On parity check (0,1)-Matrix over ℤ_p

We prove that for every prime p there exists a (0,1)-matrix M of size t_p(n, m) × n, where t_p(n, m) = O (m+ m log n/m/log min(m,p) ) such that every m columns of M are linearly independent over ℤ_p, the field of integers modulo p (and therefore over any field of characteristic p and over the real numbers field ℝ). In coding theory this matrix is a parity-check (0, 1)-matrix over ℤ_p of a linear code of minimal distance m + 1. Using the Hamming bound (for p < m) and information theoretic argument (for p ≥ m) it can be shown that the above bound is tight. We show that a random t_p(n,m) x n (0, 1)-matrix over ℤp satisfies the above with a high probability. This requires n · t_p(n, m) random bits. To reduce the number of random bits, one can use n random variables that are m-wise independent. This gives a construction with O((m² log² n)/log m) random bits. In this paper we use a new technique that gives for any m = n^c where c is a constant, a construction that uses O(m^1+ε) random bits for any constant ε. Each row in the constructed matrix is a tensor product of a (constant) d (0, l)-vectors of size 1/d.