Involutions on the Barnes-Wall lattices and their fixed point sublattices. I. (Q2471454)

This article is a continuation of the author's study on the Barnes-Wall lattices and their automorphism groups. In this article, the author studies the sublattices of the Barnes-Wall lattices which occur as fixed points of involutions. The involutions on \(\text{BW}_{2^d}\) are discussed and the isometry groups of the fixed point sublattices for all involutions of defect 1 are determined. Along the way, the author also classifies the orbits of \(\text{AGL}(d,2)\) on the Reed-Muller code \(\text{RM}(2,d)\).