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Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.


idempotent generated idempotent group ring code binary abelian group ring.

Article Details

How to Cite
Ong, K. L., & Ang, M. H. (2017). Full Identification of Idempotens in Binary Abelian Group Rings. Journal of the Indonesian Mathematical Society, 23(2), 67–75.


  1. Berman, S.D., On the Theory of Group Codes, Kibernetika, 3 (1967), 31-39.
  2. Fu,W. and Feng, T., "On Self-orthogonal Group Ring Codes", Designs, Codes and Cryptog-
  3. raphy, 50 (2009), 203-214.
  4. Hurley, B. and Hurley, T., "Systems of MDS Codes from Units and Idempotents", Discrete
  5. Mathematics, 335 (2014), 81-91.
  6. Hurley, P. and Hurley, T., "Codes from Zero-divisors and Units in Group Rings", Int. J.
  7. Information and Coding Theory, 1 (2009), 57-87.
  8. Jitman, S., Ling, S., Liu, H. W. and Xie, X., "Abelian Codes in Principal Ideal Group
  9. Algebras", IEEE Transactions on Information Theory, 59 (2013), 3046-3058.
  10. McLoughlin, I. and Hurley, T., "A Group Ring Construction of The Extended Binary Golay
  11. Code", IEEE Transactions in Information Theory, 54 (2008), 4381-4383.
  12. Ong, K.L., Ang, M.H., "Study of Idempotents in Cyclic Group Rings over F2", AIP Confer-
  13. ence Proceedings, 1739, 020011 (2016), doi: 10.1063/1.4952491.
  14. Tan, Z. S., Ang, M. H. and Teh, W. C., "Group Ring Codes over a Dihedral Group", Accepted
  15. by Malaysian Journal of Mathematical Sciences, (2015)
  16. Wong, Denis C.K. and Ang, M.H., "Group Algebra Codes Dened over Extra Special p-
  17. group", Journal of Algebra, Number Theory and Applications, 78 (2013), 19-27.
  18. Wong, Denis C.K. and Ang, M.H., "A family of MDS abelian group Codes", Far East Journal
  19. of Mathematical Sciences, 78 (2013), 19-27.
  20. Wong, Denis C.K. and Ang, M.H., "Group Codes Dene Over Dihedral Groups of Small
  21. Order", Malaysian Journal of Mathematical Sciences, 7(S) (2013), 101-116.