Generalized Hadamard matrices which are developed modulo a group
Let \(N\) and \(G\) be finite groups of orders \(n=| N|\) and \(g=| G|\), respectively, and let \(\text{EA}(q)\) be the elementary abelian group of order \(q\), \(q\) a prime power. Also, let \(\text{EA}(n)\) be the group of order \(n\) which is the direct product of elementary abelian groups. This paper discusses the construction and existence of generalized Hadamard matrices \(\text{GH}(n,G)\) which are developed modulo a group \(N\). In particular, it is shown that the existence of a \(\text{GH}(n,G)\) developed modulo \(N\) is equivalent to the existence of a solution to a certain equation over the group ring of \(G\backslash N\) over the integers. Finally, some remarks on error-correcting coding theory are made. It is shown that every element of \(G\) must appear in any \(\text{GH}(n,G)\) developed modulo a group, if \(n\geq g(g-1)\).
scientific article; zbMATH DE number 89880
scientific article