Few-cosine spherical codes and Barnes-Wall lattices

Using Barnes-Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. It has 64 vectors and cosines −37 ,− 1 7 , 1 7 . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are 0,±14 ,−1. Its automorphism group has shape 21+8·GL(4, 2). The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters (16, 256, 6), a Nordstrom-Robinson code, and gives a context for determining its automorphism group, which has form 2:Alt7.