Zero-sum square matrices

Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the sum of each row of B and the sum of each column of B is a multiple of p. Let M(p, k) denote the least integer in for which every square matrix of order at least m has a square submatrix of order k which is zero-sum mod p. In this paper we supply upper and lower bounds for M(p, k). In particular, we prove that lim sup M(2, k)/k ≤ 4, lim inf M(3, k)/k ≤ 20, and that M(p, k) ≥ k√2/2e exp(1/e)^p/2. Some nontrivial explicit values are also computed.