The Invariants of the Clifford Groups 

Gabriele Nebe(*)
Abteilung Reine Mathematik der Universitaet Ulm 
89069 Ulm, Germany 

E. M. Rains and N. J. A. Sloane 
Information Sciences Research, AT&T Shannon Labs 
180 Park Avenue, Florham Park, NJ 07932-0971, U.S.A. 

(*)Most of this work was carried out during G. Nebe's visit to
AT&T Labs in the Summer of 1999 

December 6, 1999; revised September 8, 2000 


ABSTRACT

The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m
(m not 3 ) is a subgroup of index 2 in a certain ``Clifford group'' SC_m
of structure 2_{+}^{1+2m}.O^{+}(2m,2).  This group and its complex 
analogue CC_m of structure (2_+^{1+2m} \zentr Z_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's 1996 result
that the space of invariants for SC_m of degree 2k is spanned by the complete weight
enumerators of the codes C \otimes 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 L_m and SC_m:
let M be the Z[sqrt2]-lattice with Gram matrix
    [   2    sqrt2 ] 
    [ sqrt2    2   ]
Then L_m is the rational part of M^{\otimes m}, and SC_m = Aut (M^{\otimes m}).

Also, if C is a binary self-dual code not generated by vectors of weight 2,
then SC_m is precisely the automorphism group of the complete weight
enumerator of C \otimes F_{2^m}.
There are analogues of all these results for the complex group
CC_m, with ``doubly-even self-dual code'' instead of ``self-dual code''.


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



For the full version, see
http://www.research.att.com/~njas/doc/cliff1.pdf  or
http://www.research.att.com/~njas/doc/cliff1.ps



A slightly different version of this paper appeared in
Designs, Codes and Cryptography, Vol. 24 (2001), pp. 99-121.