On realization complexity of linear Boolean transformations by schemes of depth 3 (Q1406385)

Denote by \(L^a(M)\) the realization complexity of the system of functions prescribed by the matrix \(M\) which is performed by schemes of depth \(a\), \( L_T(M)\) is the complexity of the trivial realization of the system of functions prescribed by the matrix \(M\). The author presents the construction of a sequence of matrices without the rectangles \(Q_n\) for which the estimate \[ \frac{L^3(Q_n)}{L_T(Q_n)} \lesssim \frac 7{12} \] is valid.