Some new constructions of isodual and LCD codes over finite fields
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May 2019, 13(2): 267-280. doi: 10.3934/amc.2019018
Some classes of LCD codes and self-orthogonal codes over finite fields
Xia Li 1,2, , Feng Cheng 1, , Chunming Tang 3,, and Zhengchun Zhou 1,
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, China
School of Mathematics and Information, China West Normal University, Nanchong 637002, China
* Corresponding author: Chunming Tang
Received March 2018 Revised September 2018 Published February 2019
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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: Linear codes, self-orthogonal codes, LCD codes, weight distributions, Krawtchouck polynomial.
Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.
Citation: Xia Li, Feng Cheng, Chunming Tang, Zhengchun Zhou. Some classes of LCD codes and self-orthogonal codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 267-280. doi: 10.3934/amc.2019018
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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
$ \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
Table 3. The weight distribution of $ \mathcal C_{\overline{D}_ t} $ for $ 1\leq t\leq m $
$ \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
Table 4. The weight distribution of $ {\mathcal C}_{\overline{D}_{t}} $ for $ 1\leq t\leq m $
$ \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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