On authomorphisms of extremal type II codes
ISMAEL GUTIÉRREZ GARCÍAa *, DARWIN VILLAR SALINASb
aUniversidad del Norte, Departamento de Matemátcias y Estadística, Barranquilla, Colombia.
bRWTH-Aachen University, Department of Mathematics, Aachen, Germany.
Abstract.In this article we present some techniques to determine the types of automorphisms of extremal doubly even binary self-dual codes, also called extremal type II codes, with parameters [24, 12, 8], [48, 24, 12] and [120, 60, 24]. We aim to obtain information about the automorphism group considering the exclusion of some prime numbers from its order.
Keywords: Binary codes, self-dual codes, doubly even codes, extremal codes and automorphisms of codes.
MSC2010: 11T71, 20B25, 94B60.
Sobre automorfismos de códigos extremales de tipo II
]]> Resumenen el presente artículo se muestran algunas técnicas para obtener tipos de automorfismos de los códigos binarios auto-duales, doblemente pares y extremales, también denominados extremales de tipo II, con parámetros [24, 12, 8], [48, 24, 12] y [120, 60, 24]. El objetivo central es obtener información sobre el correspondiente grupo de automorfismos a partir de la exclusión de algunos números primos de su orden.Texto Completo disponible en PDF
References
[1] Assmus E. Jr., Mattson H. Jr., and Turyn R., "Research to develop the algebraic theory of codes", Air force Cambridge Res. Lab., Bedford, MA, Report AFCRL-67-0365 (1967). [ Links ]
[2] Borello M., "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), no. 12, 7240-7245. [ Links ]
[3] Borello M. and Willems W., "Automorphism of order 2p in binary self-dual extremal codes of length a multiple of 24", IEEE Trans. Inform. Theory 59 (2013), no. 6, 3378-3383. [ Links ]
]]>[4] Borello M., Dalla Volta F. and Nebe G., "The automorphism group of a self-dual [72, 36, 16] code does not contain S3, A4 or D8", arXiv:1303.4899 [ Links ]
5] Bouyuklieva S., De la Cruz J. and Willems W., "On the automorphism group of a binary self-dual [120, 60, 24] code", Appl. Algebra Engrg. Comput. 24 (2013), no. 3-4, 201-214. [ Links ]
[6] Bouyuklieva S., O'Brien E. and Willems W., "The automorphism group of a binary selfdual doubly-even [72, 36, 16] code is solvable", IEEE Trans. Inform. Theory 52 (2006), no. 2, 4244-4248. [ Links ]
[7] De la Cruz J. and Willems W., "On extremal self-dual codes of Length 96", IEEE Trans. Inform. Theory 57 (2011), no. 10, 6820-6823. [ Links ]
[8] Dontcheva R., "On the doubly-even self-dual codes of length 96", IEEE Trans. Inform. Theory 48 (2002), no. 7, 557-560. [ Links ]
[9] Gleason A.M., "Weight polynomials of self-dual codes and the MacWilliams identities", 1970 Actes du Congrès International des Mathématiciens, vol 3 (1971), 211-215. [ Links ]
[10] Huffman W.C., "Automorphisms of codes with applications to extremal doubly even codes of length 48", IEEE Trans. Inform. Theory 28 (1982), 511-521. [ Links ]
[11] Leon J.S, Masley J.M. and Pless V., "Duadic codes", IEEE Trans. Inform. Theory 30 (1984), 709-714. [ Links ]
[12] MacWilliams F.J. and Sloane N.J.A., The theory of error-correcting codes, North-Holland, Mathematical Library Amsterdam, 1977. [ Links ]
[13] MacWilliams F., Odluzko A., Sloane N. andWard H., "Self-dual codes over GF(4)", Journal of Combinatorial Theory 25A (1978), 288-318. [ Links ]
[14] Mallows C., and Sloane N., "An upper bound for self-dual codes", Information and Control 22 (1973), 188-200. [ Links ]
[15] O'Brien E. and Willems W., "On the automorphism group of a binary self-dual doubly even [72, 36, 16] code", IEEE Trans. Inform. Theory 57 (2011), no. 7, 4445-4451. [ Links ]
[16] Rains E., "Shadow bounds for self-dual codes", IEEE Trans. Inform. Theory 44 (1998), 134-139. [ Links ]
[17] Sloane N., "Is there a (72; 36) d = 16 self-dual code?", IEEE Trans. Inform. Theory IT-19 (1973), no. 2, 251-251. [ Links ]
[18] Yorgov V.Y. and Huffman W.C., "A [72, 36, 16] doubly even code does not have an automorphism of order 11", IEEE Trans. Inform. Theory 33 (1987), no. 5, 749-752. [ Links ]
[19] Zhang S., "On the nonexsitence of extremal self-dual codes", Discrete Appl. Math. 91 (1999), no. 1-3, 277-286. [ Links ]
*Corresponding author: E-mail: isgutier@uninorte.edu.co.
Received: 16 April 2013, Accepted: 02 September 2013.