Purpose: To investigate more fully, than what was done in the past, the construction of symmetric Hadamard matrices of "propus type", a symmetric variation of the GoethalsSeidel array characterized by necessary symmetry of one of the blocks and equality of two other blocks out of the total of four blocks. Methods: Analytic theory of equations for parameters of difference families used in the propus construction of symmetric Hadamard matrices, based on the theorems of Liouville and Dixon. Numerical method, due to the authors, for the search of two or three cyclic blocks to construct Hadamard matrices of two-circulant or propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function. Results: A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters. In addition to the novelty of this collection, we have obtained new symmetric Hadamard matrices of orders 92, 116 and 156. For the order 156, no symmetric Hadamard matrices were known previously. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network "Internet" together with executable on-line algorithms.

Abstract:We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders v with odd v ≤ . In this paper we cover the cases v = , , , , . The odd integers v < for which no symmetric Hadamard matrices of order v are known are the following:, , , , , , , , ,By using the propus construction, we found several symmetric Hadamard matrices of order v for v = , , .

No abstract

Cretan matrices are orthogonal matrices with elements ≤ 1. These may have application in forming some new materials. There is a search for Cretan matrices, especially with high determinant, for all orders. These have been found by both mathematical and computational methods.This paper highlights the differences between theoretical and computational solutions to finding Cretan matrices.It has been shown that the incidence matrix of a symmetric balanced incomplete block design can be used to form Cretan(v; 2) matrices. We give families of Cretan matrices constructed using Hadamard related difference sets.

There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v = 77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided cases to 17.