ON MARRERO’S Jm-HADAMARD MATRICES

Abstract

In this paper, generalizing Marrero's construction, we introduce the concept of $J_m-$Hadamard matrices, and by allowing permutations, we construct other $2^mm!-1$ $J_m$-Hadamard matrices from a given one of order $mt$; previous construction generated only other $2^m-1$ ones. We also generalize Craigen's construction of products of two Hadamard matrices to those of several Hadamard matrices and a $J_m$-Hadamard matrix, yielding generalizations of Craigen's results. Furthermore, we introduce the $J_m$-class $CJ_m$ for $m=2$ or $4k$ and study the partially ordered set $\mathfrak{M}$ of $J_m-$classes $CJ_m$. Our main result shows that $CJ_8\subsetneqq CJ_4\subsetneqq CJ_2$.