The MacWilliams Identity for the Skew Rank Metric

TL;DR

This paper extends the MacWilliams Identity to codes in the skew rank metric, providing a new functional transformation formula and deriving moments of the skew rank distribution for skew-symmetric matrix codes.

Contribution

It develops a $q$-analog MacWilliams Identity for skew-symmetric matrix codes under the skew rank metric, introducing a skew-$q$ algebra and using generalized Krawtchouk polynomials.

Findings

Derived a new functional transformation form of the MacWilliams Identity for skew rank metric.

Established moments of the skew rank distribution for these codes.

Introduced a skew-$q$ algebra framework for analyzing skew-symmetric matrix codes.

Abstract

The weight distribution of an error correcting code is a crucial statistic in determining it's performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $q$-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$q$ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.