Constructions of optimal rank-metric codes from automorphisms of rational function fields

Rakhi Pratihar; Tovohery Hajatiana Randrianarisoa

doi:10.3934/amc.2022034

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2023, Volume 17, Issue 1: 262-287. Doi: 10.3934/amc.2022034

This issue Previous Article On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes Next Article New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes

Constructions of optimal rank-metric codes from automorphisms of rational function fields

Rakhi Pratihar^1, and
Tovohery Hajatiana Randrianarisoa^2, ,

1.
UiT - The Arctic University of Norway, Tromsø, Norway
2.
Florida Atlantic University, Florida, USA

^* Corresponding author: Tovohery Hajatiana Randrianarisoa
^* Corresponding author: Tovohery Hajatiana Randrianarisoa

Received: August 2021

Revised: February 2022

Early access: May 5, 2022

Published online: May 5, 2022

This work was supported by the University Grant Commission, Govt. of India (Sr. No. 2061641156), the Research Council of Norway grant No. 280731 and the Swiss SNF grant No. 181446.

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Abstract

We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.

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Mathematics Subject Classification: Primary: 94B05, 94B60; Secondary: 15A03, 15A54.

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Figure 1. $ \mathcal{F} = \{2, 2, 4, 4, 6, 6\} $

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