On Sparse Parity Chack Matrices (Extended Abstract)

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 fixed integers k ≥ 2 and r ≥ 1, we show the probabilistic lower bound N(m, k, r) = Ω(mkr/2(k−1)); for k a power of 2, we prove the upper bound N(m, k, r) = O(n[kr/(k−1)]/2), which matches the lower bound for infinitely many values of r. We give some explicit constructions.

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