The weight distribution of a class of p-ary cyclic codes and their applications

Lanqiang Li; Shixin Zhu; Li Liu

doi:10.3934/amc.2019008

Article Contents

2019, Volume 13, Issue 1: 137-156. Doi: 10.3934/amc.2019008

This issue Previous Article Further improvement of factoring

$ N = p^r q^s$

with partial known bits Next Article Cyclic DNA codes over

$ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$

The weight distribution of a class of p-ary cyclic codes and their applications

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

* Corresponding author: Shixin Zhu
* Corresponding author: Shixin Zhu

Received: May 2018

Revised: August 2018

Published: December 2018

This research is supported in part by the National Natural Science Foundation of China under Project 61572168, Project 61772168 and Project 11501156, the Natural Science Foundation of Anhui Province under Grant 1808085MA15 and the Key University Science Research Project of Anhui Province under Grant KJ2018A0497.

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Abstract

Cyclic codes over finite field have been studied for decades due to their wide applications in communication and storage systems. However their weight distributions are known only in a few cases. In this paper, we investigate a class of $ p$-ary cyclic codes whose duals have three zeros, where $ p$ is an odd prime. The weight distributions of the class of cyclic codes for all distinct cases are determined explicitly. The results indicate that these codes contain five-weight codes, seven-weight codes and eleven-weight codes. Some of these codes are optimal. Moreover, the covering structures of the class of codes are considered and being used to construct secret sharing schemes.

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

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Table Ⅰ. Weight distribution of the cyclic code C for odd m in Theorem 3.1

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p^{m-1}$	$ (p^m-1)(p^{m-1}+1)$
$(p-1)p^{m-1}-1$	$ (p^m-1)(p^{m-1}+1)(p-1)$
$(p-1)p^{m-1}-p^{\frac{m-1}{2}}$	$\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-1}{2}})(p-1) $
$ (p-1)p^{m-1}-p^{\frac{m-1}{2}}-1$	$\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}-p^{\frac{m-1}{2}}\big)(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}$	$\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-1}{2}})(p-1)$
$ (p-1)p^{m-1}+p^{\frac{m-1}{2}}-1$	$\frac{1}{2}(p^m-1)\big((p-1)p^{m-1}+p^{\frac{m-1}{2}}\big)(p-1) $
$ p^{m}-1$	$p-1$

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Table Ⅱ. Weight distribution of the cyclic code C for even m in Theorem 3.1

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p^{m-1}$	$p^m-1 $
$(p-1)p^{m-1}-1$	$(p^m-1)(p-1) $
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$	$\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}}) $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$	$\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$	$\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$	$\frac{1}{2}(p^m-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$	$\frac{1}{2}(p^m-1)(p^{m-1}-p^{\frac{m-2}{2}})(p-1) $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$	$\frac{1}{2}(p^m-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})(p-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$	$\frac{1}{2}(p^m-1)(p^{m-1}+(p-1)p^{\frac{m-2}{2}})$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$	$\frac{1}{2}(p^m-1)(p^{m-1}+p^{\frac{m-2}{2}})(p-1)$
$ p^{m}-1$	$p-1 $

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Table Ⅲ. Weight distribution of the cyclic code C for even m in Theorem 3.2

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p^{m-1}$	$(p^m-1)(1+p^{m-d}-p^{m-2d}) $
$(p-1)p^{m-1}-1$	$ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})$	$(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}-p^{\frac{m+2d-2}{2}})-1$	$(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$	$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$	$(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}$	$(p-1)(p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m+2d-2}{2}}-1$	$(p-1)((p-1)p^{m-2d-1}+ p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$	$(p^{m-1}-(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$	$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ p^{m}-1$	$p-1 $

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Table Ⅳ. Weight distribution of the cyclic code C for even m in Theorem 3.2

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p^{m-1}$	$ (p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)p^{m-1}-1$	$ (p-1)(p^m-1)(1+p^{m-d}-p^{m-2d})$
$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})$	$(p^{m-1}+(p-1)p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}-p^{\frac{m-2}{2}})-1$	$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}$	$(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$(p-1)p^{m-1}-p^{\frac{m+2d-2}{2}}-1$	$(p-1)((p-1)p^{m-2d-1}-p^{\frac{m-2d-2}{2}})\dfrac{(p^m-1)}{p^d+1}$
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}$	$(p-1)(p^{m-1}-p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)p^{m-1}+p^{\frac{m-2}{2}}-1$	$(p-1)((p-1)p^{m-1}+p^{\frac{m-2}{2}})\dfrac{p^d(p^m-1)}{p^d+1} $
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})$	$(p^{m-2d-1}-(p-1)p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1}$
$ (p-1)(p^{m-1}+p^{\frac{m+2d-2}{2}})-1$	$(p-1)(p^{m-2d-1}+p^{\frac{m-2d-2}{2}})\dfrac{p^m-1}{p^d+1} $
$ p^{m}-1$	$p-1 $

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Table Ⅴ. Weight distribution of the cyclic code C for even m in Corollary 1

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p^{m-1}$	$(p^m-1) $
$(p-1)p^{m-1}-1$	$ (p-1)(p^m-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}$	$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1)$
$(p-1)p^{m-1}-p^{\frac{m-2}{2}}-1$	$(p-1)((p-1)p^{m-1}-p^{\frac{m-2}{2}})(p^d-1) $
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})$	$(p^{m-1}-(p-1)p^{\frac{m-2}{2}})(p^d-1)$
$ (p-1)(p^{m-1}+p^{\frac{m-2}{2}})-1$	$(p-1)(p^{m-1}+p^{\frac{m-2}{2}})(p^d-1) $
$ p^{m}-1$	$p-1 $

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Table Ⅵ. Weight distribution of the cyclic code C for even m in Corollary 2

Hamming Weight $i$	Frequency $A_i$
$0$	$1$
$(p-1)p$	$(p^2-1) $
$p(p-1)-1$	$ 2(p-1)(p^2-1)$
$(p-1)p-2$	$(p-1)^2(p^2-p-1) $
$ p^2-2$	$(p-1)(p^2-1) $
$ p^2-1$	$2(p-1) $

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