Distribution of cardinalities of row spaces of Boolean matrices of order \(n\) (Q1586835)

Let \(A\) be a Boolean matrix of size \(n\times n\). Denote by \(R(A)\) the subspace spanned by the rows of \(A\). It is obvious that the cardinality \#\(R(A)\leq 2^n\). In the paper under review the author obtains two intervals of gaps for the cardinalities \#\(R(A)\in [1,2^{n-1}]\), proving the following: for \(n\geq 7\) the cardinality \#\(R(A)\notin (2^{n-1}-2^{n-5}, 2^{n-1}-2^{n-6})\cup (2^{n-1}-2^{n-6}, 2^{n-1})\).