Artículo Original
Publicado 2019-02-19
Palabras clave
- códigos,
- distancia de Hamming,
- peso,
- álgebra de grupo,
- ideal
- código de grupo ...Más
Cómo citar
Polcino Milies, C. (2019). Álgebras de grupo y teoría de códigos: una breve reseña. Revista Integración, Temas De matemáticas, 37(1), 153–166. https://doi.org/10.18273/revint.v37n1-2019008
Resumen
Estudiamos códigos construidos a partir de ideales de álgebras de grupo y estamos particularmente interesados en sus dimensiones y pesos. Introducimos inicialmente un tipo especial de idempotentes y estudiamos los ideales que generan. Usamos esta información para mostrar que existen grupos abelianos no cíclicos que son más convenientes que los cíclicos. Finalmente, discutimos brevemente algunos resultados sobre códigos no abelianos.
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Referencias
[1] Arora S.K., Pruthi M., “Minimal cyclic codes of length 2p^n”, Finite Fields Appl. 5 (1999), No. 2, 177–187.
[2] Assuena S. and Polcino Milies C., “Good codes from metacyclic groups”, Contemporary Math., to appear.
[3] Berlekamp E.R., Key papers in the development of Coding Theory, I.E.E.E. Press, New York, 1974.
[4] Berman S.D., “On the theory of group codes”, Kibernetika 3 (1967), No. 1, 31–39.
[5] Berman S.D., “Semisimple cyclic and abelian codes II”, Kibernetika 3 (1967), No. 3, 17–23.
[6] Bernhardt F., Landrock P. and Manz O., “The extended Golay codes considered as ideals”, J. Combin. Theory Ser. A 55 (1990), No. 2, 235–246.
[7] Chalom G., Ferraz R. and Polcino Milies C., “Essential idempotents and simplex codes”, J. Algebra Comb. Discrete Struct. Appl. 4 (2017), No. 2, 181–188.
[8] Charpin P., “The Reed-Solomon code as ideals in a modular algebra”, C.R. Acad. Sci. Paris, Ser. I. Math. 294 (1982), 597–600.
[9] Dougherty S., Gildea J., Taylor R. and Tylyshchak A., “Group rings, G-codes and constructions of self-dual and formally self-dual codes”, Des. Codes Cryptogr. 86 (2018), No. 9, 2115–2138.
[10] Drensky V. and Lakatos P., “Monomial ideals, group algebras and error correcting codes”, in Lect. Notes in Comput. Sci. 257, Springer, Berlin (1989), 181–188.
[11] Dutra F.S., Ferraz R.A., Polcino Milies C., “Semisimple group codes and dihedral codes”, Algebra Discrete Math. (2009), No. 3, 28–48.
[12] Ferraz R., Guerreiro M. and Polcino Milies C., “G-equivalence in group algebras and minimal abelian codes”, IEEE Trans. on Inform. Theory 60 (2014), No. 1, 252–260.
[13] Ferraz R. and Polcino Milies C., “Idempotents in group algebras and minimal abelian codes”, Finite Fields Appl. 13 (2007), No. 2, 382–393.
[14] Ferraz R., Polcino Milies C. and Taufer E., “Left ideals in matrix rings over finite fields”, preprint, arXiv:1711.09289.
[15] García Pillado C., González S., Martínez C., Markov V. and Nechaev A., “Group codes over non-abelian groups”, J. Algebra Appl. 12 (2013), No. 7, 20 pp.
[16] Grassl M., Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de/BKLC/index.html [21 December 2018]
[17] Hamming R.W., “Error-detecting and error-correcting codes”, Bell System Tech. J. 29 (1950), No. 2, 147–160.
[18] Hamming R.W., Interview, February 1977.
[19] Keralev A. and Solé P., “Error-correcting codes as ideals in group rings”, in Contemporary. Math. 273, Amer. Math. Soc. (2001), 11–18.
[20] Landrock P. and Manz O., “Classical codes as ideals in group algebras”, Des. Codes Cryptogr. 2 (1992), No. 3, 273–285.
[21] MacWilliams F.J., “Binary codes which are ideals in the group algebra of an abelian group”, Bell System Tech. J. 49 (1970), 987–1011.
[22] Polcino Milies C. and de Melo F., “On Cyclic and Abelian Codes”, IEEE Trans. Inform. Theory 59 (2013), No. 11, 7314–7319.
[23] Polcino Milies C. and Sehgal S.K., An introduction to group rings, Algebras and Applications, Kluwer Academic Publishers, Dortrecht, 2002.
[24] Poli A., “Codes dans les algebras de groups abelienes (codes semisimples, et codes modulaires)”, in Information Theory (Proc. Internat. CNRS Colloq., Cachan 1977) Colloq. Internat. CNRS 276 (1978), 261–271.
[25] Prange E., Cyclic error-correcting codes in two symbols, AFCRC-TN-57-103, USAF, Cambridge Research Laboratories, New York, 1957.
[26] Pruthi M. and Arora S.K., “Minimal codes of prime power length”, Finite Fields Appl. 3 (1997), No. 2, 99–113.
[27] Sabin R.E., “On determining all codes in semi-simple group rings”, in Lecture Notes in Comput. Sci. 673, Springer (1993), 279–290.
[28] Sabin R.E. and Lomonaco S.J., “Metacyclic Error-correcting Codes”, Appl. Algebra Engrg. Comm. Comput. 6 (1995), No. 3, 191–210.
[29] Shannon C.E., “A mathematical theory of communication”, Bell System Tech. J. 27 (1948), 379–423.
[30] Thompson T.M., From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, Washington, 1983.
[2] Assuena S. and Polcino Milies C., “Good codes from metacyclic groups”, Contemporary Math., to appear.
[3] Berlekamp E.R., Key papers in the development of Coding Theory, I.E.E.E. Press, New York, 1974.
[4] Berman S.D., “On the theory of group codes”, Kibernetika 3 (1967), No. 1, 31–39.
[5] Berman S.D., “Semisimple cyclic and abelian codes II”, Kibernetika 3 (1967), No. 3, 17–23.
[6] Bernhardt F., Landrock P. and Manz O., “The extended Golay codes considered as ideals”, J. Combin. Theory Ser. A 55 (1990), No. 2, 235–246.
[7] Chalom G., Ferraz R. and Polcino Milies C., “Essential idempotents and simplex codes”, J. Algebra Comb. Discrete Struct. Appl. 4 (2017), No. 2, 181–188.
[8] Charpin P., “The Reed-Solomon code as ideals in a modular algebra”, C.R. Acad. Sci. Paris, Ser. I. Math. 294 (1982), 597–600.
[9] Dougherty S., Gildea J., Taylor R. and Tylyshchak A., “Group rings, G-codes and constructions of self-dual and formally self-dual codes”, Des. Codes Cryptogr. 86 (2018), No. 9, 2115–2138.
[10] Drensky V. and Lakatos P., “Monomial ideals, group algebras and error correcting codes”, in Lect. Notes in Comput. Sci. 257, Springer, Berlin (1989), 181–188.
[11] Dutra F.S., Ferraz R.A., Polcino Milies C., “Semisimple group codes and dihedral codes”, Algebra Discrete Math. (2009), No. 3, 28–48.
[12] Ferraz R., Guerreiro M. and Polcino Milies C., “G-equivalence in group algebras and minimal abelian codes”, IEEE Trans. on Inform. Theory 60 (2014), No. 1, 252–260.
[13] Ferraz R. and Polcino Milies C., “Idempotents in group algebras and minimal abelian codes”, Finite Fields Appl. 13 (2007), No. 2, 382–393.
[14] Ferraz R., Polcino Milies C. and Taufer E., “Left ideals in matrix rings over finite fields”, preprint, arXiv:1711.09289.
[15] García Pillado C., González S., Martínez C., Markov V. and Nechaev A., “Group codes over non-abelian groups”, J. Algebra Appl. 12 (2013), No. 7, 20 pp.
[16] Grassl M., Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de/BKLC/index.html [21 December 2018]
[17] Hamming R.W., “Error-detecting and error-correcting codes”, Bell System Tech. J. 29 (1950), No. 2, 147–160.
[18] Hamming R.W., Interview, February 1977.
[19] Keralev A. and Solé P., “Error-correcting codes as ideals in group rings”, in Contemporary. Math. 273, Amer. Math. Soc. (2001), 11–18.
[20] Landrock P. and Manz O., “Classical codes as ideals in group algebras”, Des. Codes Cryptogr. 2 (1992), No. 3, 273–285.
[21] MacWilliams F.J., “Binary codes which are ideals in the group algebra of an abelian group”, Bell System Tech. J. 49 (1970), 987–1011.
[22] Polcino Milies C. and de Melo F., “On Cyclic and Abelian Codes”, IEEE Trans. Inform. Theory 59 (2013), No. 11, 7314–7319.
[23] Polcino Milies C. and Sehgal S.K., An introduction to group rings, Algebras and Applications, Kluwer Academic Publishers, Dortrecht, 2002.
[24] Poli A., “Codes dans les algebras de groups abelienes (codes semisimples, et codes modulaires)”, in Information Theory (Proc. Internat. CNRS Colloq., Cachan 1977) Colloq. Internat. CNRS 276 (1978), 261–271.
[25] Prange E., Cyclic error-correcting codes in two symbols, AFCRC-TN-57-103, USAF, Cambridge Research Laboratories, New York, 1957.
[26] Pruthi M. and Arora S.K., “Minimal codes of prime power length”, Finite Fields Appl. 3 (1997), No. 2, 99–113.
[27] Sabin R.E., “On determining all codes in semi-simple group rings”, in Lecture Notes in Comput. Sci. 673, Springer (1993), 279–290.
[28] Sabin R.E. and Lomonaco S.J., “Metacyclic Error-correcting Codes”, Appl. Algebra Engrg. Comm. Comput. 6 (1995), No. 3, 191–210.
[29] Shannon C.E., “A mathematical theory of communication”, Bell System Tech. J. 27 (1948), 379–423.
[30] Thompson T.M., From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, Washington, 1983.