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. Their code has 64 vectors and cosines − 3 7 , − 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 , ± 1 4 , − 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 .