Some classes of LCD codes and self-orthogonal codes over finite fields

Xia Li; Feng Cheng; Chunming Tang; Zhengchun Zhou

doi:10.3934/amc.2019018

Article Contents

2019, Volume 13, Issue 2: 267-280. Doi: 10.3934/amc.2019018

This issue Previous Article A new construction of rotation symmetric bent functions with maximal algebraic degree Next Article Some new constructions of isodual and LCD codes over finite fields

Some classes of LCD codes and self-orthogonal codes over finite fields

1.
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, China
3.
School of Mathematics and Information, China West Normal University, Nanchong 637002, China

^* Corresponding author: Chunming Tang
^* Corresponding author: Chunming Tang

Received: February 28, 2018

Revised: August 31, 2018

Published: January 2019

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Abstract

Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general $ p $-ary case, where $ p $ is an odd prime. Based on the extended construction, several classes of $ p $-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

Keywords:

Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.

Citation:

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Table 1. The weight distribution of $ {\mathcal C}_{D_{t}} $ for $ 1\leq t\leq m $

Weight	Multiplicity
$ 0 $	$ 1 $ time
$ \frac{(p-1)n-(p-1)K_{t}(k,m)}{p} $ for $ k=1,2,\cdots,m-1 $	$ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})}{p} $	$ (p-1)^m $ times

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Table 2. The weight distribution of $ {\mathcal C}_{D_{t}} $ for $ 1\leq t\leq m $

Weight	Multiplicity
$ 0 $	$ 1 $ time
$ \frac{\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $	$ (p-1)m $ times
$ \frac{(p-1)n-(p-1)K_{t}(k-1,m-1)+(p-1)}{p} $ $ k=2,\cdots,m $	$ (p-1)^k{m\choose k} $ times

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Table 3. The weight distribution of $ \mathcal C_{\overline{D}_ t} $ for $ 1\leq t\leq m $

Weight	Multiplicity
$ 0 $	$ 1 $ time
$ \frac{(p-1)n-(p-1) \left ( K_{t}(k,m)+ K_{m}(k,m) \right )}{p} $ for $ k=1,2,\cdots,m-1 $	$ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})+(p-1)((p-1)^m-(-1)^m)}{p} $	$ (p-1)^m $ times

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Table 4. The weight distribution of $ {\mathcal C}_{\overline{D}_{t}} $ for $ 1\leq t\leq m $

Weight	Multiplicity
$ 0 $	$ 1 $ time
$ \frac{p(p-1)^{m}+\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $	$ (p-1)m $ times
$ \frac{(p-1)n-(p-1)\left ( K_{t}(k-1,m-1)+K_{m}(k,m) -1\right )}{p} $ $ k=2,\cdots,m $	$ (p-1)^k{m\choose k} $ times

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