Power Hadamard matrices and Plotkin-optimal $p^k$-ary codes

Abstract

A power Hadamard matrix H(x) is a square matrix of dimension n with entries from Laurent polynomial ring, where f is some Laurent polynomial of degree greater than 0. In the first part of this work, some new results on power Hadamard matrices are studied, where we mainly extend the work of Craigen and Woodford. In the second part, codes obtained from Butson-Hadamard matrices are discussed and some bounds on the minimum distance of these codes are proved. In particular, we show that the code obtained from a Butson-Hadamard matrix meets the Plotkin bound under a non-homegeneous weight.