Ternary Primitive LCD BCH codes

Xinmei Huang; Qin Yue; Yansheng Wu; Xiaoping Shi

doi:10.3934/amc.2021014

Article Contents

2023, Volume 17, Issue 3: 644-659. Doi: 10.3934/amc.2021014

This issue Previous Article Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes Next Article Quantum states associated to mixed graphs and their algebraic characterization

Ternary Primitive LCD BCH codes

1.
Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, 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, Jiangsu, 211100, China
4.
School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
5.
Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China

Received: December 2020

Early access: May 2021

Published: June 2023

This paper was supported by the National Natural Science Foundation of China (No.61772015), the Foundation of Science and Technology on Information Assurance Laboratory (No.KJ-17-010) and the Foundation of Jinling Institute of Technology (No.JIT-B-202016, No.JIT-FHXM-2020). Y. Wu was sponsored by NUPTSF (No. NY220137). (Corresponding author:Yansheng Wu.).

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Abstract

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.

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Mathematics Subject Classification: Primary: 11T23, 94B05; Secondary: 11L05.

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Full Text(HTML)

Weight	Frequency
0	1
$ \frac{2}3\cdot(3^m-3^{\frac m2}) $	$ \frac{3^m-1}{2} $
$ \frac{2}3\cdot(3^m+3^{\frac m2}) $	$ \frac{3^m-1}{2} $

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Weight	Frequency
0	1
$ \frac{2}3 \cdot(3^m-3^{\frac{m}2}) $	$ \frac{3^m-1}2 $
$ \frac{2}3 \cdot(3^m+3^{\frac{m}2}) $	$ \frac{3^m-1}2 $
$ \frac{1}3(2\cdot3^m+3^\frac{m}2)-1 $	$ 3^m-1 $
$ \frac{1}3(2\cdot3^m-3^\frac{m}2)-1 $	$ 3^m-1 $
$ 3^m-1 $	2

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Weight	Frequency
0	1
$ \frac12\cdot(3^m-1) $	12
$ \frac34\cdot(3^m-1) $	8
$ 3^m-1 $	6

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