On Sparse Parity Check Matrices

Authors:
Hanno Lefmann;Pavel Pudlák;Petr Savicky
Affiliations:
Universität Dortmund, FB Informatik, LS II, D-44221 Dortmund, Germany;Mathematical Institute, Academy of Sciences, Zitná 25, CZ–11567 Praha 1, Czech Republic;Institute of Computer Science, Academy of Sciences, Pod vodárenskou vezí 2, CZ-18200 Praha 8, Czech Republic
Venue:
Designs, Codes and Cryptography
Year:
1997

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Abstract

We consider the extremal problem to determine the maximalnumber N(m,k,r) of columns of a 0-1 matrix withm rows and at most r ones in each columnsuch that each k columns are linearly independentmodulo 2. For fixed integers k\geq 1and r≥ 1, we shall prove the probabilistic lowerbound N(m,k,r) = Ω (m^(kr/2(k-1))); for ka power of 2, we prove the upper bound N(m,k,r)=O(m^{\lceilkr/(k-1)\rceil /2}) which matches the lower bound forinfinitely many values of r. We give some explicitconstructions.