On the generalised rank weights of quasi-cyclic codes

Enhui Lim; Frédérique Oggier

doi:10.3934/amc.2022010

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2024, Volume 18, Issue 1: 192-205. Doi: 10.3934/amc.2022010

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On the generalised rank weights of quasi-cyclic codes

Enhui Lim and
Frédérique Oggier^,

Division of Mathematical Sciences, Nanyang Technological University, Singapore

^* Corresponding author
^* Corresponding author

Received: October 2021

Revised: January 2022

Early access: March 2022

Published: February 2024

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Abstract

Generalised rank weights were formulated in analogy to Wei's generalised Hamming weights, but for the rank metric. In this paper we study the generalised rank weights of quasi-cyclic codes, a special class of linear codes usually studied for their properties in error correction over the Hamming metric. By using the algebraic structure of quasi-cyclic codes, a new upper bound on the generalised rank weights of quasi-cyclic codes is formulated, which is tighter than the known Singleton bound. Additionally, it is shown that the first generalised rank weight of self-dual $ 1 $-generator quasi-cyclic codes is almost completely determined by the choice of $ {\mathbb F}_{q^{m}} $.

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Mathematics Subject Classification: Primary: 11T71.

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