Some new symmetric Hadamard matrices

Abstract

Introduction: It is conjectured that the symmetric Hadamard matrices of order 4v exist for all odd integers v>0. In recent years, their existence has been proven for many new orders by using a special method known as the propus construction. This construction uses difference families Xk (k=1, 2, 3, 4) over the cyclic group Zv (integers mod v) with parameters (v; k1, k2, k3, k4; λ) where X1 is symmetric, X2=X3, and k1+2k2+k4=v+λ. It is also conjectured that such difference families (known as propus families) exist for all parameter sets mentioned above excluding the case when all the ki are equal. This new conjecture has been verified for all odd v≤53. Purpose: To construct many new symmetric Hadamard matrices by using the propus construction and to provide further support for the above-mentioned conjecture. Results: The first examples of symmetric Hadamard matrices of orders 4v are presented for v=127 and v=191. The systematic computer search for symmetric Hadamard matrices based on the propus construction has been extended to cover the cases v=55, 57, 59, 61, 63. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information.