An algorithm for constructing and classifying the space of small integer weighing matrices

In this paper we describe an algorithm for generating all the possible $PIW(m,n,k)$ - integer $m\times n$ Weighing matrices of weight $k$ up to Hadamard equivalence. Our method is efficient on a personal computer for small size matrices, up to $m\le n=12$, and $k\le 50$. As a by product we also improved the $\mathit{\text{nsoks}}$ \cite{riel2006nsoks} algorithm to find all possible representations of an integer $k$ as a sum of $n$ integer squares. We have implemented our algorithm in $\mathtt{\text{Sagemath}}$ and as an example we provide a complete classification for \ $n=m=7$ and $k=25$. Our list of $IW(7,25)$ can serve as a step towards finding the open classical weighing matrix $W(35,25)$.