Hadamard matrices from Goethals — Seidel difference families with a repeated block

Abstract

Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki  λ  v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v  51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.