On unit weighing matrices with small weight

TL;DR

This paper investigates the structure and classification of unit weighing matrices of small weights, revealing how their counts depend on decompositions of their order and highlighting complexities for weights above 4.

Contribution

It provides a detailed analysis of UW(n,4) matrices and introduces new insights into their enumeration and structure, especially for weights 2, 3, and 4.

Findings

Number of inequivalent UW(n,4) depends on decompositions of n

Identifies two sporadic complex cases for weights greater than 4

Provides structural insights into small-weight unit weighing matrices

Abstract

We study the structure of unit weighing matrices of order n and weights 2, 3 and 4. We show that the number of inequivalent unit weighing matrices UW(n,4) depends on the number of decompositions of n into sums of non-negative multiples of some specific positive integers. We also show two interesting sporadic cases in order to show the complexities involved for weights larger than 4.