Research and Innovation Articles
Published 2019-02-19
Keywords
- code,
- Hamming distance,
- weight,
- group algebra,
- ideal
- group code ...More
How to Cite
Polcino Milies, C. (2019). Group algebras and coding theory: a short survey. Revista Integración, Temas De matemáticas, 37(1), 153–166. https://doi.org/10.18273/revint.v37n1-2019008
Abstract
We study codes constructed from ideals in group algebras and we are particularly interested in their dimensions and weights. First we introduced a special kind of idempotents and study the ideals they generate.
We use this information to show that there exist abelian non-cyclic groups that give codes which are more convenient than the cyclic ones. Finally, we discuss briefly some facts about non-abelian codes.
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References
[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.