Construction of minimal non-abelian left group codes

详细信息    查看全文

• 作者：Gabriela Olteanu ; Inneke Van Gelder
• 关键词：Left group codes ; Linear codes ; Primitive idempotents ; Wedderburn decomposition ; Finite group algebras ; 94B05 ; 16S34 ; 20C05
• 刊名：Designs, Codes and Cryptography
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：75
• 期：3
• 页码：359-373
• 全文大小：261 KB
• 参考文献：1.Artin E.: Galoissche Theorie. Verlag Harri Deutsch, Zurich (1973).
  2.Baart R., Boothby T., Cramwinckel J., Fields J., Joyner D., Miller R., Minkes E., Roijackers E., Ruscio L., Tjhai C.: GUAVA鈥攁 GAP package.Version 3.12 (21/05/2012).http://鈥媤ww.鈥媑ap-system.鈥媜rg/鈥婸ackages/鈥媑uava.鈥媓tml .
  3.Bernal J., del R铆o 脕., Sim贸n J.: An intrinsical description of group codes. Des. Codes Cryptogr. 51(3), 289鈥?00 (2009).
  4.Broche O., Herman A., Konovalov A., Olivieri A., Olteanu G., del R铆o 脕., Van Gelder I.: Wedderga鈥擶edderburn Decomposition of Group Algebras. Version 4.6.0 (2013). http://鈥媤ww.鈥媍s.鈥媠t-andrews.鈥媋c.鈥媢k/鈥媋lexk/鈥媤edderga , http://鈥媤ww.鈥媑ap-system.鈥媜rg/鈥婸ackages/鈥媤edderga.鈥媓tml .
  5.Broche O., del R铆o 脕.: Wedderburn decomposition of finite group algebras. Finite Fields Appl. 13(1), 71鈥?9 (2007).
  6.Brouwer A.: Bounds on the size of linear codes. In: Pless V.S., Huffman W. (eds.) Handbook of Coding Theory, chap. 4, pp. 295鈥?61. Elsevier, Amsterdam (1998).
  7.Cheng Y., Sloane N.: Codes from symmetry groups and a [32,17,8] code. SIAM J. Discret. Math. 2, 28鈥?7 (1989).
  8.Gao S.: Normal bases over finite fields. Ph.D. thesis, University of Waterloo, Waterloo (1993).
  9.The GAP Group: GAP鈥擥roups, Algorithms, and Programming, Version 4.6.5 (2013). http://鈥媤ww.鈥媑ap-system.鈥媜rg .
  10.Garc铆a Pillado C., Gonz谩lez S., Martin茅z C., Markov V., Nechaev A.: Group codes over non-abelian groups. J. Algebra Appl. 12(7), 1350037 (20 pages) (2013).
  11.Grassl M.: Searching for linear codes with large minimum distance. In: Bosma W., Cannon J. (eds.) Discovering Mathematics with Magma鈥擱educing the Abstract to the Concrete, Algorithms and Computation in Mathematics, vol. 19, pp. 287鈥?13. Springer, Heidelberg (2006).
  12.Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://鈥媤ww.鈥媍odetables.鈥媎e (2007). Accessed 15 Sept 2013.
  13.Jespers E., Leal G., Paques A.: Central idempotents in the rational group algebra of a finite nilpotent group. J. Algebra Appl. 2(1), 57鈥?2 (2003).
  14.Jespers E., del R铆o 脕., Olteanu G., Van Gelder I.: Group rings of finite strongly monomial groups: central units and primitive idempotents. J. Algebra 387, 99鈥?16 (2013).
  15.Lenstra H.W.: Finding isomorphisms between finite fields. Math. Comput. 56(193), 329鈥?47 (1991).
  16.L眉neburg H.: On a little but useful algorithm. In: Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes, pp. 296鈥?01. Springer, London (1986).
  17.Olivieri A., del R铆o 脕., Sim贸n J.: On monomial characters and central idempotents of rational group algebras. Commun. Algebra 32(4), 1531鈥?550 (2004).
  18.Olteanu G., Van Gelder I.: Finite group algebras of nilpotent groups: a complete set of orthogonal primitive idempotents. Finite Fields Appl. 17(2), 157鈥?65 (2011).
  19.Passman D.: Infinite Crossed Products. Pure and Applied Mathematics, vol. 135. Academic Press, Boston (1989).
  20.Reiner I.: Maximal Orders. Academic Press, London (1975).
  21.Roman S.: Field Theory, Graduate Texts in Mathematics, vol. 158. Springer, New York (2006).
  22.Sabin R., Lomonaco S.: Metacyclic error-correcting codes. Appl. Algebra Eng. Commun. Comput. 6(3), 191鈥?10 (1995).
  23.Yamada T.: The Schur Subgroup of the Brauer Group. Lecture Notes in Mathematics, vol. 397. Springer, Berlin (1973).
  
• 作者单位：Gabriela Olteanu (1)
  Inneke Van Gelder (2)
  
  1. Department of Statistics-Forecasts-Mathematics, Babe艧-Bolyai University, Str. T. Mihali 58-60, 400591, Cluj-Napoca, Romania
  2. Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Coding and Information Theory
  Data Structures, Cryptology and Information Theory
  Data Encryption
  Discrete Mathematics in Computer Science
  Information, Communication and Circuits
  
• 出版者：Springer Netherlands
• ISSN：1573-7586

文摘

Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra \({\mathbb F}G\) for a large class of groups \(G\). As an illustration of our methods, alternative constructions to some best linear codes over \({\mathbb F}_2\) and \({\mathbb F}_3\) are given. Furthermore, we give constructions of non-abelian left group codes.