On Sparse Parity Check Matrices

We consider the extremal problem to determine the maximal number N(m; k; r) of columns of a 0-1-matrix with m rows and at most r ones in each column such that each k columns are linearly independent modulo 2. For each xed k 1 and r 1, we shall prove a probabilistic lower bound N(m; k; r) = (m kr=2(k?1)); for k a power of 2, we prove an upper bound N(m; k; r) = O(m dkr=(k?1)e=2) which matches the lower bound for innnitely many values of r. We give some explicit constructions.