Isometry classes of indecomposable linear codes

  • Harald Fripertinger
  • Adalbert Kerber
Submitted Contributions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 948)


In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian. We describe these classes as orbits and we demonstrate how they can be enumerated using cycle index polynomials. The necessary tools are already incorporated in SYMMETRICA, a (public domain) computer algebra package devoted to representation theory and combinatorics of symmetric groups and of related classes of groups. Moreover, we describe how systems of representatives of these classes can be evaluated using double coset methods.


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  1. 1.
    N.G. De Bruijn. Pólya's Theory of Counting. In E.F. Beckenbach, editor, Applied Combinatorial Mathematics, chapter 5, pages 144–184. Wiley, New York, 1964.Google Scholar
  2. 2.
    L.E. Dickson. Linear Groups. Dover Publications, Inc., New York, 1958.Google Scholar
  3. 3.
    B. Elspas. The Theory of Autonomous Linear Sequential Networks. IRE Transactions on Circuit Theory, CT-6:45–60, 1959.Google Scholar
  4. 4.
    H. Fripertinger. Enumeration of isometry-classes of linear (n, k)-codes over GF(q) in Symmetrica. Bayreuther Math. Schriften, 49:215–223, 1995.Google Scholar
  5. 5.
    H. Fripertinger. Cycle indices of linear, affine and projective groups. To be published.Google Scholar
  6. 6.
    J.A. Green. The characters of the finite general linear groups. Trans. Amer. Math. Soc., 80:402–447, 1955.Google Scholar
  7. 7.
    J.W.P. Hirschfeld. Projective Geometries over Finite Fields. Clarendon Press, Oxford, 1979. ISBN 0-19-853526-0.Google Scholar
  8. 8.
    A. Kerber. Algebraic Combinatorics via Finite Group Actions. B. I. Wissenschaftsverlag, Mannheim, Wien, Zürich, 1991. ISBN 3-411-14521-8.Google Scholar
  9. 9.
    J.P.S. Kung. The Cycle Structure of a Linear Transformation over a Finite Field. Linear Algebra and its Applications, 36:141–155, 1981.Google Scholar
  10. 10.
    H. Lehmann. Das Abzähltheorem der Exponentialgruppe in gewichteter Form. Mitteilungen aus dem Mathem. Seminar Giessen, 112:19–33, 1974.Google Scholar
  11. 11.
    H. Lehmann. Ein vereinheitlichender Ansatz für die REDFIELD — PÓLYA — de BRUIJNSCHE Abzähltheorie. PhD thesis, Universität Giessen, 1976.Google Scholar
  12. 12.
    R. Lidl and H. Niederreiter. Finite Fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, London, Amsterdam, Don Mills — Ontario, Sydney, Tokyo, 1983. ISBN 0-201-13519-1.Google Scholar
  13. 13.
    G. Pólya. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Mathematica, 68:145–254, 1937.Google Scholar
  14. 14.
    D. Slepian. Some Further Theory of Group Codes. The Bell System Technical Journal, 39:1219–1252, 1960.Google Scholar
  15. 15.
    St. Weinrich. Konstruktionsalgorithmen für diskrete Strukturen und ihre Implementierung. Master's thesis, Universität Bayreuth, July 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Harald Fripertinger
    • 1
  • Adalbert Kerber
    • 2
  1. 1.Institut für MathematikKarl-Franzens-Univ. GrazGraz
  2. 2.Lehrstuhl II für MathematikUniv. BayreuthBayreuth

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