Parameters of LCD BCH codes with two lengths

Haode Yan; Hao Liu; Chengju Li; Shudi Yang

doi:10.3934/amc.2018034

Article Contents

2018, Volume 12, Issue 3: 579-594. Doi: 10.3934/amc.2018034

This issue Previous Article A first step towards the skew duadic codes Next Article A connection between sumsets and covering codes of a module

Parameters of LCD BCH codes with two lengths

1.
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2.
Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
3.
Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
4.
School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

^* Corresponding author: Chengju Li
^* Corresponding author: Chengju Li

Received: August 2017

Revised: March 2018

Published: July 2018

The work was supported by the National Natural Science Foundation of China (NSFC) under Grant 11701179, the Shanghai Sailing Program under Grant 17YF1404300, the Foundation of Science and Technology on Information Assurance Laboratory under Grant KJ-17-007, the National NSFC under Grant 11701317, the Shanghai Natural Science Foundation under Grant 17ZR1408400, and the National Key R&D Program of China under Grant 2017YFB0802302.

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Abstract

In this paper, we study LCD BCH codes over the finite field GF $(q)$ with two types of lengths $n$ , where $n = q^l+1$ and $n = (q^l+1)/(q+1)$ . Several classes of LCD BCH codes are given and their parameters are determined or bounded by exploring the cyclotomic cosets modulo $n$ . For $n = q^l+1$ , we determine the dimensions of the codes with designed distance $δ$ , where $q^{\lfloor\frac{l+1}{2}\rfloor}+1 ≤ δ ≤q ^{\lfloor\frac{l+3}{2}\rfloor}+1$ . For $n = (q^l+1)/(q+1)$ , the dimensions of the codes with designed distance $δ$ are presented, where $2 ≤ δ ≤q ^\frac{l-1}{2}+1$ .

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

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