Hermitian congruence and the existence of completion of generalized Hadamard matrices
Some problems on existence, construction and application of generalized Hadamard matrices (GHM) in the group ring image \[ R=Q(G)/(\sum_{g\in G}g), \] where G is a finite group, are considered. The matrix equation \[ AA^*=A^*A=mI_ h+\mu J_ h \] in the cyclotomic field and in the group ring R is investigated. It is shown that for \(m=h\) and \(\mu =0\) the obtained results yield a nonexistence theorem for GHMs. Necessary and sufficient conditions for the existence of those matrices are given. The results of Marshall Hall an Hadamard completions are extended to GHMs. The application of GHM to coding theory (construction of codes attaining the Plotkin bound) is also considered. A theorem on the automorphism group of a GHM is given.
scientific article; zbMATH DE number 4091519
scientific article