A Note on New Semi-Regular Divisible Difference Sets

We give a C0051JUCtion for new families of semi-regular divisible difference sets. The construction iJ a variation of McFarland's scheme [Sl tor nonc;yclic difference SCIS. • Let G be a group of order 11111 and N a subgroup of G of order n. If D is a k-subset of G then Dis a {m, n, le , >.1, >.2) divisible difference set in G relative to N provided that the differences dd'1 fur d, d ' ED, d ;e d ', contain every nonidentity element of N exactly >.1 times and every element of G\N exactly >.2 times. If k > >1 and k2 = mn~, then the divisible difference set is caUed semi-regular. Families of semi-regular divisible difference sets with >.1 -;it 0 are rare, as mentioned in [4]. If >.1 = >.2 then D is a (mn, k, >-1) difference set in G. One way to check if a subset of a group is a divisible difference set is to use the group ring equation. If we abuse notation by writing D = Edwd and v <-1> = tdEDd 1 then the definition of a divisible difference set is equivalent to the equation vv<-1> = k + >.1(N 1) + >..2(G N) in the group ring Z[G] (see [2), (3) for examples of this technique) . In th.is paper, we wiU construct semi-regular divisible diffe rence sets with new sets of parameters. The construction is similar to those fuund in [l], [2], [3] . We start with the group E = EA(qd+1), the elementary Abelian group with qd+I elements, where q is a prime power. We will view E as a vector space of dimension d + l over GF(q). A hyperplane of E is a subspace of dimension d; a standard counting argument shows that E contains (qd+I l)/(q 1) hyperplanes. Label these hyperplanes H1 for i = 1, .. . , (qd+I I)/ (q 1) and note that EIH1 = EA(q) for each i. Suppose EA(q) supports a (q, k', >..') difference set. Then for each i form the set D1 = uJ:1aiJH1 C E (regarding each aiJ H1 as a subset of£), where {a1JH1: j = l, ... , k'} is a (q, k', >.' ) difference set in EIH1 (regarding each a11H; as an element of E/H1) . Suppose M is an Abelian group containing a (m, (qd+I l)/(q l~, >..w) difference set {b1: i = I, ... , (qd+I l)/(q 1)}. Theo form the set D = u~!i+ -l)l(q1) b1D1 c M x E. The set D thus constructed is a divisible difference set: *This work is partially supponcd by NSA grant # MDA 904-92-H-3067. 380 J. DAVIS AND J. JEDWAB ll!EOREM 1. Let q be a prime power. If there exists a (m, (qdI l)l(q I) , A") difference set in an Abelian group Mand a (q, k ', A') difference set in EA(q), then there exists a (m, qd+I, qd((qd+I l)l(q l))k', qd(((qd+1 l)/(q l))k' qd(k' A')), q d-1k'2 A ") divisible difference set in G = M x EA(qd+I ) relaJive to EA(q~+1) . Proof We work in the group ring Z[E]. For hyperplanes H;, Hr of Ethe expression H;H;· in Z[E] is equal to qdH, ifi = i ' and qd1Eifi ¢ i '. Since {a1i H1} is a (q, k', A') difference set in EIHi. it follows that in Z[E] we have D;Dt 'J = qd1J.ra11aij! H1 = qd(k' H1 + A'(E H1)). Also note that f.;H, = (qd+I 1)/(q 1) + (q l )l(q 1)(£ l). The proof involves a separation into cases based on i = i ' and i ¢ i ': = qd ~ (k'H1 + A'(E H1)) + qd1k'2E ~ b1bp I ,,.,. = qdk' q + qd q k' qd(k' A') (£ I) d+I I [ d-..1 I ~ q 1 q-1 A divisible difference set with the parameters of Theorem 1 can also be constructed in any group containing a normal subgroup isomorphic to E, using a similar adaption of the above method to that introduced by Dillon [3] to modify the scheme of McFarland [5]. The parameters in Theorem 1 are not semiregular in general, but they are in the special case when the difference set {b1} is trivially the whole of M: COROLLARY 1. If m = (qd+I l)/(q 1), t/um the divisible difference set is semi-regular. Proof. J?= <("((qd+I _ l)/(q _ l))lk12 = ((qd+I _ l)/(q _ l})(qd+l)(qdlk'2((qd+I _ 1)/ (q 1))) = mnA2• C For example, if we choose q = 7, k' = 3, A' = I, and d = 1, then the corollary shows the existence of a (8, 49, 168, 70, 72) semi-regular divisible difference set in M x EA(49), where Mis any group of order 8 (including nonabelian). Known existence results for (q, k', A') difference sets in EA(q) provide many examples for the construction of Corollary 1. In the case q • 3 (mod 4) there exists a (q, (q 1)/2, (q 2)/4) difference set. In the case q = 1 (mod 4) there are examples such as (13, 4, 1) and (73, 9, 1) in the projective planes, as well as others such as (37, 9, 2). A NOTE ON NEW SEMI-REGULAR DIVISIBLE DIFFERENCE SETS 381