Gr\"obner bases and cocyclic Hadamard matrices

TL;DR

This paper develops an improved polynomial-based method to identify cocyclic Hadamard matrices, significantly reducing computational complexity and enabling the explicit construction of larger matrices up to order 124.

Contribution

It introduces an alternative polynomial ideal leveraging cocyclic matrix structure, decreasing complexity from exponential in t^2 to exponential in n, and provides procedures for specific group cases.

Findings

Reduced computational complexity for finding cocyclic Hadamard matrices.

Successfully constructed larger matrices up to order 124.

Enhanced polynomial methods for matrix characterization.

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.