The conorm code of an AG-code

María Chara; Ricardo A. Podestá; Ricardo Toledano

doi:10.3934/amc.2021018

Article Contents

2023, Volume 17, Issue 3: 714-732. Doi: 10.3934/amc.2021018

This issue Previous Article On ideal and weakly-ideal access structures Next Article On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $

The conorm code of an AG-code

1.
Researcher of CONICET at Facultad de Ingeniería Química, (UNL), Santiago del Estero 2829, (3000) Santa Fe, Argentina
2.
FaMAF – CIEM (CONICET), Universidad Nacional de Córdoba, Av. Medina Allende 2144, (5000) Córdoba, Argentina
3.
Facultad de Ingeniería Química, (UNL), Santiago del Estero 2829, (3000) Santa Fe, Argentina

^* Corresponding author: María Chara
^* Corresponding author: María Chara

Received: November 2020

Revised: March 2021

Early access: June 2021

Published: June 2023

Partially supported by CONICET, UNL CAI+D 2016, SECyT-UNC, CSIC

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Abstract

Given a suitable extension $ F'/F $ of algebraic function fields over a finite field $ \mathbb{F}_q $, we introduce the conorm code $ \operatorname{Con}_{F'/F}( \mathcal{C}) $ defined over $ F' $ which is constructed from an algebraic geometry code $ \mathcal{C} $ defined over $ F $. We study the parameters of $ \operatorname{Con}_{F'/F}( \mathcal{C}) $ in terms of the parameters of $ \mathcal{C} $, the ramification behavior of the places used to define $ \mathcal{C} $ and the genus of $ F $. In the case of unramified extensions of function fields we prove that $ \operatorname{Con}_{F'/F}( \mathcal{C})^\perp = \operatorname{Con}_{F'/F}( \mathcal{C}^\perp) $ when the degree of the extension is coprime to the characteristic of $ \mathbb{F}_q $. We also study the conorm of cyclic algebraic-geometry codes and we show that some repetition codes, Hermitian codes and all Reed-Solomon codes can be represented as conorm codes.

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Mathematics Subject Classification: Primary: 94B27; Secondary: 94B15, 14H05.

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References

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