Some new symmetric Hadamard matrices

Аннотация

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 X_k (k=1, 2, 3, 4) over the cyclic group Z_v (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 k_i 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.