On One Finite Matrix Group and Codes on Euclidean Spheres

Starting from a finite group of orthogonal $2^n\times 2^n$ and $2p^n\times 2p^n$ matrices over the field of real numbers, we construct new orbit codes on a Euclidean sphere. Some of these codes have more than twice as many points as codes with the same code distance obtained by the standard procedure from second-order Reed–Muller codes.