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New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm

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  • In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $ k^{th} $-range neighbours, and search for binary $ [72, 36, 12] $ self-dual codes. In particular, we present six generator matrices of the form $ [I_{36} \ | \ \tau_6(v)], $ where $ I_{36} $ is the $ 36 \times 36 $ identity matrix, $ v $ is an element in the group matrix ring $ M_6(\mathbb{F}_2)G $ and $ G $ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $ [72, 36, 12] $ self-dual codes directly over the finite field $ \mathbb{F}_2 $. We construct 1471 new Type I binary $ [72, 36, 12] $ self-dual codes with the rare parameters $ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $ in their weight enumerators.

    Mathematics Subject Classification: Primary: 94B05, 16S34, 68W50.

    Citation:

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  • Figure 1.  Flowchart of the Neighbourhood-Virus Optimisation Algorithm

    Table 1.  New Type I $ [72, 36, 12] $ Codes from $ \mathcal{G}_i $

    Generator Matrix $ r_{A_1} $ $ r_{A_2} $ $ r_{A_3} $ $ r_{A_4} $ $ r_{A_5} $ $ r_{A_6} $ $ \gamma $ $ \beta $ $ |Aut(C_i)| $
    $ \mathcal{G}_1 $ $ (1, 1, 1, 1, 1, 1) $ $ (0, 1, 0, 0, 0, 0) $ $ (1, 1, 1, 0, 1, 0) $ $ (0, 0, 0, 1, 1, 0) $ $ (1, 0, 1, 0, 1, 1) $ $ (1, 1, 1, 0, 0, 1) $ $ 0 $ $ 165 $ $ 72 $
    $ \mathcal{G}_2 $ $ (1, 0, 0, 0, 1, 1) $ $ (0, 0, 0, 0, 0, 1) $ $ (0, 1, 0, 1, 0, 0) $ $ (0, 1, 0, 0, 1, 1) $ $ (1, 1, 1, 1, 1, 0) $ $ (1, 0, 0, 1, 0, 1) $ $ 0 $ $ 315 $ $ 72 $
    $ \mathcal{G}_3 $ $ (1, 1, 0, 1, 0, 0) $ $ (1, 0, 1, 1, 1, 1 ) $ $ (1, 0, 0, 1, 1, 0) $ $ (1, 0, 0, 1, 1, 1) $ $ (0, 0, 1, 1, 0, 1 ) $ $ (0, 0, 0, 1, 1, 1) $ $ 0 $ $ 255 $ $ 36 $
    $ \mathcal{G}_4 $ $ (0, 1, 0, 1, 1, 1 ) $ $ (1, 1, 0, 1, 1, 0) $ $ (1, 1, 1, 0, 1, 0 ) $ $ (1, 0, 0, 1, 1, 1) $ $ ( 0, 0, 1, 0, 1, 0) $ $ (1, 1, 0, 0, 1, 0) $ $ 0 $ $ 309 $ $ 72 $
    $ \mathcal{G}_5 $ $ (1, 1, 0, 1, 0, 0) $ $ (0, 0, 0, 1, 0, 1 ) $ $ (1, 1, 0, 1, 0, 0) $ $ (1, 0, 1, 0, 0, 1 ) $ $ (1, 1, 0, 0, 0, 1 ) $ $ (1, 1, 0, 0, 1, 0) $ $ 36 $ $ 537 $ $ 72 $
    $ \mathcal{G}_6 $ $ (1, 0, 0, 0, 0, 0 ) $ $ (0, 0, 1, 1, 0, 1) $ $ (1, 0, 0, 1, 1, 1) $ $ (1, 1, 0, 0, 0, 1 ) $ $ (0, 1, 1, 0, 0, 1 ) $ $ (1, 0, 0, 1, 1, 0 ) $ $ 0 $ $ 231 $ $ 36 $
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  • [1] W. BosmaJ. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.
    [2] J. R. Cuevas, H. J. Wang, Y. C. Lai and Y. C. Liang, Virus optimization algorithm: A novel metaheuristic for solving continuous optimization problems, The 10th Asia Pacific Industrial Engineering Management System Conference, (2009), 2166–2174.
    [3] S. T. Dougherty, The neighbor graph of binary self-dual codes, Des. Codes and Cryptogr., 90 (2022), 409-425.  doi: 10.1007/s10623-021-00985-2.
    [4] S. T. Dougherty, J. Gildea and A. Kaya, Quadruple bordered constructions of self-dual codes from group rings over frobenius rings, Cryptogr. Commun., 12, (2020), 127–146. doi: 10.1007/s12095-019-00380-8.
    [5] S. T. DoughertyJ. GildeaA. Korban and A. Kaya, Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68, Adv. Math. Commun., 14 (2020), 677-702.  doi: 10.3934/amc.2020037.
    [6] S. T. DoughertyJ. GildeaA. Korban and A. Kaya, New extremal self-dual binary codes of length 68 via composite construction, $\mathbb{F}_2+u\mathbb{F}_2$ lifts, extensions and neighbors, Int. J. Inf. Coding Theory, 5 (2018), 211-226. 
    [7] S. T. DoughertyJ. GildeaA. Korban and A. Kaya, Composite matrices from group rings, composite $G$-codes and constructions of self-dual codes, Des. Codes Cryptogr., 89 (2021), 1615-1638.  doi: 10.1007/s10623-021-00882-8.
    [8] S. T. DoughertyJ. GildeaA. KorbanA. KayaA. Tylyshchak and B. Yildiz, Bordered constructions of self- dual codes from group rings and new extremal binary self-dual codes, Finite Fields Appl., 57 (2019), 108-127.  doi: 10.1016/j.ffa.2019.02.004.
    [9] S. T. DoughertyJ. GildeaR. Taylor and A. Tylshchak, Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.
    [10] S. T. DoughertyT. A. Gulliver and M. Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.  doi: 10.1109/18.641574.
    [11] S. T. DoughertyJ.-L. Kim and P. Solé, Double circulant codes from two class association schemes, Adv. Math. Commun., 1 (2007), 45-64.  doi: 10.3934/amc.2007.1.45.
    [12] S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Group matrix ring codes and constructions of self-dual codes, Applicable Algebra in Engineering, Communication and Computing, 2021. doi: 10.1007/s00200-021-00504-9.
    [13] J. Gildea, A. Kaya, A. Korban and B. Yildiz, New extremal binary self-dual codes of length 68 from generalized neighbours, Finite Fields Appl., 67 (2020), 101727, 12 pp. doi: 10.1016/j.ffa.2020.101727.
    [14] J. GildeaA. KayaR. Taylor and B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl., 51 (2018), 71-92.  doi: 10.1016/j.ffa.2018.01.002.
    [15] T. A. Gulliver and M. Harada, On double circulant doubly-even self-dual $[72, 36, 12]$ codes and their neighbors, Austalas. J. Comb., 40 (2008), 137-144. 
    [16] A. Korban, All known Type I and Type II $[72, 36, 12]$ binary self-dual codes, available online at https://sites.google.com/view/adriankorban/binary-self-dual-codes.
    [17] A. Korban, S. Sahinkaya and D. Ustun, A novel genetic search scheme based on nature - Inspired evolutionary algorithms for self-dual codes, arXiv: 2012.12248, in submission.
    [18] A. Korban, S. Sahinkaya and D. Ustun, New singly and doubly even binary $[72, 36, 12]$ self-dual codes from $M_2(R)G$- group matrix rings, Finite Fields Appl., 76 (2021), Paper No. 101924, 20 pp. doi: 10.1016/j.ffa.2021.101924.
    [19] A. Korban, S. Sahinkaya and D. Ustun, An application of the virus optimization algorithm to the problem of finding extremal binary self-dual codes, arXiv: 2103.07739, in submission.
    [20] A. Korban, S. Sahinkaya and D. Ustun, New extremal binary self-dual codes of length 72 from composite group matrix rings and the neighbour method integrated to the virus optimisation algorithm, in submission.
    [21] A. Korban, S. Sahinkaya and D. Ustun, New type i binary $[72, 36, 12]$ self-dual codes from composite matrices and $R_1$ lifts, Advances in Mathematics of Communications, 2021. doi: 10.3934/amc.2021034.
    [22] A. Korban, S. Sahinkaya and D. Ustun, Generator matrices for the manuscript "New type I binary $[72, 36, 12]$ self-dual codes from $M_6(\mathbb{F}_2)G$ - Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm", available online at https://sites.google.com/view/serap-sahinkaya/generator-matrices.
    [23] E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inf. Theory, 44 (1998), 134-139.  doi: 10.1109/18.651000.
    [24] M. ShiD. HuangL. Sok and P. Solé, Double circulant self-dual and LCD codes over Galios rings, Adv. Math. Commun., 13 (2019), 171-183.  doi: 10.3934/amc.2019011.
    [25] M. ShiL. Qian and P. Solé, On self-dual negacirculant codes of index two and four, Des. Codes Cryptogr., 86 (2018), 2485-2494.  doi: 10.1007/s10623-017-0455-0.
    [26] M. ShiL. SokP. Solé and S. Çalkavur, Self-dual codes and orthogonal matrices over large finite fields, Finite Fields Appl., 54 (2018), 297-314.  doi: 10.1016/j.ffa.2018.08.011.
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