Linear Codes and the Existence of a Reversible Hadamard Difference Set inZ2×Z2×Z45

Abstract

Linear codes overGF(5) are utilized for the construction of a reversible abelian Hadamard difference set inZ2×Z2×Z45. This is the first example of an abelian Hadamard difference set in a group of order divisible by a primep≡1 (mod 4). Applying the Turyn composition theorem, one obtains abelian difference sets and Hadamard matrices of Williamson type of order 4×54n×p4n11×…×p4nttwheren, n1, …, ntare arbitrary non-negative integers and eachpiis a prime,pi≡3 (mod 4).