The Invariants of the Clifford Groups

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” $$\mathcal{C}_m$$ of structure 2 + 1+2 . 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 L m 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 L m 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.”