Abstract
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 N q(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 N q(m, k, r) = Ω(m kr/(2(k−1))), and N q(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 N q(m, k, r) = Θ(m kr/(2(k−1))), while for any even k ≥ 4 and gcd(k − 1, r) = 1 we have N q(m, k, r) = Ω(m kr/(2(k−1)) · (logm)1/(k−1)). For char (GF(q)) > 2 we prove that N q(m,4,r) = Θ(m ⌈4r/3⌉/2), while for q = 2l we only have N q(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.
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Lefmann, H. (2003). Sparse Parity-Check Matrices over Finite Fields. In: Warnow, T., Zhu, B. (eds) Computing and Combinatorics. COCOON 2003. Lecture Notes in Computer Science, vol 2697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45071-8_13
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DOI: https://doi.org/10.1007/3-540-45071-8_13
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