Involutions on the the Barnes - Wall lattices and their fixed point sublattices , I . version 18
نویسنده
چکیده
We study the sublattices of the rank 2d Barnes-Wall lattices BW2d which occur as fixed points of involutions. They have ranks 2d−1 (for dirty involutions) or 2d−1 ± 2k−1 (for clean involutions), where k, the defect, is an integer at most d2 . We discuss the involutions on BW2d and determine the isometry groups of the fixed point sublattices for all involutions of defect 1. Transitivity results for the Bolt-Room-Wall group on isometry types of sublattices extend those in [PO2d]. Along the way, we classify the orbits of AGL(d, 2) on the Reed-Muller codes RM(2, d) and describe cubi sequences for short codewords, which give them as Boolean sums of codimension 2 affine subspaces.
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