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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)

Abstract

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 

Notes

Acknowledgements

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.

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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

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