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The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$

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  • A computer calculation with $M$AGMA shows that there is no extremal self-dual binary code $\mathcal{C}$ of length $72$ whose automorphism group contains the symmetric group of degree $3$, the alternating group of degree $4$ or the dihedral group of order $8$. Combining this with the known results in the literature one obtains that $Aut(\mathcal{C})$ has order at most $5$ or is isomorphic to the elementary abelian group of order $8$.
    Mathematics Subject Classification: Primary: 94B05, 20B25.


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  • [1]

    C. Aguilar Melchor and P. Gaborit, On the classification of extremal $[36,18,8]$ binary self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4743-4750.doi: 10.1109/TIT.2008.928976.


    E. F. Assmuss and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151.doi: 10.1016/S0021-9800(69)80115-8.


    M. Borello, The automorphism group of a self-dual $[72,36,16]$ binary code does not contain elements of order $6$, IEEE Trans. Inform. Theory, 58 (2012), 7240-7245.doi: 10.1109/TIT.2012.2211095.


    W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbol. Comput., 24 (1997), 235-265.doi: 10.1006/jsco.1996.0125.


    S. Bouyuklieva, On the automorphisms of order $2$ with fixed points for the extremal self-dual codes of length $24m$, Des. Codes Cryptogr., 25 (2002), 5-13.doi: 10.1023/A:1012598832377.


    S. Bouyuklieva, On the automorphism group of a doubly even $(72,36,16)$ code, IEEE Trans. Inform. Theory, 50 (2004), 544-547.doi: 10.1109/TIT.2004.825252.


    L. E. Danielsen and M. G. Parker, On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Combin. Theory Ser. A, 112 (2006), 1351-1367.doi: 10.1016/j.jcta.2005.12.004.


    T. Feulner and G. Nebe, The automorphism group of an extremal $[72,36,16]$ code does not contain $Z_7$, $Z_3\times Z_3$, or $D_{10}$, IEEE Trans. Inform. Theory, 58 (2012), 6916-6924.doi: 10.1109/TIT.2012.2208176.


    W. C. Huffman, Automorphisms of codes with application to extremal doubly even codes of length $48$, IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.doi: 10.1109/TIT.1982.1056499.


    C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.doi: 10.1016/S0019-9958(73)90273-8.


    G. Nebe, An extremal $[72,36,16]$ binary code has no automorphism group containing $Z_2\times Z_4$, $Q_8$, or $Z_{10}$, Finite Fields Appl., 18 (2012), 563-566.doi: 10.1016/j.ffa.2011.12.001.


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


    N. J. A. Sloane, Is there a $(72; 36)$ $d = 16$ self-dual code?, IEEE Trans. Inform. Theory, 2 (1973), 251.

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