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. Their code has 64 vectors and cosines -37,-17,17. 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 2^1^+^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 2^4:Alt"7.
['Robert L. Griess']
Few-cosine spherical codes and Barnes--Wall lattices