Rank weight hierarchy of some classes of polynomial codes

• Main Authors: Ducoat, Jérôme, Oggier, Frédérique
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2023
• Subjects: Science::Mathematics; Coding Theory; Rank Metric
• Online Access: https://hdl.handle.net/10356/163959
• Institution: Nanyang Technological University

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.