Explicit weighing matrices $W(n,k)$ for all $n \equiv 0 \pmod 4$ with $4 \le n \le 108$, and all $1 \le k \le n$

Source abstract

We present an explicit weighing matrix $W(n,k)$ for every order $n\equiv 0\pmod 4$ with $4\le n\le 108$ and every weight $1\le k\le n$: 1512 matrices in total, each checked in exact integer arithmetic. Of these, 961 come from classical constructions (identity, direct sums, Kronecker products, Paley matrices and skew-Hadamard doubling) and 551 are Goethals-Seidel matrices, found by exhaustive search (24 matrices, $n\le 24$) or by simulated annealing on the periodic autocorrelation cost (527 matrices, $n\ge 28$). The note is self-contained as far as the methods are concerned: it presents all the construction methods used and the algorithm that selects among them, explains the algorithm step by step, proves every identity on which the constructions rely, and illustrates each method with a matrix taken from the catalogue. It does not reproduce the 1512 matrices, which are available in the repository. It also gives a short parity obstruction to Goethals-Seidel matrices $W(4m,4m-1)$ with $m$ even, and lists the generating sequences of thirteen matrices whose parameters appear as unresolved in historical tables consulted by the author; their status in the current literature is still being checked, and no claim of priority is made.