Two classes of cyclic extended double-error-correcting Goppa codes

Yanyan Gao; Qin Yue; Xinmei Huang; Yun Yang

doi:10.3934/amc.2022003

Article Contents

2024, Volume 18, Issue 1: 105-115. Doi: 10.3934/amc.2022003

This issue Previous Article Galois LCD codes over rings Next Article Additive polycyclic codes over

$ \mathbb{F}_{4} $

induced by binary vectors and some optimal codes

Two classes of cyclic extended double-error-correcting Goppa codes

1.
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211167, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China
3.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China
4.
Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, China

Corresponding author: Qin Yue
Corresponding author: Qin Yue

Received: July 2021

Revised: November 2021

Early access: February 2022

Published: February 2024

This paper was supported by National Natural Science Foundation of China (Nos. 61772015, 11961050, 12101277), Fundation of Nanjing Institute of Technology (No. CKJB202007), the Guangxi Natural Science Foundation(No. 2020GXNSFAA159053), Foundation of Science and Technology on Information Assurance Laboratory (No. KJ-17-010)

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Abstract

Let $ \Bbb F_{2^m} $ be a finite extension of the field $ \Bbb F_2 $ and $ g(x) = x^2+\alpha x+1 $ a quadratic polynomial over $ \Bbb F_{2^m} $. In this paper, two classes of cyclic extended double-error-correcting Goppa codes are proposed. We obtain the following two classes of Goppa codes: (1) cyclic extended Goppa code with the irreducible polynomial $ g(x) $ and $ L = \Bbb F_{2^m}\cup \{\infty\} $; (2) cyclic extended Goppa code with the reducible polynomial $ g(x) $ and $ |L'| = 2^m-1 $. In addition, the parameters of above cyclic extended Goppa codes are given.

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Mathematics Subject Classification: 94B05.

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