Gröbner bases and cocyclic Hadamard matrices

Abstract

Hadamard ideals were introduced in 2006 as a set of nonlin-ear polynomial equations whose zeros are uniquely related toHadamard matrices with one or two circulant cores of a given or-der. Based on this idea, the cocyclic Hadamard test enables us todescribe a polynomial ideal that characterizes the set of cocyclicHadamard matrices over a fixed finite group Gof order 4t. Nev-ertheless, the complexity of the computation of the reduced Gröb-ner basis of this ideal is 2O(t2), which is excessive even for very small orders. In order to improve the efficiency of this polynomialmethod, we take advantage of some recent results on the innerstructure of a cocyclic matrix to describe an alternative polyno-mial ideal that also characterizes the aforementioned set of cocyclicHadamard matrices over G. The complexity of the computation de-creases 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.