On Parity Check (0, 1)-Matrix over Zp

We prove that for every prime p ≤ poly(n) there exists a (0, 1)-matrix M of size tp(n,m) × n where tp(n,m) = O ( m+ m log n m log min(m, p) ) such that every m columns of M are linearly independent over Zp, 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 Zp 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. To reduce the number of random bits, we use n random variables that are m-wise independent. This gives O((m log n)/ logm) random bits. We then use a new technique to extend this result to a (0, 1)-matrix of size sp(n,m, d) × n where sp(n,m, d) = O(t(n,m)) and each row in the matrix is a tensor product of a constant d (0, 1)-vectors of size n. This, for m = n where c < 1 is any constant, gives O(m ) random bits for any constant . 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((m log n)/ logm) weighings. Previous algorithm, [CK08], solves the problem (with the same complexity) only for weights between n−a and n 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], solves the problem only for weights between n−a and n for constants a and b and finds 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.