New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

Fanghui Ma; Jian Gao; Fang-Wei Fu

doi:10.3934/amc.2019027

Article Contents

2019, Volume 13, Issue 3: 421-434. Doi: 10.3934/amc.2019027

This issue Previous Article Exponential generalised network descriptors Next Article A unified polynomial selection method for the (tower) number field sieve algorithm

New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

1.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2.
School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China
3.
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, Hubei 430062, China
4.
Hunan Key Laboratory of Mathematical Modeling and Analysis in Engineering, School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China

^* Corresponding author: Jian Gao
^* Corresponding author: Jian Gao

Received: June 2018

Published: August 2019

This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 61571243, 11701336, 11626144 and 11671235), the Fundamental Research Funds for the Central Universities of China, the Scientific Research Fund of Hubei Key Laboratory of Applied Mathematics (Hubei University)(Grant No. HBAM201804), and the Scientific Research Fund of Hunan Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology)(Grant No. 2018MMAEZD09)

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Abstract

Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu $. In this paper, we construct new non-binary quantum codes from ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $. We give the structure of ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.

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

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Table 1. New quantum codes $ [[n,k,d]]_q $ from $ (1-v-uv) $-constacyclic codes over $ R $

$ n $	$ g_0(x) $	$ g_1(x) $	$ g_2(x) $	$ g_3(x) $	$ \Phi (\mathscr{C}) $	$ [[n,k,d]] $	$ [[n',k',d']] $
$ 18 $	$ 101 $	$ 1221 $	$ 2211 $	$ 12021 $	$ [72,60,4]_{3} $	$ [[72,48,4]]_{3} $	$ [[72,48,2]]_{3} $ (ref. [5])
$ 10 $	$ 411 $	$ 131 $	$ 441 $	$ 131 $	$ [40,32,3]_{5} $	$ [[40,24,3]]_{5} $	$ [[40,24,2]]_{5} $ (ref. [4])
$ 11 $	$ 114431 $	$ 411421 $	$ 431441 $	$ 431441 $	$ [44,24,8]_{5} $	$ [[44,4,8]]_{5} $	$ [[44,4,5]]_{5} $ (ref. [4])
$ 15 $	$ 11 $	$ 41 $	$ 41 $	$ 41 $	$ [60,56,2]_{5} $	$ [[60,52,2]]_{5} $	$ [[60,48,2]]_{5} $ (ref. [4])
$ 22 $	$ 232121 $	$ 411421 $	$ 431441 $	$ 411421 $	$ [88,68,4]_{5} $	$ [[88,48,4]]_{5} $	$ [[88,48,2]]_{5} $ (ref. [4])
$ 24 $	$ 30201 $	$ 21 $	$ 31 $	$ 431 $	$ [96,88,4]_{5} $	$ [[96,80,4]]_{5} $	$ [[96,80,2]]_{5} $ (ref. [4])
$ 28 $	$ 2423411 $	$ 4213421 $	$ 4312431 $	$ 4213421 $	$ [112,88,4]_{5} $	$ [[112,64,4]]_{5} $	$ [[112,64,2]]_{5} $ (ref. [4])
$ 33 $	$ 14244431021 $	$ 411421 $	$ 431441 $	$ 13043141211 $	$ [132,104,4]_{5} $	$ [[132,72,4]]_{5} $	$ [[132,72,2]]_{5} $ (ref. [5])
$ 35 $	$ 121 $	$ 131 $	$ 131 $	$ 1111111 $	$ [140,128,3]_{5} $	$ [[140,116,3]]_{5} $	$ [[140,112,2]]_{5} $ (ref. [4])
$ 36 $	$ 321 $	$ 31 $	$ 21 $	$ 31 $	$ [144,139,2]_{5} $	$ [[144,134,2]]_{5} $	$ [[144,128,2]]_{5} $ (ref. [4])
$ 45 $	$ 11 $	$ 41 $	$ 41 $	$ 41 $	$ [180,176,2]_{5} $	$ [[180,172,2]]_{5} $	$ [[180,156,2]]_{5} $ (ref. [5])
$ 4 $	$ w1 $	$ 21 $	$ w^21 $	$ w^61 $	$ [16,12,3]_{9} $	$ [[16,8,3]]_{9} $	$ [[16,8,2]]_{9} $ (ref. [4])
$ 6 $	$ 2w^61 $	$ 121 $	$ 111 $	$ 121 $	$ [24,16,4]_{9} $	$ [[24,8,4]]_{9} $	$ [[24,8,2]]_{9} $ (ref. [4])
$ 20 $	$ 2w0w^71 $	$ w^21 $	$ w^61 $	$ w^21 $	$ [80,73,4]_{9} $	$ [[80,66,4]]_{9} $	$ [[80,48,4]]_{9} $ (ref. [4])
$ 4 $	$ 501 $	$ 51 $	$ 81 $	$ 51 $	$ [16,11,4]_{13} $	$ [[16,6,4]]_{13} $	—
$ 6 $	$ 61 $	$ 31 $	$ 41 $	$ 91 $	$ [24,20,3]_{13} $	$ [[24,16,3]]_{13} $	—

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