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" C_m of structure 2_+^(1+2m). O^+(2m,2). This group and its complex analogue 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 C_m degree 2k is spanned by the complete weight enumerators of the codes C⊗F_(2m), where Cranges 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 C_m: let M be the Z[√2]-lattice with Gram matrix [2 √2 √2 2}. Then L_m is the rational part of M^(⊗ m), and C_m = Aut(M^(⊗m)). Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of C⊗F_(2m). There are analogues of all these results for the complex group X_m, with "doubly-even self-dual code" instead of "self-dual code."

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