Primitive idempotents in group algebras and minimal abelian codes

Original Research
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Abstract

In this paper, the minimal abelian codes for several classes of non-cyclic abelian groups are constructed by explicitly determining a complete set of primitive idempotents in the corresponding group algebras. Furthermore, we acquire the minimum Hamming distances and the dimensions of these minimal abelian codes in the group algebra \({\mathbb {F}}_q(G_{\ell _1} \, \times \, G_{\ell _2})\), where \(G_{\ell _1} \, \times \, G_{\ell _2}\) is the direct product of two cyclic groups and \(\ell _1, \, \ell _2\) are prime divisors of \(q-1\).

Keywords

Finite field Primitive idempotent Abelian code 

Mathematics Subject Classification

17C27 94B60 
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Notes

Acknowledgments

The author is very grateful to the reviewers and the editor for their valuable comments and suggestions to improve the quality of this paper.

References

  1. 1.
    Arora, S.K., Batra, S., Cohen, S.D., Pruthi, M.: The primitive idempotents of a cyclic group algebra. Southeast Asian Bull. Math. 26, 197–208 (2002)MathSciNetGoogle Scholar
  2. 2.
    Arora, S.K., Pruthi, M.: Minimal cyclic codes of length \(2p^n\). Finite Fields Appl. 5, 177–187 (1999)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Batra, S., Arora, S.K.: Some cyclic codes of length \(2p^n\). Des. Codes Cryptogr. 61, 41–69 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bakshi, G.K., Raka, M.: Minimal cyclic codes of length \(p^nq\). Finite Fields Appl. 9, 432–448 (2003)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Berman, S.D.: Semiple cyclic and abelian codes. II. Kybernetika 3, 21–30 (1967)Google Scholar
  6. 6.
    Berman, S.D.: On the theory of group codes. Kybernetika 3, 31–39 (1967)Google Scholar
  7. 7.
    Chabanne, H.: Permutation decoding of abelian codes. IEEE Trans. Inf. Theory 38, 1826–1829 (1992)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, B., Liu, H., Zhang, G.: A class of minimal cyclic codes over finite fields. Des. Codes Cryptogr. (2013). doi: 10.1007/s10623-013-9857-9 Google Scholar
  9. 9.
    Jitman, S., Ling, S., Liu, H., Xie, X.: Abelian codes in principal ideal group algebra. IEEE Trans. Inf. Theory 59, 3046–3058 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (2008)MATHGoogle Scholar
  11. 11.
    Macwilliams, F.J.: Binary codes which are ideals in the group algebra of an abelian group. Bell Syst. Tech. J. 44, 987–1011 (1970)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pruthi, M.: Cyclic codes of length \(2^m\). Proc. Indian Acad. Sci. Math. Sci. 111, 371–379 (2001)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Pruthi, M., Arora, S.K.: Minimal cyclic codes of prime power length. Finite Fields Appl. 3, 99–113 (1997)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rajan, B.S., Siddiqi, M.U.: Transform domain characterization of abelian codes. IEEE Trans. Inf. Theory 38(6), 1817–1821 (1992)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Sabin, R.E.: On determining all codes in semi-simple group rings. Lecture Notes Comput. Sci. 673, 279–290 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Singh, R., Pruthi, M.: Primitive idempotents of irreducible quadratic residue cyclic codes of length \(p^nq^m\). Int. J. Algebra 5, 285–294 (2011)MathSciNetGoogle Scholar
  17. 17.
    Wan, Z.: Lectures on Finite Fields and Galois Rings. World Scientific Publishing, Singapore (2003)MATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZaozhuang UniversityZaozhuangPeople’s Republic of China

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