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Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 173–182 | Cite as

Self-dual codes better than the Gilbert–Varshamov bound

  • Alp Bassa
  • Henning Stichtenoth
Article

Abstract

We show that every self-orthogonal code over \({\mathbb {F}}_q\) of length n can be extended to a self-dual code, if there exists self-dual codes of length n. Using a family of Galois towers of algebraic function fields we show that over any nonprime field \({\mathbb {F}}_q\), with \(q\ge 64\), except possibly \(q=125\), there are infinite families of self-dual codes, which are asymptotically better than the asymptotic Gilbert–Varshamov bound.

Keywords

Self-dual codes Algebraic geometry codes Gilbert–Varshamov Bound Tsfasman–Vladut–Zink Bound Towers of function fields Asymptotically good codes Quadratic forms Witt’s Theorem 

Mathematics Subject Classification

14G50 94B27 94B65 15A63 11T71 

References

  1. 1.
    Bassa A., Beelen P., Garcia A., Stichtenoth H.: Galois towers over non-prime finite fields. Acta Arith. 164, 163–179 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    MacWilliams F.J., Sloane N.J.A., Thompson J.G.: Good self-dual codes exist. Discret. Math. 3, 153–162 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Pless V., Pierce J.N.: Self-dual codes over GF(q) satisfy a modied Varshamov–Gilbert bound. Inf. Control 23, 35–40 (1973).CrossRefzbMATHGoogle Scholar
  4. 4.
    Pless V.: On Witt’s theorem for nonalternating symmetric bilinear forms over a field of characteristic 2. Proc. Am. Math. Soc. 15, 979–983 (1964).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Pless V.: On the uniqueness of the Golay codes. J. Comb. Theory 5, 215–228 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Serre J.-P.: A Course in Arithmetic. Springer Verlag, New York-Heidelberg (1973).CrossRefzbMATHGoogle Scholar
  7. 7.
    Stichtenoth H.: Self-dual Goppa codes. J. Pure Appl. Algebra 55(1–2), 199–211 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Stichtenoth H.: Algebraic Function Fields and Codes, Graduate Texts in Mathematics 254. Springer-Verlag, Berlin (2009).Google Scholar
  9. 9.
    Stichtenoth H.: Transitive and self-dual codes attaining the Tsfasman-Vlăduţ-Zink bound. IEEE Trans. Inform. Theory 52(5), 2218–2224 (2006).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesBoğaziçi UniversityBebekTurkey
  2. 2.Sabancı University, MDBFTuzlaTurkey

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