On codes with unique decoding to the nearest

The paper is devoted to the problem No. 7 of the NSUCRYPTO 2023 Olympiad on the description of an interesting class of codes: with unique decoding to the nearest codeword in the Hamming metric. The possibility of representing the code $C$ as a matroid is proved and an intermediate solution is given in the case when a set of bits $B$ such that there is a two-bit word containing it that is decoded not at $z = 0$ coincides with all positions of the code bits.