Extremal graphs with no C4,s, or C10,s
Journal of Combinatorial Theory Series B
On construction of k-wise independent random variables
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
On Sparse Parity Check Matrices
Designs, Codes and Cryptography
Analysis of low density codes and improved designs using irregular graphs
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
The Algorithmic Aspects of Uncrowded Hypergraphs
SIAM Journal on Computing
MODp-tests, almost independence and small probability spaces
Random Structures & Algorithms
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For fixed positive integers k, q, r with q a prime power and large m, we investigate matrices with m rows and a maximum number Nq(m, k, r) of columns, such that each column contains at most r nonzero entries from the finite field GF(q) and each k columns are linearly independent over GF(q). For even k we prove the lower bounds Nq(m, k, r) = Ω(mkr/(2(k-1))), and Nq(m, k, r) = Ω(m(k-1)r/(2(k-2))) for odd k ≥ 3. For k = 2i and gcd(k-1, r) = k-1 we obtain Nq(m, k, r) = Θ(mkr/(2(k-1))), while for any even k ≥ 4 and gcd(k - 1, r) = 1 we have Nq(m, k, r) = Ω(mkr/(2(k-1)) ċ (log m)1/(k-1)). For char (GF(q)) 2 we prove that Nq(m, 4, r) = Θ(m⌈4r/3⌉/2), while for q = 2l we only have Nq(m, 4, r) = O(m⌈4r/3⌉/2). We can find matrices, fulfilling these lower bounds, in polynomial time. Our results extend and complement earlier results from [4,14], where the case q = 2 was considered.