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 $-3/7, -1/7, 1/7$. We construct the {\it Optimism Code}, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are $0, 1/4, -1/4, -1$. Its automorphism group has shape $2^{1+8}{\cdot}GL(4,2)$. 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 $2^4{:}Alt_7$.