On binary LCD BCH codes of length $ \frac{{{2^m} + 1}}{3} $

Yuzhe Liu; Xiaoshan Kai; Shixin Zhu

doi:10.3934/amc.2022053

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2024, Volume 18, Issue 4: 995-1009. Doi: 10.3934/amc.2022053

This issue Previous Article Constructions for several classes of few-weight linear codes and their applications Next Article Linear complementary dual codes and double circulant codes over a semi-local ring

On binary LCD BCH codes of length $ \frac{{{2^m} + 1}}{3} $

School of Mathematics, Hefei University of Technology, Hefei 230601, China

^*Corresponding author: Xiaoshan Kai
^*Corresponding author: Xiaoshan Kai

Received: February 2022

Revised: June 2022

Early access: July 2022

Published: August 2024

The research was supported by the Natural Science Foundation of China under Grants 61972126, U21A20428, 12171134 and 61802102, and the Fundamental Research Funds for the Central Universities of China under Grant PA2019GDZC0097

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Abstract

BCH codes are a special subclass of cyclic codes and have many important applications in data storage and communication systems. In this paper, we investigate the structure of binary linear complementary dual (LCD) BCH codes with length $ n = \frac{{{2^m} + 1}}{3} $, where $ m \geq 7 $ is an odd integer. By exploring cyclotomic cosets modulo $ n $, we determine the dimension of LCD BCH codes for designed distance $ \delta $ in the range $ 2 \le \delta \le{2^{\frac{{m + 1}}{2}}} $. Furthermore, we compute the first five largest coset leaders modulo $ n $ and construct some binary LCD BCH codes. We also present two families of optimal binary linear codes from LCD BCH codes.

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Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 94B15.

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