The structure of weighing matrices having large weights
An important ingredient of this paper is the concept named a signed group, which is, the reviewer believes, worthwhile to be restated here. A signed group \(S\) of order \(s\) is a trio consisting of a group \(G\) of order \(2s\), a central involution of \(G\) denoted by \(-1\) and a transversal \(P\), which is called positive, of \(G\) with respect to \(\langle -1\rangle\). The signed group ring \(\mathbb{Z}[S]\) is the factor ring of the group algebra \(\mathbb{Z}[G]\) by the ideal defined by the identification of \(- 1\) with the number \(-1\). A signed group weighing matrix \(W\) of order \(n\) and weight \(w\) defined over \(S\), which is denoted by \(\text{SW}(n, w, S)\), is a matrix of order \(n\), with all nonzero entries in \(S\) (as elements of \(\mathbb{Z}[S]\)), which satisfies \(WW^*= wI\). Further, let \(\text{SP}_n\) be the group of all signed permutation matrices of degree \(n\). Only the representations of \(G\) to \(\text{SP}_n\) such that \(R(- 1)= -R(1)\) are considered. Now a relevant relation between signed group weighing matrices and ordinary weighing matrices is stated as follows (Lemma 9): Let the degree of \(R\) be \(m\) and let \(V= W(m, v)\) be a weighing matrix of order \(m\) and weight \(v\). If \(W= \text{SW}(n, w, S)\), then \(R(W)(I_n\times V)= W(mn, vw)\), and vice versa. There are many interesting construction (existence) theorems proved in this paper. Here are a few of them (according to the whim of the reviewer). If there is a \(W(n, n- 2)\), then there is an Hadamard matrix of order \(2n\) if \(n\equiv 0\pmod 4\), and order \(4n\) if \(n\equiv 2\pmod 4\) (Theorem 12). If there is a \(W(4n+ 4, 4n+ 1)\), then there is an Hadamard matrix of order \(32(n+ 1)\) (Theorem 19). If there are Golay sequences of length \(g\), then there is an Hadamard matrix of order \(2^7\) \(({g\over 2}\neq 7)\) (Theorem 22).
scientific article; zbMATH DE number 751328
scientific article