Rank weight hierarchy of some classes of polynomial codes

Abstract

We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field $\FF_{q^m}$, $q$ a prime power, $m \geq 2$. We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of $[n,n-1]$ constacyclic codes. We characterize polynomial codes of $r$th rank weight $r$, and in particular of irst rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be.