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(m kr/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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Lefmann, H., Pudlák, P., Savický, P. (1996). On sparse parity check matrices (extended abstract). In: Cai, JY., Wong, C.K. (eds) Computing and Combinatorics. COCOON 1996. Lecture Notes in Computer Science, vol 1090. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61332-3_137
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DOI: https://doi.org/10.1007/3-540-61332-3_137
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