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New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes

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  • A lift of binary self-dual codes to the ring $R_2$ is described. By using this lift, a family of self-dual codes over $R_2$ of length $17$ are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length $68$ are obtained. For the first time in the literature, extremal binary codes of length $68$ with $\gamma=4$ and $\gamma = 6$ in $W_{68,2}$ have been obtained. In addition to these, six new codes with $\gamma = 0$ and fourteen new codes with $\gamma = 2$ in $W_{68,2}$ have also been found.
    Mathematics Subject Classification: Primary: 94B05; Secondary: 94B99.


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  • [1]

    W. Bosma, J. 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.


    S. Bouyuklieva, Some Optimal self-orthogonal and self-dual codes, J. Discrete Math., 287 (2004), 1-10.doi: 10.1016/j.disc.2004.06.010.


    S. Bouyuklieva and I. Boukliev, Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 44 (1998), 323-328.doi: 10.1109/18.651059.


    S. Bouyuklieva, N. Yankov and J.-L. Kim, Classification of binary self-dual [48,24,10]-codes with an automorphism odd prime order, Finite Fields Appl., 18 (2012), 1104-1113.doi: 10.1016/j.ffa.2012.08.002.


    J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.doi: 10.1109/18.59931.


    S. T. Dougherty, A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Infrom. Theory, 43 (1997), 2036-2047.doi: 10.1109/18.641574.


    S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 4-26.doi: 10.1016/j.ffa.2009.11.003.


    S. T. Dougherty, J.-L. Kim and H. Liu, Constructions of self-dual codes over finite commutative chain rings, J. Inform. Coding Theory, 1 (2010), 171-190.doi: 10.1504/IJICOT.2010.032133.


    S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.doi: 10.1016/j.ffa.2010.11.002.


    S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, European J. Pure Appl. Math., 6 (2013), 89-106.


    P. Gaborit and A. Otmani, Experimental constructions of self-dual codes, Finite Fields Appl., 9 (2003), 372-394.doi: 10.1016/S1071-5797(03)00011-X.


    S. Karadeniz and B. Yildiz$R_2$-generator matrices for extremal self-dual codes of length 68, available online at http://www.fatih.edu.tr/~akaya/NewSelf-dual68.pdf


    S. Karadeniz and B. Yildiz, Double-circulant and double-bordered-circulant constructions for self-dual codes over $R_2$, Adv. Math. Commun., 6 (2012), 193-202.doi: 10.3934/amc.2012.6.193.


    S. Karadeniz and B. YildizNew extremal binary self-dual codes of length $64$ as $R_3$-lifts of the extended binary Hamming code, submitted.


    H. H. Kim, H. Lee, J. B. Lee and Y. Lee, Construction of self-dual codes with an automorpihsm of of order $p$, Adv. Math. Commun., 5 (2011), 23-36.doi: 10.3934/amc.2011.5.23.


    A. MunemasaDatabase of self-dual codes, available online at http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/data/2/18.magma


    T. Nishimura, A new extremal self-dual code of length 64, IEEE Trans. Inform. Theory, 50 (2004), 2173-2174.doi: 10.1109/TIT.2004.833359.


    E. M. Rains, Shadow bounds for self dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.doi: 10.1109/18.651000.


    H. P. Tsai, P. Y. Shih, R. Y. Wuh, W. K. Su and C. H. Chen, Construction of self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 3826-3831.doi: 10.1109/TIT.2008.926454.


    J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.doi: 10.1353/ajm.1999.0024.


    B. Yildiz and S. Karadeniz, Linear codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, Des. Codes Crypt., 54 (2010), 61-81.doi: 10.1007/s10623-009-9309-8.


    B. Yildiz and S. Karadeniz, Self-dual codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, J. Franklin Inst., 347 (2010), 1888-1894.doi: 10.1016/j.jfranklin.2010.10.007.

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