The key space of the McEliece–Sidelnikov cryptosystem

The McEliece cryptosystem is one of the oldest public-key cryptosystems. It was proposed by R. J. McEliece in 1978. The McEliece cryptosystem is based on a certain NP-hard problem in the coding theory. In this paper we consider a generalisation of the McEliece cryptosystem proposed by V. M. Sidelnikov in 1994. The McEliece–Sidelnikov cryptosystem is based on $u$-fold use of the Reed–Muller codes $RM(r,m)$.
This research is devoted to questions related to the space of equivalent secret keys of the new cryptosystem, that is, secret keys generating identical public keys. We study the structure of the set of public keys of the McEliece–Sidelnikov cryptosystem for an arbitrary number u of blocks. For the case of two blocks we describe all equivalence classes of secret keys with representatives of a special form.