Thin circulant matrices and lower bounds on the complexity of some Boolean operators

We prove a lower bound Ω ( k+l k2l2 N k+l+2 kl ) on the maximal possible weight of a (k, l)-free (that is, free of all-ones k × l submatrices) Boolean circulant N × N matrix. The bound is close to the known bound for the class of all (k, l)-free matrices. As a consequence, we obtain new bounds for several complexity measures of Boolean sums’ systems and a lower bound Ω(N2 log N) on the monotone complexity of the Boolean convolution of order N .