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 \(k \geqslant 1\) 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.
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Lefmann, H., Pudlák, P. & Savicky, P. On Sparse Parity Check Matrices. Designs, Codes and Cryptography 12, 107–130 (1997). https://doi.org/10.1023/A:1008233013327
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DOI: https://doi.org/10.1023/A:1008233013327