Sparse Parity-Check Matrices over Finite Fields

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 = 2^l 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.