Self-Orthogonal Codes and the Topology of Spinor Groups

  • Jay A. Wood
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 20)


Maximal doubly-even self-orthogonal binary linear codes correspond to the maximal elementary abelian 2-groups of the spinor group Spin(n). We will describe the correspondence and discuss various techniques from the algebraic topology of Spin(n) which may be useful in studying self-orthogonal codes. In particular, Quillen’s results in equivariant cohomology theory coupled with some Morse theory may allow one to address certain questions on the minimum weight of doubly-even self-orthogonal codes.

Key words

self-orthogonal codes spinor groups flat connections equivariant cohomology Morse theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Michael F. Atiyah and Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London, A 308 (1982), pp. 523–615.Google Scholar
  2. [2]
    Michael F. Atiyah, Raoul Bott and Arnold Shapiro, Clifford Modules, Topology, 3(Supp 1) (1964), pp. 3–38.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Armand Borel, Seminar on Transformation Groups, Annals of Mathematics Studies 46, Princeton University Press, Princeton, N. J., 1960.zbMATHGoogle Scholar
  4. [4]
    Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Mathematical Journal, 13 (1961), pp. 216–240.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Armand Borel, On the p-rank of compact connected Lie groups, preprint 1987.Google Scholar
  6. [6]
    Armand Borel and Jean-Pierre Serre, Sur certains sous-groupes des groupes de Lie compacts, Commentarii Mathematici Helvetici, 27 (1953), pp. 128–139.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Raoul Bott, Nondegenerate critical manifolds, Annals of Mathematics, 60 (1954), pp. 248–261.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Raoul Bott, The stable homotopy of the classical groups, Annals of Mathematics, 70 (1959), pp. 313–337.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Theodor Bröcker and Tammo tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics 98, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985.zbMATHGoogle Scholar
  10. [10]
    John H. Conway and Vera Pless, On the enumeration of self-dual codes, Journal of Combinatorial Theory, Series A, 28 (1980), pp. 26–53.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Simon K. Donaldson, An application of gauge theory to the topology of 4-manifolds, Journal of Differential Geometry, 18 (1983), pp. 279–315.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Theodore Prankel, Critical submanifolds of the classical groups and Stiefel manifolds, in Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, Stewart S. Cairns, editor, Princeton University Press, Princeton, N. J., 1965, pp. 37–53.Google Scholar
  13. [13]
    Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications 1, Springer-Verlag, New York, Heidelberg, Berlin, 1984.zbMATHGoogle Scholar
  14. [14]
    Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics 80, Academic Press, New York, San Francisco, London, 1978.zbMATHGoogle Scholar
  15. [15]
    Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Interscience Tracts in Pure and Applied Mathematics 15, Interscience Publishers, New York, London, Sydney, 1963/1969.zbMATHGoogle Scholar
  16. [16]
    Saunders MacLane, Homology, Grundlehren der mathematischen Wissenschaften 114, Springer- Verlag, Berlin, Göttingen, Heidelberg, 1963.zbMATHGoogle Scholar
  17. [17]
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Mathematical Library 16, North-Holland, Amsterdam, New York, Oxford, 1978.Google Scholar
  18. [18]
    Haynes Miller, Stable splittings of Stiefel manifolds, Topology, 24 (1985), pp. 411–419.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    John W. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, Princeton, N. J., 1969.Google Scholar
  20. [20]
    John W. Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, Princeton, N. J., 1974.zbMATHGoogle Scholar
  21. [21]
    Vera Pless, A classification of self-orthogonal codes over GF(2), Discrete Mathematics, 3 (1972), pp. 209–246.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Vera Pless and N. J. A. Sloane, On the classification and enumeration of self-dual codes, Journal of Combinatorial Theory, Series A, 18 (1975), pp. 313–335.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Daniel Quillen, The spectrum of an equivariant cohomology ring I,II, Annals of Mathematics, 94 (1971), pp. 549–602.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Daniel Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Mathematische Annalen, 194 (1971), pp. 197–212.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, N. J., 1951.zbMATHGoogle Scholar
  26. [26]
    George W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics 61, Springer-Verlag, New York, Heidelberg, Berlin, 1978.zbMATHGoogle Scholar
  27. [27]
    Jay A. Wood, Spinor groups and algebraic coding theory, Journal of Combinatorial Theory, Series A (to appear).Google Scholar
  28. [28]
    Jay A. Wood, Flat connections, spinor groups and error-correcting codes, submitted to the Proceedings of the 1988 Northwestern International Conference on Algebraic Topology.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1990

Authors and Affiliations

  • Jay A. Wood
    • 1
  1. 1.Department of MathematicsBowdoin CollegeBrunswickAustralia

Personalised recommendations