A Search For Solvable Weighing Matrices

Abstract

A weighing matrix $W(n,k)$ of order $n$ with weight $k$ is an $n\times n$ matrix with entries from $\{0,1,-1\}$ which satisfies $W{W}^T=k{I}_n$. Such a matrix is group-developed if its rows and columns can be indexed by elements of a finite group $G$ so that $w_{g,h}=w_{gf,hf}$ for all $g,h,f$ in $G$. Group-developed weighing matrices are a natural generalization of perfect ternary arrays and Hadamard matrices. They are closely related to difference sets.

We describe a search for weighing matrices with order 60 and weight 25, developed over solvable groups. There is one known example of a $W(60,25)$ developed over a non-solvable group; no solvable examples are known.

We use techniques from representation theory, including a new viewpoint on complementary quotient images, to restrict solvable examples. We describe a computer search strategy which has eliminated two of twelve possible cases. We summarize plans to complete the search.