Computer construction conjecture for symmetric hadamard matrices

We consider and compare methods for computer construction of variously structured plug-in matrices (suitable matrices) used to construct skew-Hadamard matrices from the Goethals-Seidel array and symmetric Hadamard matrices from the Balonin-Seberry array. We call symmetric analogue matrices of suitable matrices for computer construction of skew-Hadamard matrices luchshie matrices (luchshie is the Russian plural for \u27best\u27). We provide tables of known inequivalent luchshie matrices of order 4, n \u3c 53, and symmetric Hadamard matrices of order 4n, n \u3c 400. We propose the conjecture that there exist luchshie (±1) matrices of order odd t for all t. Hence, there exists a symmetric Hadamard matrix of order At for every odd t