# American Institute of Mathematical Sciences

November  2013, 7(4): 503-510. doi: 10.3934/amc.2013.7.503

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

 1 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, 20125 Milan, Italy, Italy 2 Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

Received  March 2013 Revised  July 2013 Published  October 2013

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 $8$.
Citation: Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
##### References:
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##### References:
 [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.  doi: 10.1109/TIT.2008.928976.  Google Scholar [2] E. F. Assmuss and H. F. Mattson, New $5$-designs,, J. Combin. Theory, 6 (1969), 122.  doi: 10.1016/S0021-9800(69)80115-8.  Google Scholar [3] 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.  doi: 10.1109/TIT.2012.2211095.  Google Scholar [4] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language,, J. Symbol. Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar [5] 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.  doi: 10.1023/A:1012598832377.  Google Scholar [6] S. Bouyuklieva, On the automorphism group of a doubly even $(72,36,16)$ code,, IEEE Trans. Inform. Theory, 50 (2004), 544.  doi: 10.1109/TIT.2004.825252.  Google Scholar [7] 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.  doi: 10.1016/j.jcta.2005.12.004.  Google Scholar [8] 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.  doi: 10.1109/TIT.2012.2208176.  Google Scholar [9] W. C. Huffman, Automorphisms of codes with application to extremal doubly even codes of length $48$,, IEEE Trans. Inform. Theory, IT-28 (1982), 511.  doi: 10.1109/TIT.1982.1056499.  Google Scholar [10] C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes,, Inform. Control, 22 (1973), 188.  doi: 10.1016/S0019-9958(73)90273-8.  Google Scholar [11] 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.  doi: 10.1016/j.ffa.2011.12.001.  Google Scholar [12] E. M. Rains, Shadow bounds for self-dual codes,, IEEE Trans. Inform. Theory, 44 (1998), 134.  doi: 10.1109/18.651000.  Google Scholar [13] N. J. A. Sloane, Is there a $(72; 36)$ $d = 16$ self-dual code?,, IEEE Trans. Inform. Theory, 2 (1973).   Google Scholar
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