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

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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  • $ \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.

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

    Citation:

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

    $p$ $[\alpha,\beta]$ Generators $(\alpha+\beta,p^k,d_L)$ $[n,k,d]$
    $3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=x+1$ $(8,3^7,4)$ $[12,7,4]$
    $3$ $[4,4]$ $f=x^3+2x^2+x+2,l=x^2+x+2,g_1=x+2,g_2=1$ $(8,3^8,3)$ $[12,8,3]$
    $5$ $[6,3]$ $f=x^5+4x^4+x^3+4x^2+x+4,l=x^4+3x^3+x+3,g_1=x^2+x+1,g_2=1$ $(9,5^5,6)$ $[12,5,6]$
    $7$ $[2,6]$ $f=x^2+6,l=4x+6,g_1=x+5,g_2=x+4$ $(8,7^{10},4)$ $[14,10,4]$
    $3$ $[5,5]$ $f=x^5+2,l=x^4+2x^3+x+2,g_1=x+2,g_2=1$ $(10,3^9,4)$ $[15,9,4]$
    $5$ $[5,5]$ $f=x^5+4,l=x^4+2x^3+4x^2+x+3,g_1=x^2+3x+1,g_2=1$ $(10,5^8,6)$ $[15,8,6]$
    $5$ $[6,12]$ $f=x^3+3x^2+2x+4,l=4x^2+3x+2,g_1=x+4,g_2=x+3$ $(18,5^{25},4)$ $[30,25,4]$
     | Show Table
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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) $
     | Show Table
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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 $(118, 59^{171}, 3)$ $[[177,165, \geq3]]_{59}$ -
     | Show Table
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