NDLI: On parity check (0, 1)-matrix over $Z_{p}$

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On parity check (0, 1)-matrix over $Z_{p}$

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On parity check (0, 1)-matrix over $Z_{p}$

Content Provider	ACM Digital Library
Author	Mazzawi, Hanna Bshouty, Nader H.
Abstract	We prove that for every prime p there exists a (0, 1)-matrix M of size $t_{p}(n,$ m) x n, where [EQUATION] 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 $O(m^{1+∈})$ 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 $n^{1/d}.$ 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((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 $n^{b}$ 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 $Z_{p}$ 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.
Starting Page	1383
Ending Page	1394
Page Count	12
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2011-01-23
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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