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Constructions of Partial Difference Sets and Relative Difference Sets Using Galois Rings II

✍ Scribed by Yu Qing Chen; D.K. Ray-Chaudhuri; Qing Xiang

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 745 KB
Volume: 76
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1996.0100

No coin nor oath required. For personal study only.

✦ Synopsis

In a previous paper, [Des., Codes and Cryptogr. 8 (1996), 215 227]; we used Galois rings to construct partial difference sets, relative difference sets and a difference set. In the present paper, we first generalize and improve the construction of partial difference sets in [Des., Codes and Cryptogr. 8 (1996), 215 227]; also we obtain a family of relative difference sets from these partial difference sets. Second, we construct a class of relative difference sets in (Z 4 ) 2m+1 Ä (Z 4 ) r Ä (Z 2 Ä Z 2 ) s , r+s=m, r, s 0 with respect to a subgroup (Z 2 ) 2m+1 . These constructions make use of character sums from Galois rings, and relate relative difference sets to Hadamard difference sets.

1996 Academic Press, Inc.

1. Introduction

Let G be a finite group of order

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 Â D and [d &1 : d # D]=D is equivalent to a strongly regular Cayley graph, such a PDS is called regular. The study of partial difference sets is closely related to partial geometries, Schur rings, strongly regular Cayley graphs and two-weight codes. The recent survey of Ma [5] contains very detailed descriptions of these connections.

Assume that v=mn and that G contains a normal subgroup N of order n.

d 1 {d 2 , represent each element in G "N exactly * times and each nonidentity element in N zero time. If G=H_N, where H is article no. 0100

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