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

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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

Abstract
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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 $8$.

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. 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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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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