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

Abstract

The automorphism group of the Barnes-Wall lattice Lm in dimension 2m(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 

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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

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