Odin and Shadow Cretan matrices accompanying primes and their powers

Introduction: Cretan matrices – orthogonal matrices, consisting of the elements 1 and –b (real number), are an ideal object for visual application finite-dimensional mathematics. These include, in particular, Hadamard and, with expansion number elements, conference matrices. The most convenient research apparatus is to use field theory multiplicative Galois groups, which especially important new types Purpose: To study symmetries investigate two odd even orders, distinguished by symmetries, respectively, differ significantly from previously known Mersenne, Euler Fermat Results: Formulas levels given matrices: Odin bicycles (with a border) orders 4t 3 shadow 2 4 described. For character sizes equal prime numbers powers primes, existence matrix special types, doubly symmetric, skew-symmetric respect signs elements) symmetric cyclic blocks, proved. It shown that their case: Mersenne existing absence symmetry all selected without exception. Practical relevance: Оrthogonal sequences methods effective finding finite fields groups direct practical importance problems noise-immune coding, compression masking video information.