Tokyo Journal of Mathematics

On Some Properties of the Hyper-Kloosterman Codes

Koji CHINEN

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Abstract

The hyper-Kloosterman code was first defined as a trace code by Chinen-Hiramatsu [1]. In this article, two basic parameters of it, the minimum distance and the dimension are estimated. Analysis of the dimension shows that it is one of few examples of trace codes, of which the dimensions do not reduce when taking the trace, and are determined explicitly. It is also shown that the hyper-Kloosterman code can be realized as a quasi-cyclic code. It implies a method of explicit construction of quasi-cyclic codes of a new type.

Article information

Source
Tokyo J. Math., Volume 26, Number 1 (2003), 55-65.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208682

Digital Object Identifier
doi:10.3836/tjm/1244208682

Mathematical Reviews number (MathSciNet)
MR1981999

Zentralblatt MATH identifier
1041.11082

Subjects
Primary: 11T71: Algebraic coding theory; cryptography
Secondary: 11T23: Exponential sums 94B40: Arithmetic codes [See also 11T71, 14G50]

Citation

CHINEN, Koji. On Some Properties of the Hyper-Kloosterman Codes. Tokyo J. Math. 26 (2003), no. 1, 55--65. doi:10.3836/tjm/1244208682. https://projecteuclid.org/euclid.tjm/1244208682


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References

  • K. Chinen and T. Hiramatsu, Hyper-Kloosterman sums and their applications to the coding theory, Appl. Algebra Engrg. Comm. Comput., 12 (2001), 381–390.
  • T. Hiramatsu, Uniform distribution of the weights of the Kloosterman codes, SUT J. Math., 31 (1995), 29–32.
  • G. Lachaud, Distribution of the weights of the dual of the Melas code, Discrete Math., 79 (1989/90), 103–106.
  • D. H. and E. Lehmer, The cyclotomy of hyper-Kloosterman sums, Acta Arith., 14 (1968), 89–111.
  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North Holland (1977).
  • C. M. Melas, A cyclic code for double error correction, IBM J. Res. Devel., 4 (1960), 364–366.
  • Séminair de géométrie algébrique du Bois-marie SGA4$\frac12$, Lecture Notes in Math. 569, Springer (1977).
  • H. Stichtenoth, Algebraic Function Fields and Codes, Springer (1993).
  • R. L. Townsend and E. L. Weldon, Jr., Self-orthogonal quasi-cyclic codes, IEEE Trans. Inform. Theory, IT-13 No. 2 (1967), 183–195.
  • J. Wolfmann, The weights of the dual code of the Melas code over GF(3), Discrete Math., 74 (1989), 327–329. \endthebibliography