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

We prove that for every prime p there exists a (0, 1)-matrix M of size tp(n,m) × n, where tp(n,m) = O(m+ mlog nm logmin(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 field of real numbers 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 the information theoretic argument (for p ≥ m) it can be shown that the above bound is tight. This solves the following open problems: (1) Coin weighing problem: Suppose that n coins are given among which there are at most m counterfeit coins of arbitrary weights. There is a nonadaptive algorithm that finds the counterfeit coins and their weights in t(n,m) = O((mlog n)/ logm) weighings. A previous result [S. S. Choi and J. H. Kim, in Proceedings of the 2008 ACM International Symposium on Theory of Computing, ACM, New York, 2008, pp. 749-758] proves the existence of a nonadaptive algorithm that solves the problem (with the same complexity) only for weights between n^-a and n^b for constants a and b and finds the counterfeit coins but not their weights. (2) Reconstructing graph from additive queries: Suppose that G is an unknown weighted graph with n vertices and at most m edges. There exists a nonadaptive algorithm that finds the edges of G and their weights in O(t(n,m)) additive queries. Previous results [S. S. Choi and J. H. Kim, in Proceedings of the 2008 ACM International Symposium on Theory of Computing, ACM, New York, 2008, pp. 749-758; N. H. Bshouty and H. Mazzawi, in Proceedings of the 20th International Conference on Algorithmic Learning Theory, Springer, Berlin, 2009, pp. 97-109] prove the existence of a nonadaptive algorithm that solves the problem only for weights between n^-a and n^b for constants a and b and finds the edges but not their weights. (3) 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 whose output is the sum of the messages. Then all messages can be sent (encoded and decoded) with O(t(n,m)) transmissions. Previous algorithms [E. Biglieri and L. Gÿorfi, Multiple Access Channels: Theory and Practice, NATO Security through Sci. Ser. D: Inform. Commun. Security, IOS Press, Amsterdam, 2007] 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.