Linear Codes and the Existence of a Reversible Hadamard Difference Set inZ2×Z2×Z45
✍ Scribed by M. van Eupen; Vladimir D. Tonchev
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 284 KB
- Volume
- 79
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
Linear codes over GF(5) are utilized for the construction of a reversible abelian Hadamard difference set in Z 2 _Z 2 _Z 4 5 . This is the first example of an abelian Hadamard difference set in a group of order divisible by a prime p#1 (mod 4). Applying the Turyn composition theorem, one obtains abelian difference sets and Hadamard matrices of Williamson type of order 4_5 4n _p 4n1 1 _ } } } _p 4nt t where n, n 1 , ..., n t are arbitrary non-negative integers and each p i is a prime, p i #3 (mod 4).
1997 Academic Press
1. Introduction
We assume familiarity with the basics of combinatorial designs theory and coding theory (cf., e.g. [4, 5, 9]). We use the notation [n, k, d ] q for a linear code of length n, dimension k, and minimum distance d over GF(q), and w A1 1 A w Aw 2 2 } } } for the weight enumerator of a code with A w 1 nonzero words of weight w 1 =d, A w 2 nonzero words of weight w 2 , etc. A t-weight code is a code with t nonzero weights. A code is projective if its dual distance is at least 3
contains each nonidentity element of G exactly * times. An abelian difference set is a difference set in an abelian group G. A multiplier is an automorphism of G that preserves the set of translates [Dg | g # G]. A difference set which is fixed by a multiplier &1 is called reversible. A Hadamard (also a Menon) difference set (HDS) is a difference set with parameters (4m 2 , 2m 2 &m, m 2 &m) for some integer m. Two recent surveys on Hadamard difference sets and their applications are [2, 3].