On the Existence of Hermitian Self-Dual Extended Abelian Group Codes

  • Lilibeth Dicuangco-ValdezEmail author
  • Pieter Moree
  • Patrick Solé
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 115)


Split group codes are a class of group algebra codes over an abelian group. They were introduced by Ding et al. (IEEE Trans. Inform. Theory IT-46:485–495, 2000) as a generalization of the cyclic duadic codes. For a prime power q and an abelian group G of order n such that gcd(n, q) = 1, consider the group algebra \(\mathbb{F}_{q^{2}}[G^{{\ast}}]\) of \(\mathbb{F}_{q^{2}}\) over the dual group G of G. We prove that every ideal code in \(\mathbb{F}_{q^{2}}[G^{{\ast}}]\) whose extended code is Hermitian self-dual is a split group code. We characterize the orders of finite abelian groups G for which an ideal code of \(\mathbb{F}_{q^{2}}[G^{{\ast}}]\) whose extension is Hermitian self-dual exists and derive asymptotic estimates for the number of non-isomorphic abelian groups with this property.

Mathematics Subject Classification

11N64 94B05 11N37 



The first author gratefully acknowledges financial support from the University of the Philippines and from the Philippine Council for Advanced Science and Technology Research and Development through the Department of Science and Technology.

The second author would like to thank Alexander Ivić for pointing out reference [16] to him.

The second and the third author like to thank Bernhard Heim for the invitation to participate in the Automorphic Forms conference in Oman. Heim proved again that he is a Grandmaster in conference organization.


  1. 1.
    L. Dicuangco, P. Moree, P. Solé, The lengths of Hermitian self-dual extended duadic codes. J. Pure Appl. Algebra 1, 223–237 (2007)CrossRefGoogle Scholar
  2. 2.
    C. Ding, D.R. Kohel, S. Ling, Split group codes. IEEE Trans. Inform. Theory IT-46, 485–495 (2000)CrossRefMathSciNetGoogle Scholar
  3. 3.
    S.R. Finch, Mathematical Constants, Encyclopedia of Mathematics and Its Applications, vol. 94 (Cambridge University Press, Cambridge, 2003)Google Scholar
  4. 4.
    G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (The Clarendon Press, Oxford University Press, New York, 1979)zbMATHGoogle Scholar
  5. 5.
    A. Ivić, On the number of abelian groups of a given order and on certain related multiplicative functions. J. Number Theory 16, 119–137 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    E. Krätzel, Die maximale Ordnung der Anzahl der wesentlich verschiedenen abelschen Gruppen n-ter Ordnung. Quart. J. Math. Oxford Ser. 21, 273–275 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J.S. Leon, J.M. Masley, V.S. Pless, Duadic codes. IEEE Trans. Inform. Theory IT-30(5), 709–714 (1984)CrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Moree, Artin’s primitive root conjecture: a survey. Integers 12A, A13, 100 pp (2012)Google Scholar
  9. 9.
    P. Moree, J. Cazaran, On a claim of Ramanujan in his first letter to Hardy. Expos. Math. 17, 289–311 (1999)zbMATHMathSciNetGoogle Scholar
  10. 10.
    J.-L. Nicolas, On highly composite numbers, in Ramanujan Revisited (Urbana-Champaign, Ill., 1987) (Academic, Boston, 1988), pp. 215–244Google Scholar
  11. 11.
    R.W.K. Odoni, A problem of Rankin on sums of powers of cusp-form coefficients. J. Lond. Math. Soc. 44(2), 203–217 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    V. Pless, Q-codes. J. Combin. Theory Ser. A 43, 258–276 (1986)Google Scholar
  13. 13.
    A.G. Postnikov, Introduction to Analytic Number Theory. Translations of Mathematical Monographs, vol. 68 (AMS, Providence, 1988)Google Scholar
  14. 14.
    S. Ramanujan, Collected Papers (Chelsea, New York, 1962)Google Scholar
  15. 15.
    J.J. Rushanan, Duadic codes and difference sets. J. Combin. Theory Ser. A 57, 254–261 (1991)Google Scholar
  16. 16.
    O. Robert, P. Sargos, Three-dimensional exponential sums with monomials. J. Reine Angew. Math. 591, 1–20 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    W. Schwarz, E. Wirsing, The maximal number of non-isomorphic abelian groups of order n. Arch. Math. (Basel) 24, 59–62 (1973)Google Scholar
  18. 18.
    M. Smid, Duadic codes. IEEE Trans. Inform. Theory IT-33(3), 432–433 (1987)CrossRefMathSciNetGoogle Scholar
  19. 19.
    H.N. Ward, Quadratic residue codes and divisibility, in Handbook of Coding Theory, ed. by V.S. Pless, W.C. Huffman (Elsevier Science, Amsterdam, 1998), pp. 827–870Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Lilibeth Dicuangco-Valdez
    • 1
    Email author
  • Pieter Moree
    • 2
  • Patrick Solé
    • 3
  1. 1.Institute of MathematicsUniversity of the PhilippinesDiliman, Quezon CityPhilippines
  2. 2.Max-Planck-InstitutBonnGermany
  3. 3.Department of Comelec, CNRSLTCI, Telecom- ParistechParisFrance

Personalised recommendations