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Designs, Codes and Cryptography

Erschienen in:

01.06.2015

Construction of minimal non-abelian left group codes

verfasst von: Gabriela Olteanu, Inneke Van Gelder

Erschienen in: Designs, Codes and Cryptography | Ausgabe 3/2015

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Abstract

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.

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Artin E.: Galoissche Theorie. Verlag Harri Deutsch, Zurich (1973).

Baart R., Boothby T., Cramwinckel J., Fields J., Joyner D., Miller R., Minkes E., Roijackers E., Ruscio L., Tjhai C.: GUAVA—a GAP package.Version 3.12 (21/05/2012).http://www.gap-system.org/Packages/guava.html.

Bernal J., del Río Á., Simón J.: An intrinsical description of group codes. Des. Codes Cryptogr. 51(3), 289–300 (2009).

Broche O., Herman A., Konovalov A., Olivieri A., Olteanu G., del Río Á., Van Gelder I.: Wedderga—Wedderburn Decomposition of Group Algebras. Version 4.6.0 (2013). http://www.cs.st-andrews.ac.uk/alexk/wedderga, http://www.gap-system.org/Packages/wedderga.html.

Broche O., del Río Á.: Wedderburn decomposition of finite group algebras. Finite Fields Appl. 13(1), 71–79 (2007).

Brouwer A.: Bounds on the size of linear codes. In: Pless V.S., Huffman W. (eds.) Handbook of Coding Theory, chap. 4, pp. 295–461. Elsevier, Amsterdam (1998).

Cheng Y., Sloane N.: Codes from symmetry groups and a [32,17,8] code. SIAM J. Discret. Math. 2, 28–37 (1989).

Gao S.: Normal bases over finite fields. Ph.D. thesis, University of Waterloo, Waterloo (1993).

The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.6.5 (2013). http://www.gap-system.org.

10.

García Pillado C., González S., Martinéz 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—Reducing the Abstract to the Concrete, Algorithms and Computation in Mathematics, vol. 19, pp. 287–313. Springer, Heidelberg (2006).

12.

Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. Online available at http://www.codetables.de (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), 57–62 (2003).

14.

Jespers E., del Río Á., Olteanu G., Van Gelder I.: Group rings of finite strongly monomial groups: central units and primitive idempotents. J. Algebra 387, 99–116 (2013).

15.

Lenstra H.W.: Finding isomorphisms between finite fields. Math. Comput. 56(193), 329–347 (1991).

16.

Lüneburg H.: On a little but useful algorithm. In: Proceedings of the 3rd International Conference on Algebraic Algorithms and Error-Correcting Codes, pp. 296–301. Springer, London (1986).

17.

Olivieri A., del Río Á., Simón J.: On monomial characters and central idempotents of rational group algebras. Commun. Algebra 32(4), 1531–1550 (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), 157–165 (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), 191–210 (1995).

23.

Yamada T.: The Schur Subgroup of the Brauer Group. Lecture Notes in Mathematics, vol. 397. Springer, Berlin (1973).

Titel: Construction of minimal non-abelian left group codes
verfasst von: Gabriela Olteanu
Inneke Van Gelder
Publikationsdatum: 01.06.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 3/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9922-z

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