Coding information by Walsh matrices

The representation of the general linear group $\mathrm{GL}(n, 2) $ by the automorphism subgroup $\mathrm{GL}(N, 2) $ under the multiplicative notation in its action in the space $ \mathbb{R} ^ N $, where $ N = 2 ^ n $, is considered. Each matrix as an element of the group $\mathrm{GL}(n, 2) $ defines ordering: the group $ \mathbb{Z}_2 ^ n $ and its group of characters, which are popular in digital processing of information in the form of discrete Walsh functions. On the basis of the fast Walsh transform and this correspondence the authors created a software prototype of an automatic output signal coding system. The essence of the proposed software product is the number of possible permutations, which is calculated by the formula $(2^n-2^0)(2^n-2^1)\ldots (2^n-2^{n-1})$ for $n$-th order matrices. Based on the program, it is possible to organize a multi-channel system of reconfigurable decoders when transmitting hidden information over open communication channels.