Lower bounds for circuits with MOD_m gates

Let CC_o(n)[m] be the class of circuits that have size o(n) and in which all gates are MOD[m] gates. We show that CC [m] circuits cannot compute MOD_q in sub-linear size when m, q > 1 are co-prime integers. No non-trivial lower bounds were known before on the size of CC [m] circuits of constant depth for computing MOD_q. On the other hand, our results show circuits of type MAJ o CC_o(n)[m] need exponential size to compute MOD_q . Using Bourgain´s recent breakthrough result on estimates of exponential sums, we extend our bound to the case where small fan-in AND gates are allowed at the bottom of such circuits i.e. circuits of type MAJ o CC[m] o AND _{epsiv log n}, where epsiv > 0 is a sufficiently small constant. CC [m] circuits of constant depth need superlinear number of wires to compute both the AND and MOD_q functions. To prove this, we show that any circuit computing such functions has a certain connectivity property that is similar to that of superconcentration. We show a superlinear lower bound on the number of edges of such graphs extending results on superconcentrators