The Invariants of the Clifford Groups |
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Authors: | Gabriele Nebe E M Rains N J A Sloane |
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Institution: | (1) Abteilung Reine Mathematik der Universität Ulm, 89069 Ulm, Germany;(2) Information Sciences Research, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ, 07932-0971, U.S.A.;(3) Information Sciences Research, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ, 07932-0971, U.S.A |
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Abstract: | The automorphism group of the Barnes-Wall lattice L
m in dimension 2
m
(m ; 3) is a subgroup of index 2 in a certain Clifford group
of structure 2
+
1+2m
. O
+(2m,2). This group and its complex analogue
of structure
.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
of degree 2k is spanned by the complete weight enumerators of the codes
, 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 L
m and
: let M be the
-lattice with Gram matrix
. Then L
m is the rational part of M
m, and
= Aut(M m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then
is precisely the automorphism group of the complete weight enumerator of
. There are analogues of all these results for the complex group
, with doubly-even self-dual code instead of self-dual code.![rdquo](/content/u29302t5p3813015/xxlarge8221.gif) |
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Keywords: | Clifford groups Barnes-Wall lattices spherical designs invariants self-dual codes |
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