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Generalized quadratic-residue codes

  • J. H. van Lint
Part of the International Centre for Mechanical Sciences book series (CISM)

Abstract

At the 1975 CISM Summer School on Information Theory P. Camion (cf. [1]) introduced “global quadratic abelian codes” which are a generalization of (classical) quadratic residue codes (QR-codes). A year earlier H.N. Ward (cf. [8]) had used symplectic geometry to introduce a generalization of QR-codes. Both presentations rely heavily on abstract algebra. Essentially such codes (at least in the binary case) were introduced by Ph. Del-sarte in 1971 (cf. [2]) as codes generated by the adjacency matrices of finite miquelian inversive planes. Recently J.H. van Lint and F.J. Mac-Williams (cf. [4]) showed that the methods that are used to treat QR-codes can easily be generalized to give a completely analogous treatment of the so-called generalized quadratic residue codes (GQR-codes).

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References

  1. 1.
    Camion, P., Global Quadratic Abelian Codes, in Information Theory, CISM Courses and Lectures 219, G. Longo, ed., Springer-Verlag, Wien 1975.Google Scholar
  2. 2.
    Delsarte, P.-, Majority Logic Decodable Codes Derived from Finite l’nver-sive Planes, Inf. and Control 18, 1971, 319–325.MathSciNetCrossRefGoogle Scholar
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    van Lint J.H., Coding Theory, Lecture Notes in Math. 201, Springer-Verlag, Berlin, 1971.zbMATHGoogle Scholar
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    van Lint J.H. and F.J. MacWilliams, Generalized Quadratic Residue Codes, IEEE Trans, on Information Theory 24 (1978).Google Scholar
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    MacWilliams, F.J. and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.zbMATHGoogle Scholar
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    Sands, A.D., On the Factorization of Finite Abelian Groups, Acta Math. Acad. Sci. Hung. 13, 1962, 153–159.MathSciNetCrossRefzbMATHGoogle Scholar
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    van Tilborg, H.C.A., On Weights in Codes, Report 71-WSK-03, Dept. of Mathematics, Technological University Eindhoven, 1971.zbMATHGoogle Scholar
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    Ward, H.N., Quadratic Residue Codes and Symplectic Groups, J. of Algebra 29, 1974, 150–171.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1979

Authors and Affiliations

  • J. H. van Lint
    • 1
  1. 1.Department of MathematicsEindhoven University of TechnologyThe Netherlands

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