MacWilliams identities for the generalized rank weights

TL;DR

This paper establishes MacWilliams identities for generalized rank weights of linear codes, providing formulas for weight enumerators and explicitly computing distributions for MRD codes, advancing understanding of code duality and weight structures.

Contribution

It introduces a MacWilliams-type identity for generalized rank weights and derives explicit formulas, including for MRD codes, which was not previously known.

Findings

Derived a MacWilliams identity relating a code and its dual's rank weight distributions.

Provided a formula for the rank weight enumerator polynomial.

Explicitly computed the rank distribution for MRD codes.

Abstract

We study the generalized rank weight distribution of a linear code. First, we provide a MacWilliams-type identity which relates the distributions of a code and its dual. Then, we give a formula for the enumerator polynomial. Finally, we explicitly compute the distribution of an MRD code.