A Unifying Construction for Difference Sets

Abstract

We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(2 2d+4 (2 2d+2 &1)Â3, 2 2d+1 (2 2d+3 +1)Â3, 2 2d+1 (2 2d+1 +1)Â3, 2 4d+2 ) for d 0. The construction establishes that a McFarland difference set exists in an abelian group of order 2 2d+3 (2 2d+1 +1)Â3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order p r . This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups. Academic Press

Cyclotomic integers and finite geometry

“…Here we will consider arbitrary u. McFarland and Spence difference sets are known for any prime power q and any positive integer d; see [29]. Difference sets of type (iv) are known to exist only if f is even or p ≤ 3; see [9,12,29]. However, in this section we will consider arbitrary f and p. We will first deal with Hadamard difference sets.…”

An extension of building sets and relative difference sets

“…We omit the proof of Lemma 1.3 since the proof of Theorem 4.3 in [4] works here with only notational changes.…”

“…We refer the reader to [4] for the proofs of Lemmas 1.1 and 1.2. We remark that the special parameters jGaNjY jGaNjt p Y t in Lemma 1.1 is to ensure % to be one-toone on each B i .…”

“…In a recent paper, Davis and Jedwab [4] called such a family of subsets a building set (BS). Let N`G be ®nite abelian groups and aY b 0.…”

“…A common method of constructing semi-regular RDS is to piece several subsets in an abelian group together to generate a semi-regular RDS in a larger group [4,5,8,9]. In a recent paper, Davis and Jedwab [4] called such a family of subsets a building set (BS).…”