Gröbner bases and cocyclic Hadamard matrices

Abstract

Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enables us to describe a polynomial ideal that characterizes the set of cocyclic Hadamard matrices over a fixed finite group G of order 4t. Nevertheless, the complexity of the computation of the reduced Gröbner basis of this ideal is 2O(t2), which is excessive even for very small orders. In order to improve the efficiency of this polynomial method, we take advantage of some recent results on the inner structure of a cocyclic matrix to describe an alternative polynomial ideal that also characterizes the aforementioned set of cocyclic Hadamard matrices over G. The complexity of the computation decreases in this way to 2O(t). Particularly, we design two specific procedures for looking for Zt×Z22-cocyclic Hadamard matrices and D4t-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to t≤39) are explicitly obtained.