Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests |
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Authors: | K Yu Gorbunov |
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Affiliation: | (1) Kharkevich Institute for Information Transmission Problems, RAS, Moscow, Russia |
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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 Lδ/ln(1/δ)](1 ? 2 ln ln(1/δ)/ln(1/δ)). |
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