Constructing, Classifying and Studying the Space of Small Integer Weighing Matrices

TL;DR

This paper classifies small integer weighing matrices, develops algorithms for their construction, and solves some open cases, advancing the understanding of their structure and enumeration.

Contribution

It provides a comprehensive classification of small integer weighing matrices, including symmetric and antisymmetric cases, with practical algorithms implemented in SageMath.

Findings

Complete classification of IW(n,k) matrices up to equivalence.

Solved open cases for symmetric and antisymmetric matrices.

Developed algorithms for constructing and counting IW matrices.

Abstract

Integer weighing matrices (IW-matrices for short) are integer valued orthogonal square matrices. One usecase of these is to create classical weighing matrices with various block structures. In this paper we study and classify the space $IW(n,k)$ of the integer weighing matrices of small size $n\times n$ and weight $k$. Our classification includes a full list of all inequivalent matrices up to Hadamard equivalence and automorphism groups. We then continue to a secondary classification of the symmetric and antisymmetric IW up to symmetric Hadamard equivalence. We apply this to the case of projective space weighing matrices. Next we use the classification to count the cardinality of the spaces of all $IW(n,k)$ as well as the symmetric and anti-symmetric subspace. We supply practical algorithms and implement them in \texttt{Sagemath}. Finding an (anti-)symmetric IW matrix in a given Hadamard class can be done for significantly higher orders. In particular we solve some open cases: Symmetric $W(23,16)$, $W(28,25)$ and $W(30,17)$, and an anti-symmetric $W(28,25)$. We conclude by showing a detailed classification of $IW(7,25)$. We have also improved the \texttt{NSOKS} algorithm to find all possible representations of an integer $k$ as a sum of $n$ integer squares.