On nonsingularity of circulant matrices

TL;DR

This paper proves that certain circulant matrices with specific patterns are always nonsingular when a related number is a prime power or product of two primes, and constructs singular examples otherwise.

Contribution

It establishes conditions under which circulant matrices with a fixed pattern are guaranteed to be nonsingular, and provides explicit singular examples for other cases.

Findings

Matrices are always nonsingular when 2k+1 is a prime power or product of two primes.

Constructs singular circulant matrices for other values of 2k+1.

The smallest singular matrix occurs at 2k+1=45, with a very low probability of singularity.

Abstract

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k+1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2k+1$ is either a power of a prime, or a product of two distinct primes. For any other integer $2k+1$ we construct circulant matrices having determinant $0$. The smallest singular matrix appears when $2k+1=45$. The possibility for such matrices to be singular is rather low, smaller than $10^{-4}$ in this case.