ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 1, pp. 48–56.
Pleiades Publishing, Inc., 2007.
Original Russian Text
K.Yu. Gorbunov, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 1, pp. 56–66.
Bound on the Cardinality of a Covering of an
Arbitrary Randomness Test by Frequency Tests
K. Yu. Gorbunov
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received October 17, 2006
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
, 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 0.5ln2L[δ/ ln(1/δ)](1 − 2lnln(1/δ)/ ln(1/δ)).
This paper answers a question stated in [1, Section 2.4]. Let us recall basic deﬁnitions and
problem settings from  related to the considered problem. The problem itself will be formulated
An arbitrary frequency test for binary sequences of length L is any set S of such sequences.
The speciﬁc deﬁciency δ(S) of S is (L − lb|S|)/L, where lb is the binary logarithm and |S| is
the cardinality of S.Lett denote an arbitrary element of S.Amonotonic selection rule is
any selection rule (i.e., a function deﬁned on all ﬁnite sequences and taking values 0 and 1) for
selecting a subsequence of t which, at each step i (i =1,...,L), decides (using only the head of
t of length i − 1) whether the ith symbol t
of t should be appended to the subsequence under
construction. A nonmonotonic selection rule diﬀers from a monotonic one in the following way:
it may look through bits of t in an arbitrary order (the decision on appending a bit to the sequence
under construction is made straight before reading this bit, based on the values of previously
examined bits). The speciﬁc deﬁciency of a rule conditional to S is the number
the minimum (over all t ∈ S) of the product of the length of a selected subsequence and the squared
deviation of the fraction of zeros in it from 1/2(heree is the base of the natural logarithm).
Let us brieﬂy recall the reason for this deﬁnition (see details in ). A normal rule is a rule
that always selects from a sequence of length L subsequences of the same length. Given any rule r,
one can easily construct L normal rules r
selects a subsequence of “its own”
length i and does not change the selection given by r in the cases where r also selects a subsequence
of length i. To each normal rule r
and a given deviation ε there corresponds a set T (frequency
test) of subsequences of length L for which r
selects a subsequence (of length i) with deviation of
the fraction of zeros from 1/2ofatleastε. Using the bound for the large deviation probability
(see [2, p. 93]; sometimes it is called the Chernoﬀ bound), it is easy to show that the speciﬁc
Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00122, and the Grant
of the President of the Russian Federation for State Support of Leading Scientiﬁc Schools, no. NSh-