The Reed–Muller code square and equivalence classes of McEliece–Sidelnikov cryptosystem private keys

Equivalence classes of McEliece–Sidelnikov cryptosystem private keys are studied in the work. The structure of the classes is described in the case, when the square of the code with the generator matrix $(R|HR)$, where $R$ is a generator matrix of the Reed–Muller code $\operatorname{RM}(r,m)$ of order $r$ and length $2^m$, equals the Cartesian square of the code of order $2r$ and the same length. In this case, there exists a bijection between an equivalence class and the Cartesian square of automorphism group of the code $\operatorname{RM}(r,m)$. Moreover, it is shown that the ratio of matrices $H$ causing other cases approaches zero when the code dimension approaches infinity.