On minors of maximal determinant matrices

TL;DR

This paper extends Cohn's 1965 result on Hadamard matrices to maximal determinant submatrices, showing certain size intervals are excluded, and provides data and conjectures on minors of these matrices.

Contribution

It generalizes known results about Hadamard matrices to maximal determinant matrices, introduces a conjecture on minors, and provides computational data and algorithms.

Findings

Intervals of matrix orders are excluded for minors of maximal determinant matrices.

Tables of minors for matrices up to order 21 are provided.

Evidence supports the conjecture on sums of squares of minors.

Abstract

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.