The MacWilliams identity for the skew rank metric

Izzy Friedlander; Thanasis Bouganis; Maximilien Gadouleau

doi:10.3934/amc.2023045

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2025, Volume 19, Issue 1: 140-179. Doi: 10.3934/amc.2023045

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The MacWilliams identity for the skew rank metric

1.
Department of Computer Science, Durham University, UK
2.
Department of Mathematical Sciences, Durham University, UK

^*Corresponding author: Maximilien Gadouleau
^*Corresponding author: Maximilien Gadouleau

Received: October 2022

Revised: August 2023

Early access: November 2023

Published: February 2025

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Abstract

The weight distribution of an error correcting code is a crucial statistic in determining its 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.

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Mathematics Subject Classification: Primary: 94B05, 15B33; Secondary: 20H30.

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