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Characterization of completely regular codes through p-polynomial association schemes

  • Josep Rifà
  • Llorenç Huguet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 307)

Abstract

Given a projective linear code C(n,k) with s nonzero weights we know that if C is a uniformly packed code, or more generally if C is a completely regular code, then the covering radius
of C is equal to s. Furthermore, the restriction of the Hamming scheme H(n,q) to C is an association shceme and its dual association scheme is isomorphic to its Ω-scheme, an association scheme defined on the columns of the generator matrix of C.
On the other hand there exist projective linear codes C whose restriction of H(n,q) is an association scheme although its orthogonal code C is not completely regular. In other words, there exist projective linear codes C with
= s but C is not completely regular.
In this paper we present two main results:
  1. 1.
    — C is a completely regular code if and only if
    = s and the restriction of H(n,q) on C is an association scheme.
     
  2. 2.

    — C is a completely regular code if and only if the Θ-scheme is a P-polynomial association scheme.

     

Keywords

Linear Code Association Scheme Dual Code Weight Enumerator Orthogonal Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    DELSARTE P.: "An Algebraic Approach to the Association Schemes of Coding Theory", Philips Research Rep. Supp. 10 (1975).Google Scholar
  2. [2]
    DELSARTE P.: "Four fundamental parameters of a code and their combinatorial significance", Information and control 23, 407–438. (1973).Google Scholar
  3. [3]
    GOETHALS J.M. & VAN TILBORG H.C.A.: "Uniformly packed codes" Philips Research Reports 30, 9–36. (1975).Google Scholar
  4. [4]
    RIFA J.: "Equivalències entre estructures combinatòricament regulars: Esquemes, grafs i codis". Tesi doctoral. Universitat Autònoma de Barcelona. 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Josep Rifà
    • 1
  • Llorenç Huguet
    • 1
  1. 1.Dpt. d'Informàtica. Fac. CiènciesUniversitat Autònoma de BarcelonaCatalunya

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