Sparse Parity-Check Matrices over ${GF(q)}$

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 integers k ≥ 2 we obtain the lower bounds Nq(m, k, r) = Ω(m kr/(2(k−1))), and Nq(m, k, r) = Ω(m ((k−1)r)/(2(k−2))) for odd k ≥ 3. For k = 2 we show that Nq(m, k, r) = Θ(mkr/(2(k−1))) if gcd(k − 1, r) = k − 1, while for arbitrary even k ≥ 4 with gcd(k − 1, r) = 1 we have Nq(m, k, r) = Ω(m kr/(2(k−1)) · (logm)1/(k−1)). Matrices, which fulfill these lower bounds, can be found in polynomial time. Moreover, for char (GF (q)) > 2 we obtain Nq(m, 4, r) = Θ(m d4r/3e/2), while for char (GF (q)) = 2 we can only show that Nq(m, 4, r) = O(m d4r/3e/2). Our results extend and complement earlier results from [5, 18], where the case q = 2 was considered.