Designs, Codes and Cryptography

, Volume 24, Issue 1, pp 99–122 | Cite as

The Invariants of the Clifford Groups

  • Gabriele Nebe
  • E. M. Rains
  • N. J. A. Sloane


The automorphism group of the Barnes-Wall lattice Lm in dimension 2 m (m ≠ 3) is a subgroup of index 2 in a certain “Clifford group” \(\mathcal{C}_m\) of structure 2 + 1+2m . O+(2m,2). This group and its complex analogue \(\mathcal{X}_m\) of structure \((2_ + ^{1 + 2m} YZ_8 )\).Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for \(\mathcal{C}_m\) of degree 2k is spanned by the complete weight enumerators of the codes \(C \otimes \mathbb{F}_{2^m }\), where C ranges over all binary self-dual codes of length 2k; these are a basis if m ≥ k - 1. We also give new constructions for Lm and \(\mathcal{C}_m\): let M be the \(\mathbb{Z}[\sqrt 2 ]\)-lattice with Gram matrix \(\left[ {\begin{array}{*{20}c} 2 & {\sqrt 2 } \\ {\sqrt 2 } & 2 \\ \end{array} } \right]\). Then Lm is the rational part of M⊗ m, and \(\mathcal{C}_m\) = Aut(M⊗m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then \(\mathcal{C}_m\) is precisely the automorphism group of the complete weight enumerator of \(C \otimes \mathbb{F}_{2^m }\). There are analogues of all these results for the complex group \(\mathcal{X}_m\), with “doubly-even self-dual code” instead of “self-dual code.”

Clifford groups Barnes-Wall lattices spherical designs invariants self-dual codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Bannai, S. T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, IEEE Trans. Inform. Theory, Vol. 45 (1999) pp. 1194–1205.Google Scholar
  2. 2.
    E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Australian Math. Soc., Vol. 1 (1959) pp. 47–63.Google Scholar
  3. 3.
    H.-J. Bartels, Zur Galoiskohomologie definiter arithmetischer Gruppen, J. Reine Angew. Math., Vol. 298 (1978) pp. 89–97.Google Scholar
  4. 4.
    C. H. Bennett, D. DiVincenzo, J. A. Smolin and W. K. Wootters, Mixed state entanglement and quantum error correction, Phys. Rev. A, Vol. 54 (1996) pp. 3824–3851.Google Scholar
  5. 5.
    D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge University Press (1993).Google Scholar
  6. 6.
    H. F. Blichfeldt, Finite Collineation Groups, University of Chicago Press, Chicago (1917).Google Scholar
  7. 7.
    S. Böcherer, Siegel modular forms and theta series, in Theta functions (Bowdoin 1987), Proc. Sympos. Pure Math., 49, Part 2, Amer. Math. Soc., Providence, RI, (1989), pp. 3–17.Google Scholar
  8. 8.
    B. Bolt, The Clifford collineation, transform and similarity groups III: Generators and involutions, J. Australian Math. Soc., Vol. 2 (1961) pp. 334–344.Google Scholar
  9. 9.
    B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups I, J. Australian Math. Soc., Vol. 2 (1961) pp. 60–79.Google Scholar
  10. 10.
    B. Bolt, T. G. Room and G. E. Wall, On Clifford collineation, transform and similarity groups II, J. Australian Math. Soc., Vol. 2 (1961) pp. 80–96.Google Scholar
  11. 11.
    R. Brauer, Über endliche lineare Gruppen von Primzahlgrad, Math. Annalen, Vol. 169 (1967) pp. 73–96Google Scholar
  12. 12.
    M. Broué and M. Enguehard, Une famille infinie de formes quadratiques entières; leurs groupes d'automorphismes, Ann. scient. Éc. Norm. Sup., 4e série, 6 (1973), 17–52. Summary in C. R. Acad. Sc. Paris, Vol. 274 (1972) pp. 19–22.Google Scholar
  13. 13.
    A. R. Calderbank, P. J. Cameron, W. M. Kantor and J. J. Seidel, ℤ4-Kerdock codes, orthogonal spreads and extremal Euclidean line-sets, Proc. London Math. Soc., Vol. 75 (1997) pp. 436–480.Google Scholar
  14. 14.
    A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor and N. J. A. Sloane, A group-theoretic framework for the construction of packings in Grassmannian spaces, J. Algebraic Combin., Vol. 9 (1999) pp. 129–140.Google Scholar
  15. 15.
    A. R. Calderbank, E. M. Rains, P.W. Shor and N. J. A. Sloane, Quantum error correction orthogonal geometry, Phys. Rev. Letters, Vol. 78 (1997) pp. 405–409.Google Scholar
  16. 16.
    A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, Vol. 44 (1998) pp. 1369–1387.Google Scholar
  17. 17.
    J. H. Conway, R. H. Hardin and N. J. A. Sloane, Packing lines, planes, etc.: packings in Grassmannian space, Experimental Math., Vol. 5 (1996) pp. 139–159.Google Scholar
  18. 18.
    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer-Verlag, New York (1998).Google Scholar
  19. 19.
    A. W. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Mathematics, Vol. 102 (1975) pp. 291–325.Google Scholar
  20. 20.
    W. Duke, On codes and Siegel modular forms, Intern. Math. Res. Notices, Vol. 5 (1993) pp. 125–136.Google Scholar
  21. 21.
    W. Feit, On finite linear groups in dimension at most 10, in Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Academic Press, New York (1976) pp. 397–407.Google Scholar
  22. 22.
    A. Fröhlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ. Press (1991).Google Scholar
  23. 23.
    S. P. Glasby, On the faithful representations, of degree 2n, of certain extensions of 2-groups by orthogonal and symplectic groups. J. Australian Math. Soc. Ser. A, Vol. 58 (1995) pp. 232–247.Google Scholar
  24. 24.
    A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, in Actes, Congr´es International de Mathématiques (Nice, 1970), Gauthiers-Villars, Paris, Vol. 3 (1971) pp. 211–215.Google Scholar
  25. 25.
    J.-M. Goethals and J. J. Seidel, The football, Nieuw Archief voor Wiskunde, Vol. 29 (1981) pp. 50–58. Reprinted in Geometry and Combinatorics: Selected Works of J. J. Seidel, (D. G. Corneil and R. Mathon, ed.) Academic Press (1991) pp. 363–371.Google Scholar
  26. 26.
    W. C. Huffman and D. B. Wales, Linear groups of degree nine with no elements of order seven. J. Algebra, Vol. 51 (1978) pp. 149–163.Google Scholar
  27. 27.
    B. Huppert, Endliche Gruppen I, Springer-Verlag (1967).Google Scholar
  28. 28.
    L. S. Kazarin, On certain groups defined by Sidelnikov (in Russian), Mat. Sb., Vol. 189 (No. 7, 1998) pp. 131–144; English translation in Sb. Math., Vol. 189 (1998) pp. 1087–1100.Google Scholar
  29. 29.
    A. Y. Kitaev, Quantum computations: algorithms and error correction (in Russian), Uspekhi Mat. Nauk., Vol. 52 (No. 6, 1997) pp. 53–112; English translation in Russian Math. Surveys, Vol. 52 (1997) pp. 1191–1249.Google Scholar
  30. 30.
    P. B. Kleidman and M.W. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge Univ. Press (1988).Google Scholar
  31. 31.
    J. H. Lindsey II, Finite linear groups of prime degree, Math. Annalen, Vol. 189 (1970) pp. 47–59.Google Scholar
  32. 32.
    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes North-Holland, Amsterdam (1977).Google Scholar
  33. 33.
    G. Nebe, Finite quaternionic matrix groups, Representation Theory, Vol. 2 (1998) pp. 106–223.Google Scholar
  34. 34.
    G. Nebe, E. M. Rains and N. J. A. Sloane, A simple construction for the Barnes-Wall lattices, in Forney Festschrift, (R. Blahut, eds.) to appear (2001).Google Scholar
  35. 35.
    G. Nebe, E. M. Rains and N. J. A. Sloane, Generalized self-dual codes and Clifford-Weil groups, preprint.Google Scholar
  36. 36.
    M. Oura, The dimension formula for the ring of code polynomials in genus 4, Osaka J. Math., Vol. 34 (1997) pp. 53–72.Google Scholar
  37. 37.
    V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, San Diego (1994).Google Scholar
  38. 38.
    E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory, ed. V. Pless and W. C. Huffman, Elsevier, Amsterdam (1998) pp. 177–294.Google Scholar
  39. 39.
    B. Runge, On Siegel modular forms I, J. Reine Angew. Math., Vol. 436 (1993) pp. 57–85.Google Scholar
  40. 40.
    B. Runge, On Siegel modular forms II, Nagoya Math. J., Vol. 138 (1995) pp. 179–197.Google Scholar
  41. 41.
    B. Runge, The Schottky ideal, in Abelian Varieties (Egloffstein, 1993), de Gruyter, Berlin (1995) pp. 251–272.Google Scholar
  42. 42.
    B. Runge, Codes and Siegel modular forms, Discrete Math., Vol. 148 (1996) pp. 175–204.Google Scholar
  43. 43.
    J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag (1977).Google Scholar
  44. 44.
    P. W. Shor and N. J. A. Sloane, A family of optimal packings in Grassmannian manifolds, J. Algebraic Combin., Vol. 7 (1998) pp. 157–163.Google Scholar
  45. 45.
    V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform., (1997),Vol. 33 pp. 33 35–54 (1997); English translation in Problems Inform. Transmission, Vol. 33 (1997) pp. 29–44.Google Scholar
  46. 46.
    V. M. Sidelnikov,On a finite group of matrices generating orbit codes on the Euclidean sphere, in Proceedings IEEE Internat. Sympos. Inform. Theory, Ulm, 1997, IEEE Press (1997) p. 436.Google Scholar
  47. 47.
    V. M. Sidelnikov, Spherical 7-designs in 2n-dimensional Euclidean space, J. Algebraic Combin., Vol. 10 (1999) pp. 279–288.Google Scholar
  48. 48.
    V. M. Sidelnikov, Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial (in Russian), Algebra i Analiz, Vol. 11 (No. 4, 1999) pp. 183–203.Google Scholar
  49. 49.
    B. Venkov, Réseaux et “designs” sphériques, in Réseaux euclidiens, “designs” sphériques et groupes, L'Enseignement Mathématiques Monographie Vol. 37 (Martinet, J. ed.), to appear (2001).Google Scholar
  50. 50.
    G. E. Wall, On Clifford collineation, transform and similarity groups IV: an application to quadratic forms, Nagoya Math. J., Vol. 21 (1962) pp. 199–222.Google Scholar
  51. 51.
    D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mtn. J. Math.,Vol. 2 (1972) pp. 159–168.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Gabriele Nebe
    • 1
  • E. M. Rains
    • 2
  • N. J. A. Sloane
    • 3
  1. 1.Abteilung Reine Mathematik der Universität UlmUlmGermany
  2. 2.Information Sciences Research, AT&T Shannon LabsFlorham ParkU.S.A.
  3. 3.Information Sciences Research, AT&T Shannon LabsFlorham ParkU.S.A

Personalised recommendations