The weight distributions of constacyclic codes

Fengwei Li; Qin Yue; Fengmei Liu

doi:10.3934/amc.2017039

Article Contents

2017, Volume 11, Issue 3: 471-480. Doi: 10.3934/amc.2017039

This issue Previous Article Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm Next Article Private set intersection: New generic constructions and feasibility results

The weight distributions of constacyclic codes

1.
School of Mathematics and Statistics, Zaozhuang University, Zaozhuang, Shandong 277160, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, China
3.
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China
4.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China
5.
Science and Technology on Information Assurance Laboratory, Beijing 100072, China

Received: June 2015

Published online: August 2017

The paper was supported by National Natural Science Foundation of China under Grants 11601475,61772015 and Foundation of Science and Technology on Information Assurance Laboratory under Grants KJ-15-009,6142112010202.

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Abstract

Let $\Bbb F_q$ be a finite field with $q$ elements. Suppose that $a, λ∈ \Bbb F_q^*$, $a^n=λ$ with $n|(q-1)$. In this paper, we determine the weight distribution of a class of $λ$-constacyclic codes of length $nm$ with the parity check polynomial $h(x)=(x^m-aξ^{st})(x^m-aξ^{s(t+1)})...(x^m-aξ^{s(t+r-1)})$ and $n>(r-1)m$, where $s,t, r$ are positive integers and $ξ∈ \Bbb F_q$ is a primitive n-th root of unity. Moreover, we give the weight distributions of $λ$-constacyclic codes of length $nm$ explicitly in several cases: (1) $r=1$, $n>1$; (2) $r=2$, $m=2$ and $n>2$; (3) $r=2$, $m=3$ and $n>3$; (4) $r=3$, $m=2$ and $n>4$.

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Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 11C08, 11C20.

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Table 1. Weight distribution

Weight	Frequency
0	1
$n-1$	$2n(q-1)$
$n$	$2(q-1)(q+1-n)$
$2n-2$	$n^2(q-1)^2$
$2n-1$	$2n(q-1)^2(q+1-n)$
$2n$	$(q-1)^2(q+1-n)^2$

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Table 2. Weight distribution

Weight	Frequency
0	1
$n-1$	$3n(q-1)$
$n$	$3(q-1)(q+1-n)$
$2n-2$	$3n^2(q-1)^2$
$2n-1$	$6n(q-1)^2(q+1-n)$
$2n$	$3(q-1)^2(q+1-n)^2$
$3n-3$	$n^3(q-1)^3$
$3n-2$	$3n^2(q-1)^3(q+1-n)$
$3n-1$	$3n(q-1)^3(q+1-n)^2$
$3n$	$(q-1)^3(q+1-n)^3$

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Table 3. Weight distribution

Weight	Frequency
0	1
$n-2$	$n(n-1)(q-1)$
$n-1$	$2n(q-1)(q-n+2)$
$n$	$2(q-1)^3-2(n-3)(q-1)^2+(n-2)(n-3)(q-1)$
$2n-4$	$\frac {n^2(n-1)^2}4 (q-1)^2$
$2n-3$	$n^2(q-1)^2(n-1)(q-n+2)$
$2n-2$	$n^2(q-1)^2(q-n+2)^2+n(n-1)(q-1)^2[(q-1)^2-(n-3)(q-1)+C_{n-2}^2]$
$2n-1$	$2n(q-n+2)(q-1)^4-2n(n-3)(q-n+2)(q-1)^3$
	$+n(n-2)(n-3)(q-n+2)(q-1)^2$
$2n$	$(q-1)^6+(n-3)^2(q-1)^4+\frac {(n-2)^2(n-3)^2}4 (q-1)^2+(n-2)(n-3)(q-1)^4$
	$-2(n-3)(q-1)^5-(n-2)(n-3)^2(q-1)^3$

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