On parity check (0, 1)-matrix over Zp

• Authors: Nader H. Bshouty;Hanna Mazzawi
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

We prove that for every prime p there exists a (0, 1)-matrix M of size tp(n, m) x n, where [EQUATION] 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. We show that a random tp(n, m) x n (0, 1)-matrix over Zp satisfies the above with a high probability. This requires n·tp(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 = nc where c is a constant, a construction that uses O(m1+∈) random bits for any constant c. Each row in the constructed matrix is a tensor product of a (constant) d (0, 1)-vectors of size n1/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((m log n)/log m) weighings. Previous algorithm, [CK08], solves the problem (with the same number of weighings) only for weights between n−a and nb 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 nb 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.