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Additive polycyclic codes over $ \mathbb{F}_{4} $ induced by binary vectors and some optimal codes

  • *Corresponding author: Nuh Aydin

    *Corresponding author: Nuh Aydin 
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  • In this paper, we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $ a $ in $ \mathbb{F}_{2}^{n}. $ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $ C $ is a right polycyclic code induced by a vector $ a\in \mathbb{F}_{2}^{n} $, then the Hermitian dual of $ C $ is a sequential code induced by $ a. $ As an application of these codes, we present examples of additive right polycyclic codes over $ \mathbb{F}_{4} $ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over $ \mathbb{F}_{4}. $

    Mathematics Subject Classification: Primary: 94B60, 58F17; Secondary: 94B05.

    Citation:

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  • Table 1.  Additive codes $ [n, k, d]_4 $ v.s. BKLC $ [n, k, d]_4 $ with smaller dimension

    $ [n, \frac{2k+1}{2}, d]_4 $ $ [n, k, d]_4 $ $ [n, k+1, d-1]_4 $ $ \langle \alpha g_1 +g_2, b\rangle $ Multinomial
    $ [ 7, 9/2, 3]_4 $ $ [ 7, 4, 3 ]_4 $ $ [ 7, 5, 2 ]_4 $ $ \langle\alpha (x^2 + x + 1)+x, x^3 + x^2 + 1\rangle $ $ x^7 + x^6 + x^5 + x^3 + 1 $
    $ [7, 7/2, 4]_4 $ $ [7, 3, 4]_4 $ $ [7, 4, 3]_4 $ $ \langle\alpha (x + 1)+ x^4 + x^3 + x^2 + x, x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle $ $ x^7 + 1 $
    $ [22, 37/2, 3]_4 $ $ [22, 18, 3]_4 $ $ [22, 19, 2]_4 $ $ \langle\alpha (x + 1)+ x^3 + x, x^6 + x^4 + x^3 + \\x + 1\rangle $ $ x^{22} + x^{19} + x^{15} + x^{14} + x^{13} + x^8 + \\x^7 + x^6 + x^4 + x^2 + x + 1 $
    $ [23, 39/2, 3]_4 $ $ [23, 19, 3]_4 $ $ [23, 20, 2]_4 $ $ \langle\alpha (x + 1)+ x^4 + x^2, x^6 + x^5 + 1\rangle $ $ x^{23} + x^{22} + x^{21} + x^{15} + x^{13} + x^{11} + \\x^{10} + x^9 + x^8 + x^7 + x^3 + x^2+ x + 1 $
    $ [24, 41/2 , 3]_4 $ $ [24, 20, 3]_4 $ $ [24, 21, 2]_4 $ $ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $ $ x^{24} + x^{21} + x^{20} + x^{19} + x^{18} + x^{16} +\\ x^{14} + x^8 + x^5 + x^4 + x^3 + x^2 1, $
    $ [25, 43/2, 4]_4 $ $ [25, 21, 3]_4 $ $ [25, 22, 2]_4 $ $ \langle\alpha (x + 1)+x^4 + x^2, x^6 + x^5 + 1\rangle $ $ x^{25} + x^{24} + x^{22} + x^{21} + x^{19} + x^{18} + \\x^{15} + x^{13} + x^{12} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1 $
    $ [26, 45/2, 3]_4 $ $ [26, 22, 3]_4 $ $ [26, 23, 2]_4 $ $ \langle\alpha (x + 1)+x^4 + x^3 + x, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $ $ x^{26} + x^{24} + x^{21} + x^{16} + x^{15} + x^{13} + \\x^{12} + x^{11} + x^{10} + x^9 + x^7 +x^3 + \\x^2 + 1 $
    $ [27, 47/2, 3]_4 $ $ [27, 23, 3]_4 $ $ [27, 24, 2]_4 $ $ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $ $ x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1 $
    $ [27, 47/2, 3]_4 $ $ [27, 23, 3]_4 $ $ [27, 24, 2]_4 $ $ \langle\alpha (x^2 + x + 1)+ x^2 + x, x^5 + x^4 + \\x^3 + x + 1\rangle $ $ x^{27} + x^{26} + x^{23} + x^{21} + x^{19} + x^{18} + \\x^{17} + x^{14} + x^{13} + x^{11} + x^{10} +x^7 + \\x^3 + x + 1 $
    $ [28, 49/2, 3, ]_4 $ $ [28, 24, 3]_4 $ $ [28, 25, 2]_4 $ $ \langle\alpha (x^2 + x + 1)+ x^3 + x^2 + x, x^5 + \\x^3 + x^2 + x + 1\rangle $ $ x^{28} + x^{27} + x^{24} + x^{23} + x^{22} + x^{20} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{11} + \\x^{10} + x^8 + x^7 + x^5 + x^3 + x + 1 $
    $ [29, 51/2, 3]_4 $ $ [29, 25, 3]_4 $ $ [29, 26, 2]_4 $ $ \langle\alpha (x + 1)+ x^2 + x + 1, x^6 + x^5 + \\x^4 + x + 1\rangle $ $ x^{29} + x^{25} + x^{22} + x^{21} + x^{19} + x^{15} + \\x^{14} + x^{13} + x^{11} + x^{10} + x^6 +x^3 + \\x + 1 $
    $ [30, 53/2, 3]_4 $ $ [30, 26, 3]_4 $ $ [30, 27, 2]_4 $ $ \langle\alpha (x + 1)+ x^3 + x^2 + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $ $ x^{30} + x^{27} + x^{26} + x^{25} + x^{24} + x^{20} + \\x^{19} + x^{18} + x^{16} + x^{11} + x^{10} +x^9 + \\x^8 + x^7 + x^2 + 1 $
    $ [31, 55/2, 3]_4 $ $ [31, 27, 3]_4 $ $ [31, 28, 2]_4 $ $ \langle\alpha ( x + 1)+ x^4 + x + 1, x^6 + x^5 + \\x^3 + x^2 + 1\rangle $ $ x^{31} + x^{25} + x^{23} + x^{22} + x^{20} + x^{18} + \\x^{17} + x^{16} + x^{15} + x^{14} + x^{13} +x^{12} + \\x^{10} + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 +1 $
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    Table 2.  Optimal binary linear codes $ [n, k, d]_2 $ obtained from quaternary additive codes based on $ W $, $ T $, and $ L $

    $ [n, k, d]_2 $ Map $ \langle \alpha g_1 +g_2, b\rangle $ Multinomial
    $ [ 7, 2, 4]_2^{*\circ} $ L $ \langle\alpha (x^5 + x^4 + x + 1) + 1, x^2 + x + 1 \rangle $ $ x^7 + x^4 + x^3 + 1 $
    $ [ 10, 4, 4]_2 $ L $ \langle\alpha (x + 1)+x + 1, x^6 + x^5 + x^4 + \\x + 1\rangle $ $ x^{10} + x^9 + x^8 + x^6 + x + 1 $
    $ [ 12, 5, 4]_2 $ L $ \langle\alpha (x^2 + x + 1)+x^2 + x + 1, x^7 + \\x^3 + x^2 + x + 1\rangle $ $ x^{12} + x^6 + x^5 + x^4 + 1 $
    $ [ 16, 9, 4]_2^* $ T $ \langle\alpha (x^7 + x^3 + x^2 + x + 1)+x^4 + \\x^3 + x^2 + x + 1, x^9 + x + 1 \rangle $ $ x^{16} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + \\x^7 + x^4 + 1 $
    $ [ 17, 9, 5]_2 $ T $ \langle\alpha (x^8 + x^5 + x^4 + x^3 + 1)+x^7 + \\x^6 + x^4 + x^3 + x, x^9 + x^8 + x^6 + \\x^5 + x^4 + x^3 + x^2 + x + 1\rangle $ $ x^{17} + x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + \\x^9 + x^8 + x^7 + x^4 + x^2 + x + 1 $
    $ [ 20, 11, 5]_2 $ W $ \langle\alpha (x+1)+x^6 + x^4 + x^2 + x, x^8 + \\x^6 + x^5 + x^4 + x^2 + x + 1\rangle $ $ x^{10} + x^7 + x^5 + x^3 + x + 1 $
    $ [ 26, 17, 4]_2 $ W $ \langle\alpha (x+1)+x^4 + x^3 + x^2, x^8 + x^5 + \\x^3 + x + 1\rangle $ $ x^{13} + x^{10} + x^6 + x^3 + x + 1 $
    $ [ 35, 24, 5]_2^* $ T $ \langle\alpha (x^{11} + x^9 + x^8 + x^6 + x^4 + x^3 + \\x^2 + x + 1)+x^{22} + x^{16} + x^{15} + x^{14} + \\x^{13} + x^{10} + x^9 + x^8 + x^7 + x^5 + x + \\1 , x^{24} + x^{21} + x^{19} + x^{16} + x^{13} + \\x^{12} + x^{10} + x^8 + x^7 + x^6 + x^2 + x +1\rangle $ $ x^{35} + x^{33} + x^{30} + x^{29} + x^{26} + x^{23} + \\x^{21} + x^{20} + x^{19} + x^{14} + x^{13} +x^{12} + \\x^{11} + x^9 + x^8 + x^7 + x^6 + x^4 + \\x^3 + x^2 + 1 $
    $ [ 49, 39, 4]_2^* $ T $ \langle\alpha ( x^{10} + x^8 + x^6 + x^4 + x^2 + x + \\1)+x , x^5 + x^4 + x^3 + x^2 + 1 \rangle $ $ x^{49} + x^{46} + x^{45} + x^{44} + x^{40} + x^{39} + \\x^{38} + x^{36} + x^{35} + x^{31} + x^{30} + x^{29} + \\x^{28} + x^{27} + x^{24} + x^{23} + x^{20} + x^{19} + \\x^{17} + x^{15} + x^{12} + x^{11} + x^9 + x^8 + \\x^4 + x + 1 $
    $ [ 62, 51, 4]_2 $ W $ \langle\alpha ( x + 1)+x^8 + x^4 + x + 1 , x^10 + \\x^9 + x^8 + x^4 + 1 \rangle $ $ x^{31} + x^{29} + x^{28} + x^{27} + x^{26} + x^{25} + \\x^{23} + x^{15} + x^{14} + x^{13} + x^{12} +x^9 + \\x^8 + x^7 + x^3 + x^2 + x + 1 $
    $ [ 98, 86, 4]_2 $ W $ \langle\alpha (x^2+ x + 1)+x^8 + x^7 + x^6 + x^3 + \\1 , x^{10} + x^8 + x^6 + x^4 + x^2 + x + 1\rangle $ $ x^{49} + x^{48} + x^{46} + x^{45} + x^{43} + x^{39} + \\x^{36} + x^{31} + x^{27} + x^{25} + x^{24} +x^{22} + \\x^{21} + x^{20} + x^{19} + x^{18} + x^{17} + x^{14} + \\x^7 + x^6 + x^3 + x + 1 $
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    Table 3.  Optimal quantum codes $ [[n, k, d]]_4 $ from self-dual/self-orthogonal/dual-containing binary linear codes obtained from quaternary additive codes

    $ [[n, k, d]]_4 $ Map $ \langle \alpha g_1 +g_2, b\rangle $ Multinomial
    $ [[ 7, 1, 3]]_4 $ T $ \langle\alpha (x^3+x^2 + 1)+ x, x^4 + x^3 + x^2 + x + 1 \rangle $ $ x^7 + x^4 + x^3 + x + 1 $
    $ [[ 10, 8, 2]]_4 $ T $ \langle\alpha (x + 1)+ 1, x^2 + x + 1\rangle $ $ x^{10} + x^9 + x^7 + x^5 + x^4 + x^2 + x + 1 $
    $ [[ 15, 7, 3]]_4 $ T $ \langle\alpha (x^4 + x + 1)+ x^2 + x + 1, x^4 + x^3 + 1 \rangle $ $ x^{15} + x^{10} + x^9 + x^7 + x^6 + x^5 + \\x^3 + x^2 + 1 $
    $ [[ 30, 26, 2]]_4 $ T $ \langle\alpha (x^2 + x + 1)+ x^8 + x^6 + 1, x^{12} + x^{11} + \\x^8 + x^7 + x^5 + x^4 + x^2 + x + 1 \rangle $ $ x^{30} + x^{29} + x^{27} + x^{25} + x^{24} + x^{23} + \\x^{22} + x^{21} + x^{20} + x^{17} + x^{13} + x^{11} + \\x^{10} + x^9 + x^6 + x^5 + x^2 + x + 1 $
    $ [[ 35, 29, 2]]_4 $ L $ \langle\alpha ( x + 1)+ x + 1, x^3 + x + 1 \rangle $ $ x^{35} + x^{34} + x^{33} + x^{31} + x^{27} + x^{25} + \\x^{24} + x^{22} + x^{20} + x^{18} + x^{17} + x^{16} + \\x^{15} + x^{11} + x^5 + x^4 + x + 1 $
    $ [[ 36, 30, 2]]_4 $ W $ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $ $ x^{18} + x^{16} + x^{13} + x^{11} + x^9 + x^8 + \\x^7 + x^4 + x^3 + 1 $
    $ [[ 48, 42, 2]]_4 $ W $ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $ $ x^{24} + x^{22} + x^{16} + x^{15} + x^{14} + x^{13} + \\x^{11} + x^{10} + x^7 + x^6 + x^4 + 1 $
    $ [[ 56, 50, 2]]_4 $ L $ \langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle $ $ x^{56} + x^{51} + x^{49} + x^{48} + x^{47} + x^{41} + \\x^{38} + x^{36} + x^{31} + x^{30} + x^{29} + x^{28} + \\x^{26} + x^{18} + x^{16} + x^{15} + x^{14} + x^{12} + \\x^{10} + x^9 + x^5 + x^2 + x + 1 $
    $ [[ 72, 66, 2]]_4 $ W $ \langle\alpha ( x + 1)+ 1, x^2 + x + 1 \rangle $ $ x^{36} + x^{35} + x^{32} + x^{31} + x^{30} + x^{29} + \\x^{26} + x^{22} + x^{21} + x^{20} + x^{18} +x^{16} + \\x^{13} + x^{10} + x^9 + x^2 + x + 1 $
    $ [[ 84, 76, 2]]_4 $ W $ \langle\alpha ( x + 1)+ 1, x^3 + x + 1 \rangle $ $ x^{42} + x^{41} + x^{40} + x^{39} + x^{37} + x^{35} + \\x^{33} + x^{30} + x^{29} + x^{28} + x^{27} +x^{25} + \\x^{21} + x^{15} + x^{14} + x^{13} + x^{11} + x^{10} + \\x^9 + x^6 + x^5 + x^2 + x + 1 $
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  • [1] T. AbualrubN. Aydin and I. Aydogdu, Optimal binary codes derived from $\mathbb{F}_{2}\mathbb{F}_{4}$-additive cyclic codes, J. Appl. Math. Comput., 64 (2020), 71-87.  doi: 10.1007/s12190-020-01344-5.
    [2] A. AlahmadiS. T. DoughertyA. Leroy and P. Sol$\acute{e }$, On the duality and direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 923-929.  doi: 10.3934/amc.2016049.
    [3] A. R. CalderbankE. M. RainsP. W. Shor and N. J. A. Sloane, Quantum error correction via codes ${\mathbb{F}_{4}}$, IEEE Trans. Inf. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.
    [4] M. Grassl, Bounds on the minimum distance of linear codes, Available from: http://www.codetables.de/.
    [5] W. C. Huffman, Additive codes over $\mathbb{F}_{4}$, Adv. Math. Commun., 2 (2008), 309-343.  doi: 10.3934/amc.2008.2.309.
    [6] M. Matsuoka, Polynomial realization of sequential codes over finite fields, SUT J. Math., 48 (2012), 47-53. 
    [7] S. R. L$\acute{o}$pez-PermouthB. R. Parra-Avila and S. Szabo, Dual generalizations of the concept of cyclicity of codes, Adv. Math. Commun., 3 (2009), 227-234.  doi: 10.3934/amc.2009.3.227.
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