Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests

Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where ∨S∨ ≤ 2 L(1−δ), for sufficiently small δ there exists a polynomial (in 1/δ) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence t ∈ S, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[δ/ln(1/δ)](1 − 2 ln ln(1/δ)/ln(1/δ)). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests

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Publisher
Nauka/Interperiodica
Copyright
Copyright © 2007 by Pleiades Publishing, Inc.
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946007010061
Publisher site
See Article on Publisher Site

Abstract

We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where ∨S∨ ≤ 2 L(1−δ), for sufficiently small δ there exists a polynomial (in 1/δ) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence t ∈ S, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[δ/ln(1/δ)](1 − 2 ln ln(1/δ)/ln(1/δ)).

Journal

Problems of Information TransmissionSpringer Journals

Published: Apr 20, 2007

References

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