Gr\"obner 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 enable 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 Gr\"obner basis of this ideal is $2^{O(t^2)}$, 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 mentioned set of cocyclic Hadamard matrices over $G$. The complexity of the computation decreases in this way to $2^{O(n)}$, where $n$ is the number of $G$-coboundaries. Particularly, we design two specific procedures for looking for $\mathbb{Z}_t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices and $D_{4t}$-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to $t \leq 31$) are explicitly obtained.