Hadamard Ideals and Hadamard Matrices From Two Circulant Submatrices

Abstract

We apply Computational Algebra methods to the construction of Hadamard matrices from two circulant submatrices, given by C. H. Yang. We associate Hadamard ideals to this construction, to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search for Hadamard matrices from two circulant submatrices for this construction, for the first eight admissible values $2,4,8,10,16,18,20,26$ and partial searches for the next three admissible values $32,34,40$. From the solutions we found, for the admissible values $26$ and $34$, we located new inequivalent Hadamard matrices of orders $52$ and $68$ with two circulant submatrices, thus improving the lower bounds for the numbers of inequivalent Hadamard matrices of orders $52$ and $68$. We also propose a heuristic decoupling of one of the equations arising from this construction, which can be used together with the PSD test to search for solutions more efficiently.