Códigos autoduales con un automorfismo de orden primo impar

Self-Dual Codes with an Automorphism of Odd Prime Order

En este artículo presentamos un resumen de algunos de los resultados mas importantes sobre códigos lineales binarios y autoduales con un automorfismo de orden primo impar que se han establecido en los últimos años. Además por medio de un automorfismo de orden 59 construimos 24 nuevos [120; 60]-códigos binarios autoduales, doblemente pares optimales.

In this paper we present a survey of the most important results on binary self-dual linear codes with an automorphism of odd prime order, which have been established in recent years. Additionally, through an automorphism of order 59, we show that there are at least 24 new binary, self dual doubly even optimal [120; 60]-codes.

Códigos autoduales con un automorfismo de orden primo impar Self-Dual Codes with an Automorphism of Odd Prime Order JAVIER DE LA CRUZ¹, ISMAEL GUTIERREZ², JORGE ROBINSON³

¹Universidad del Norte, Barranquilla, Colombia. Email: jdelacruz@uninorte.edu.co
²Universidad del Norte, Barranquilla, Colombia. Email: isgutier@uninorte.edu.co
³Universidad del Atlántico, Barranquilla, Colombia. Email: jorgerobinson@mail.uniatlantico.edu.co

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Resumen

En este artículo presentamos un resumen de algunos de los resultados más importantes sobre códigos lineales binarios y autoduales con un automorfismo de orden primo impar que se han establecido en los últimos años. Además por medio de un automorfismo de orden 59 construimos 24 nuevos [120,60]-códigos binarios autoduales, doblemente pares optimales.

Palabras clave: Códigos lineales, códigos binarios, códigos autoduales, códigos doblemente pares, códigos extremales, códigos optimales, automorfismos de códigos.

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2000 Mathematics Subject Classification: 53C21, 53C42.

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Abstract

In this paper we present a survey of the most important results on binary self-dual linear codes with an automorphism of odd prime order, which have been established in recent years. Additionally, through an automorphism of order 59, we show that there are at least 24 new binary, self dual doubly even optimal [120,60]-codes.

Key words: Linear codes, Binary codes, Self dual codes, Doubly even codes, Extremal codes, Automorphisms.

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Referencias

[1] J. L. Alperin and R. B. Bell, Groups and Representations, 2 edn, Springer-Verlag, New York, USA, 1995.

[2] E. F. Assmus, H. F. Mattson, and R. Turyn, 'Research to Develop the Algebraic Theory of Codes', Air force Cambridge Res. Lab., Bedford, MA, Report AFCRL-67-0365, (1967), I1-XI4.

[3] M. Borello, 'The Automorphism Group of a Self-Dual [72,36,16] Code is not an Elementary Abelian Group of Order 8', Finite Fields and Their Applications 25, (2014), 1-7.

[4] S. Bouyuklieva, 'On the Automorphism Group of a Doubly-Even (72, 32, 16) Code', IEEE Transactions on Information Theory 50, (2004), 544-547.

[5] S. Bouyuklieva, J. D. l. Cruz, and W. Willems, 'On the Automorphism Group of a Binary Self-Dual [120,60,24] Code', AAECC 24, (2013).

[6] J. H. Conway and N. J. A. Sloane, 'A New Upper Bound on the Minimal Distance of Self-Dual Codes', IEEE Transactions on Information Theory 36, 6 (1990).

[7] J. D. l. Cruz, Über die Automorphismengruppe der Extremaler Codes der Längen 96 Und 120, Ph.D. dissertation, Otto-von-Guericke Universität, Magdeburg - Germany, 2012.

[8] J. D. l. Cruz, 'On Extremal Self-Dual Codes of Length 120', Design, Codes and Cryptography, (2013). (to appear)

[9] J. D. l. Cruz and W. Willems, 'On Extremal Self-Dual Code of Length 9', IEEE Transactions on Information Theory 6, 57 (2011), 6820-6828.

[10] A. M. Gleason, 'Weight Polynomials of Codes and the MacWilliams Identities', Actes Congrès Intern. de Math. Gauthier-Villars, Paris 3, (1971), 211-215.

[11] M. Grassl, Bounds on the Minimum Distance of Linear Codes and Quantum Codes, http://www.codetables.de, (0000).

[12] W. C. Huffman, 'Automorphisms of Codes with Applications to Extremal Doubly Even Codes of Length 48', IEEE Transactions on Information Theory 28, (1982), 511-521.

[13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, UK, 2003.

[14] I. M. Isaacs, Finite Group Theory, Vol. 92, AMS, Providence, Graduate Studies in Math, 2008.

[15] X. Ma, 'Nonexistence of Extremal Doubly Even Self-Dual Codes With Large Length', Discrete Math. 185, (1998), 265-274.

[16] C. L. Mallows and N. J. A. Sloane, 'An Upper Bound for Self-Dual Codes', Information and Control 22, (1973), 188-200.

[17] E. M. Rains, 'Shadow Bounds for Self-Dual Codes', IEEE Transactions on Information Theory 44, (1998), 134-139.

[18] N. J. A. Sloane, 'Is There a [72, 36], D16 Self-Dual Code?', IEEE Transactions on Information Theory 19, (1973), 251.

[19] V. Y. Yorgov, 'Binary Self-Dual Codes with Automorphisms of Odd Order', Probl. Pered. Inform. 19, (1983), 11-24.

[20] V. Yorgov and D. Yorgov, 'The Automorphism Group of a Self Dual Binary [72,36,16] Code does not Contain Z₄', preprint, arXiv:1310.2570v2, 2013.

[21] R. Yorgova and A. Wassermann, 'Binary Self-Dual Codes with Automorphisms of Order 23', Des. Codes and Cryptography 48, (2008), 155-164.

[22] S. Zhang, 'On the Nonexistence of Extremal Self-Dual Codes', Discrete Appl. Math. 91, (1999), 277-286.

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(Recibido en octubre de 2013. Aceptado en junio de 2014)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv48n2a02,
    AUTHOR  = {de la Cruz, Javier and Gutierrez, Ismael and Robinson, Jorge},
    TITLE   = {{Códigos autoduales con un automorfismo de orden primo impar}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2014},
    volume  = {48},
    number  = {2},
    pages   = {135--163}
}

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