Upper bounds for the size and the depth of formulae for MOD-functions

Abstract We obtain new upper bounds for the size and the depth of formulae for some MOD-functions (that is, functions counting n bits modulo m). In particular, the depth of counting n bits modulo 3 is bounded by 2.8 log2 n + O(1) in the standard basis {Λ, V,—}; the size of counting modulo 5 is bounded by O(n3.22) in the same basis; the depth of counting modulo 7 is bounded by 2.93 log2 n+O(1) in the basis of all binary Boolean functions.