New Nonexistence Results on Circulant Weighing Matrices

TL;DR

This paper advances the understanding of circulant weighing matrices by proving nonexistence for 12 open cases, analyzing known matrices, and reporting on preliminary searches beyond existing tables.

Contribution

It provides new nonexistence proofs for specific open cases of circulant weighing matrices and characterizes known matrices beyond prior tables.

Findings

Nonexistence proofs for 12 open cases.

Analysis of known proper circulant weighing matrices.

Preliminary searches outside existing tables.

Abstract

A circulant weighing matrix $W = (w_{i,j})$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{0, \pm 1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave a table of existence results for such matrices with $n \leq 200$ and $k \leq 100$. In the latest version of Strassler's table given by Tan \cite{arXiv:1610.01914} there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler's table, and characterize the known proper circulant weighing matrices.