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

  • * Corresponding author: Jian Gao

    * Corresponding author: Jian Gao 

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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  • 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.

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

    Citation:

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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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