Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes

Lingyu Diao; Jian Gao; Jiyong Lu

doi:10.3934/amc.2020029

Article Contents

2020, Volume 14, Issue 4: 555-572. Doi: 10.3934/amc.2020029

This issue Previous Article Group signature from lattices preserving forward security in dynamic setting Next Article Giophantus distinguishing attack is a low dimensional learning with errors problem

Some results on $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes

1.
School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China
2.
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China
3.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410114, China
4.
College of Science, Tianjin University of Science and Technology, Tianjin, 300071, China

^* Corresponding author: Jian Gao, dezhougaojian@163.com
^* Corresponding author: Jian Gao, dezhougaojian@163.com

Received: November 30, 2018

Revised: April 30, 2019

Early access: September 2019

Published: November 2020

This research is supported by the National Natural Science Foundation of China (Grant No. 11701336, 11626144 and 11671235), the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No. 2018MMAEZD09), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)(Grant No. AM201804.)

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Abstract

$ \mathbb{Z}_p\mathbb{Z}_p[v] $-Additive cyclic codes of length $ (\alpha,\beta) $ can be viewed as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $, where $ R = \mathbb{Z}_p+v\mathbb{Z}_p $ with $ v^2 = v $. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as $ R[x] $-submodules of $ \mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1) $. We also determine the generator polynomials of the dual codes of $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes and MDSS codes are obtained from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes. Moreover, we also get some quantum codes from $ \mathbb{Z}_p\mathbb{Z}_p[v] $-additive cyclic codes.

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

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Table 1. Some optimal $ \mathbb{Z}_p\mathbb{Z}_p[v] $-linear codes $ [n,k,d] $

$p$	$[\alpha,\beta]$	Generators	$(\alpha+\beta,p^k,d_L)$	$[n,k,d]$
$3$	$[4,4]$	$f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=x+1$	$(8,3^7,4)$	$[12,7,4]$
$3$	$[4,4]$	$f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=1$	$(8,3^8,3)$	$[12,8,3]$
$5$	$[6,3]$	$f=x^5+4x^4+x^3+4x^2+x+4,l=x^4+3x^3+x+3,g_1=x^2+x+1,g_2=1$	$(9,5^5,6)$	$[12,5,6]$
$7$	$[2,6]$	$f=x^2+6,l=4x+6,g_1=x+5,g_2=x+4$	$(8,7^{10},4)$	$[14,10,4]$
$3$	$[5,5]$	$f=x^5+2,l=x^4+2x^3+x+2,g_1=x+2,g_2=1$	$(10,3^9,4)$	$[15,9,4]$
$5$	$[5,5]$	$f=x^5+4,l=x^4+2x^3+4x^2+x+3,g_1=x^2+3x+1,g_2=1$	$(10,5^8,6)$	$[15,8,6]$
$5$	$[6,12]$	$f=x^3+3x^2+2x+4,l=4x^2+3x+2,g_1=x+4,g_2=x+3$	$(18,5^{25},4)$	$[30,25,4]$

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Table 2. Some MDSS codes $ (\alpha+\beta,p^k,d_L) $

$ p $	$ [\alpha,\beta] $	Generators	$ (\alpha+\beta,p^k,d_L) $
$ 3 $	$ [3,3] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (6,3^8,2) $
$ 5 $	$ [4,4] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (8,5^{11},2) $
$ 11 $	$ [7,8] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (15,11^{22},2) $
$ 29 $	$ [12,6] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (18,29^{23},2) $
$ 37 $	$ [29,31] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (60,37^{90},2) $
$ 59 $	$ [36,68] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (104,59^{171},2) $
$ 97 $	$ [106,93] $	$ f=x-1,l=1,g_1=1,g_2=1 $	$ (199,97^{291},2) $

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Table 3. Quantum codes $ [[N,K,\geq D]]_p $}

$[\alpha, \beta]$	$f$	$g_1$	$g_2$	$(\alpha+\beta, p^k, d_L)$	$[[N, K, \geq D]]_p$	$[[N', K', D']]_p$
$[5, 5]$	1, 3, 1	1, 3, 1	1, 3, 1	$(10, 5^9, 3)$	$[[15, 3, \geq3]]_5$	-
$[20, 5]$	1, 3, 2, 3, 1	1, 3, 1	1, 3, 1	$(25, 5^{22}, 3)$	$[[30, 14, \geq3]]_5$	$[[10, 4, 3]]_5$ (ref.[12])
$[11, 11]$	1, 7, 6, 7, 1	1, 7, 6, 7, 1	1, 7, 6, 7, 1	$(22, 11^{21}, 5)$	$[[33, 9, \geq5]]_{11}$	-
$[21, 7]$	1, 6, 0, 6, 1	1, 5, 1	1, 5, 1	$(28, 7^{27}, 3)$	$[[35, 19, \geq3]]_7$	$[[18, 2, 3]]_7$ (ref.[16])
$[14, 14]$	1, 1, 5, 5, 1, 1	1, 1, 5, 5, 1, 1	1, 1, 5, 5, 1, 1	$(28, 7^{27}, 4)$	$[[42, 12, \geq4]]_7$	$[[11, 1, 4]]_7$ (ref.[11])
$[9, 18]$	1, 2, 0, 2, 1	1, 0, 2, 2, 0, 1	1, 0, 2, 2, 0, 1	$(27, 3^{31}, 3)$	$[[45, 17, \geq3]]_3$	-
$[17, 17]$	1, 13, 6, 13, 1	1, 13, 6, 13, 1	1, 13, 6, 13, 1	$(34, 17^{39}, 5)$	$[[51, 27, \geq5]]_{17}$	$[[48, 24, 5]]_{17}$ (ref.[14])
$[26, 13]$	1, 10, 2, 2, 10, 1	1, 9, 6, 9, 1	1, 9, 6, 9, 1	$(39, 13^{39}, 4)$	$[[52, 26, \geq4]]_{13}$	$[[24, 12, 4]]_{13}$ (ref.[14])
$[20, 20]$	1, 3, 2, 3, 1	1, 3, 2, 3, 1	1, 3, 2, 3, 1	$(40, 5^{48}, 3)$	$[[60, 36, \geq3]]_5$	$[[60, 56, 2]]_5$ (ref.[12])
$[22, 22]$	1, 1, 9, 9, 1, 1	1, 1, 9, 9, 1, 1	1, 1, 9, 9, 1, 1	$(44, 11^{51}, 4)$	$[[66, 36, \geq4]]_{11}$	$[[52, 28, 3]]_{11}$(ref.[16])
$[30, 30]$	1, 2, 4, 2, 1	1, 2, 4, 2, 1	1, 2, 4, 2, 1	$(60, 5^{78}, 3)$	$[[90, 66, \geq3]]_5$	$[[30, 20, 3]]_5$ (ref.[17])
$[46, 23]$	1, 22, 22, 1	1, 21, 1	1, 21, 1	$(69, 23^{85}, 3)$	$[[92, 78, \geq3]]_{23}$	$[[48, 40, \geq3]]_{23}$ (ref.[13])
$[31, 31]$	1, 29, 1	1, 29, 1	1, 29, 1	$(62, 31^{87}, 3)$	$[[93, 81, \geq3]]_{31}$	$[[52, 44, \geq3]]_{31}$ (ref.[13])
$[47, 47]$	1, 45, 1	1, 45, 1	1, 45, 1	$(94, 47^{135}, 3)$	$[[141,129, \geq3]]_{47}$	-
$[59, 59]$	1, 57, 1	1, 57, 1	1, 57, 1	$(118, 59^{171}, 3)$	$[[177,165, \geq3]]_{59}$	-

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