. (D) = ? 4i (2a) if (3a) = ?1. If b = 1, then (D) = (5a)(?2i) + (2a)2i + (4a)2 + (a)2. Therefore ( a) if (3a) = 1. (D) = 44 i ((2 a) if (3a) = ?1. If b = g, then (D) = (a)(?2i) + (4a)2i + (2a)2 + (5a)2. Therefore ( a) if (3a) = 1. (D) = 4? 4(2 i (a) if (3a) = ?1. If b = g , then (D) = (4a)(?2i) + (a)2i + (5a)2 + (2a)2. Therefore ( (2a) if (3a) = 1. (D) = 44i (a) if (3a) = ?1. 2
Summing up, we have shown that for any nonprincipal character , j (D)j = 4, hence D is a (96,20,4) di erence set in G. This completes the proof. 2
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References [1] K.T.Arasu, S.K.Sehgal \Some new di erence sets", Ohio State Mathematical Research Institute Preprints, Sept., 1993. [2] S.Boztas, R.Hammons, P.V.Kummar \4-phase sequences with near-optimum correlation properties" IEEE Trans. Inform. Theory 38 No.3 (1992) 1101-1113 [3] B.W.Brock \A new construction of circulant GH(p ; Zp)", Disc. Math. 112 (1993), 249252. [4] J.A.Davis \Constructions of relative di erence sets in p-groups", Disc. Math. 103 (1992), 7-15. [5] D.Jungnickel \Existence results for translation nets", In: Finite Geometries and designs. London Math. Soc. Lecture Notes 49, 1981, 172-196. [6] K.H.Leung, S.L.Ma \Constructions of partial di erence sets and relative di erence sets on p-groups", Bull. London Math. Soc. 22 (1990) 533-539. [7] S.L.Ma \Partial di erence sets", Disc. Math. 52 (1984),75-89 [8] S.L.Ma \A survey of partial di erence sets", Designs, Codes and Cryptography, 4 (1994) 221-261. [9] B.R.MacDonald \Finite rings with identity", New York, Marcel Dekker,1974. [10] R.L.McFarland \A family of di erence sets in non-cyclic groups", J. Combin. Theory Ser. A, 15 (1973) 1-10. [11] A.Pott \A survey on relative di erence sets" to appear in Di erence Set Conference Proceedings. [12] K.Yamamoto and M.Yamada \Hadamard di erence sets over an extension of Z=4Z ", Utilitas Mathematica 34 (1988) 169-178 2
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