Thin circulant matrixes and lower bounds on complexity of some Boolean operators

Lower estimate $\Omega(\frac{k+l}{k^2l^2}N^{2-\frac{k+l+2}{kl}})$ of the maximal possible weight of a $(k,l)$-thin (that is, free of all-ones' submatrixes of size $k\times l$) circulant matrix of order $N$ is proved. The estimate is close to the known estimate corresponding to the class of all $(k,l)$-thin matrixes. As a consequence, new estimates of several complexity measures of Boolean sums' systems and a lower estimate $\Omega(N^2\log^{-6}N)$ of monotone complexity of a Boolean convolution of order $N$ are obtained. Ill. 1, bibliogr. 11.