Rank-Metric Codes, Generalized Binomial Moments and their Zeta Functions

TL;DR

This paper introduces a new class of extremal rank-metric codes called i-BMD codes, explores their properties, and establishes connections with zeta functions and MacWilliams identities, advancing the theoretical understanding of rank-metric code invariants.

Contribution

It defines i-BMD codes, relates their invariants to code parameters, and extends the theory of rank-metric codes with new identities and zeta function connections.

Findings

i-BMD codes are a proper subclass of i-MRD codes

Established a relation between generalized rank weight enumerator and zeta functions

Proved a MacWilliams identity for generalized rank weight distributions

Abstract

In this paper we introduce a new class of extremal codes, namely the $i$-BMD codes. We show that for this family several of the invariants are determined by the parameters of the underlying code. We refine and extend the notion of an $i$-MRD code and show that the $i$-BMD codes form a proper subclass of the $i$-MRD codes. Using the class of $i$-BMD codes we then obtain a relation between the generalized rank weight enumerator and its corresponding generalized zeta function. We also establish a MacWilliams identity for generalized rank weight distributions.