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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 $D_8$

Martino Borello 1, , Francesca Dalla Volta 1, and  Gabriele Nebe 2,

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 $8$.
Keywords: Extremal self-dual code, Automorphism group..
Mathematics Subject Classification: Primary: 94B05, 20B2.
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:
[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.  Google Scholar

[2]

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.  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), 7240-7245. 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), 235-265. 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), 5-13. 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), 544-547. 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), 1351-1367. 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), 6916-6924. 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), 511-521. 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), 188-200. 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), 563-566. 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), 134-139. 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), 251.  Google Scholar

show all references

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), 4743-4750. doi: 10.1109/TIT.2008.928976.  Google Scholar

[2]

E. F. Assmuss and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151. 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), 7240-7245. 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), 235-265. 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), 5-13. 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), 544-547. 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), 1351-1367. 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), 6916-6924. 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), 511-521. 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), 188-200. 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), 563-566. 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), 134-139. 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), 251.  Google Scholar

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