On the blocking of two-dimensional affine varieties

The paper considers the problem of blocking families of subsets and proposes a construction for expanding the blocking sets of a family of two-dimensional affine manifolds in the space of bit strings when its dimension $n$ increases. Examples are given and the cardinality of the complements of the blocking sets of this family of varieties are calculated for high odd dimension. The main construction of the complement of the blocking set for $n=2m+1$ is its construction in the form of a set of elements in the form $(x,y,z)$, where $z$ is a bit, $y=x^3$ for the bit string $x$ from the complement of the blocking set in the field $\mathrm{GF}(2^m)$. The construction is applied to solve the “A secret sharing” problem of the NSUCRYPTO Olympiad not only for even, but also for an odd dimension of the space.