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Minimal ideals in finite abelian group algebras and coding theory

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Abstract

We consider abelian groups of order \(p^nq^m\) where p and q are prime rational integers under some restrictive hypotheses and determine the set of primitive idempotents of the group algebra \({{\mathbb {F}}}G\) for a finite field \({{\mathbb {F}}}\). The minimal ideals they generate can be considered as minimal codes and we determine either the respective minimum weights or bounds of these weights. We give examples showing that these bounds are actually attained in some cases.

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References

  1. Bakshi, G.K., Raka, M., Sharma, A.: Idempotent generators of irreducible cyclic codes. In: Number Theory and Discrete Geometry, RMS Lecture Notes Series, No 6, pp. 13–18 (2008)

  2. Bakshi, G.K., Raka, M.: Minimal cyclic codes of lenght \(p^n q\). Finite Fields Appl. 9(4), 432–448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bakshi, G.K., Raka, M.: Corrigendum to minimal cyclic codes of lenght \(p^n q\). Finite Fields Appl. 18, 863 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berman, S.D.: Semisimple cyclic and abelian codes II. Kybernetika 3, 21–30 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Ferraz, R., Polcino Milies, C.: Idempotents in group algebras and minimal abelian codes. Finite Fields Their Appl. 13, 382–393 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Isaacs, I.M.: Character Theory of Finite Groups. Dover, New York (1994)

    MATH  Google Scholar 

  7. Landau, E.: Elementary Number Theory, 2nd edn. Chelsea Publishing Company, New York (1966)

    Google Scholar 

  8. MacWilliams, F.J.: Binary codes which are ideals in the group algebra of an abelian group. Bell Syst. Tech. J. 49, 987–1011 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  9. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)

    MATH  Google Scholar 

  10. Milies, C.Polcino, Sehgal, S.K.: An Introduction to Group Rings. Kluwer, Dordrecht (2002)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP, CEP 05508-090, Brazil

    Gladys Chalom & Raul Antonio Ferraz

  2. Departamento de Matemática, Universidade Federal de Viçosa, Campus Universitário, Viçosa, MG, CEP 36570-000, Brazil

    Marinês Guerreiro

Authors
  1. Gladys Chalom
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  2. Raul Antonio Ferraz
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  3. Marinês Guerreiro
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Corresponding author

Correspondence to Gladys Chalom.

Additional information

Dedicated to Prof. César Polcino Milies on the occasion of his seventieth birthday.

Research supported by CAPES, PROCAD 915/2010, CNPq, Proc. 300243/79-0(RN), FAPEMIG, APQ CEX 00438-08 and FAPESP, Proc. 09/52665-0 (Brazil).

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Chalom, G., Ferraz, R.A. & Guerreiro, M. Minimal ideals in finite abelian group algebras and coding theory. São Paulo J. Math. Sci. 10, 321–340 (2016). https://doi.org/10.1007/s40863-015-0037-x

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  • DOI: https://doi.org/10.1007/s40863-015-0037-x

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