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) x n,where t(p)(n, m) = O (m + mlog 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 Z(p) 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((m(2) log(2) 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+epsilon)) random bits for any constant epsilon. Each row in the constructed matrix is a tensor product of a (constant) d (0, 1)-vectors of size n(1/d).This solves the following open problems:Coin Weighing Problem: Suppose that n coins are given among which there are at most m counterfeit coins of arbitrary weights. There is a non-adaptive algorithm that finds the counterfeit coins and their weights in t(n, m) = O((mlog n)/logm) weighings.Previous algorithm, [CK08], solves the problem (with the same number of weighings) only for weights between n(-a) and n(b) for constants a and b and finds the counterfeit coins but not their weights.Reconstructing Graph from Additive Queries: Suppose that G is an unknown weighted graph with n vertices and m edges. There exists a non-adaptive algorithm that finds the edges of G and their weights in O(t(n, m)) additive queries.Previous algorithms, [CK08, BM09], solve the problem only for weights between n(-a) and n(b) for constants a and b and find the edges but not their weights.Signature Coding Problem: Consider n stations and at most m of them want to send messages from Zp through an adder channel, that is, a channel that its output is the sum of the messages. Then all messages can be sent (encoded and decoded) with O(t(n, m)) transmissions. Previous algorithms, [BG07], run with the same number of transmissions only for messages in {0, 1}.Simple information theoretic arguments show that all the above bounds are tight.