On Sparse Parity Check Matrices

AbstractWe 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 $$k \geqslant 1$$ and $$r \geqslant 1$$ , we shall prove the probabilistic lower bound $$N(m,k,r)$$ = $$\Omega (m^{kr/2(k - 1)} )$$ ; for $$k$$ a power of $$2$$ , we prove the upper bound $$ N(m,k,r) = O(m^{\left\lceil {kr/(k - 1)} \right\rceil /2} ) $$ which matches the lower bound for infinitely many values of $$r$$ . We give some explicit constructions.