A. G. D'yachkov

Bounds on the rate of disjunctive codes

A binary code is said to be a disjunctive (s, ℓ) cover-free code if it is an incidence matrix of a family of sets where the intersection of any ℓ sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L − 1 other sets of this family. For L = ℓ = 1, both definitions coincide, and the corresponding binary code is called a disjunctive s-code. This paper is aimed at improving previously known and obtaining new bounds on the rate of these codes. The most interesting of the new results is a lower bound on the rate of disjunctive (s, ℓ) cover-free codes obtained by random coding over the ensemble of binary constant-weight codes; its ratio to the best known upper bound converges as s → ∞, with an arbitrary fixed ℓ ≥ 1, to the limit 2e−2 = 0.271 ... In the classical case of ℓ = 1, this means that the upper bound on the rate of disjunctive s-codes constructed in 1982 by D’yachkov and Rykov is asymptotically attained up to a constant factor a, 2e−2 ≤ a ≤ 1.

Almost disjunctive list-decoding codes

We say that an s-subset of codewords of a binary code X is sL-bad in X if there exists an L-subset of other codewords in X whose disjunctive sum is covered by the disjunctive sum of the given s codewords. Otherwise, this s-subset of codewords is said to be sL-good in X. A binary code X is said to be a list-decoding disjunctive code of strength s and list size L (an sL-LD code) if it does not contain sL-bad subsets of codewords. We consider a probabilistic generalization of sL-LD codes; namely, we say that a code X is an almost disjunctive sL-LD code if the fraction of sL-good subsets of codewords in X is close to 1. Using the random coding method on the ensemble of binary constant-weight codes, we establish lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes. For this ensemble, the obtained lower bounds are tight and show that the capacity of almost disjunctive sL-LD codes is greater than the zero-error capacity of disjunctive sL-LD codes.

Superimposed designs and codes for nonadaptive search of mutually obscuring defectives

Let t be the total number of samples in a population which have an unknown subset of /spl les/s defective samples, 2/spl les/s<t. We consider the nonadaptive group testing (D.Z. Du, et al., 1993) for the model in which a group test is positive if and only if that group test contains exactly one defective sample. Otherwise, the group test is negative. This model (P. Damaschke, 1998) has been referred to as group testing for mutually obscuring (MO) defectives.

On a class of codes for the insertion-deletion metric

We study a class of q-ary codes for the insertion-deletion distance function in the space of q-ary n-sequences. For q = 4, the codes arise from the potentialities of molecular biology. With the help of random coding arguments we obtain a lower bound on the code rate.

Reverse-complement similarity codes for DNA sequences

We introduce three definitions of quaternary codes which are based on a biologically motivated measure of sequence similarity for quaternary n-sequences, extending Hamming similarity. The corresponding codes are used in bio-molecular experiments with DNA sequences. One of the codes is based on a distance function, extending Hamming distance. We discuss upper and lower bounds on the rates of these codes.

Superimposed codes for multiple accessing of the OR-channel

We study an application of superimposed codes for multiple accessing of the OR-channel.

Cover-free codes and separating system codes

We give some relations between the asymptotic rates of cover-free (CF) codes, separating system (SS) codes and completely separating system (CSS) codes. We also provide new upper bounds on the asymptotic rate of SS codes based on known results for CF and CSS codes. Finally, we derive a random coding bound for the asymptotic rate of SS codes and give tables of numerical values corresponding to our improved upper bounds.

Asymptotics of the Shannon and Renyi entropies for sums of independent random variables

With the help of the local limit theorem we investigate the asymptotics of the Shannon and Renyi (1961) entropies for sums of independent identically distributed random variables.

On multistage learning a hidden hypergraph

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph H_un = H(V, E) by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family F(t, s, ℓ) of localized hypergraphs for which the total number of vertices |V| = t, the number of edges |E| ≤ s, s ≪ t, and the cardinality of any edge |e| ≤ ℓ, ℓ ≪ t. Our goal is to identify all edges of H_un ∈ F(t, s, ℓ) by using the minimal number of tests. We develop an adaptive algorithm that matches the information theory bound, i.e., the total number of tests of the algorithm in the worst case is at most sℓ log₂ t(1+o(1)). We also discuss a probabilistic generalization of the problem.

On a hypergraph approach to multistage group testing problems

Group testing is a well known search problem that consists in detecting up to s, s ≪ t, defective elements of the set [t] = {1, . . . , t} by carrying out tests on properly chosen subsets of [t]. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests. In this paper we consider multistage group testing. We propose a general idea how to use a hypergraph approach to searching defective elements. For the case s = 2 and t → ∞, we design an explicit construction, which makes use of 2 log2 t(1 + o(1)) tests in the worst case and consists of 4 stages. For the general case of fixed s > 2 and t → ∞, we provide an explicit construction, which uses (2s - 1) log2 t(1+o(1)) tests and consists of 2s - 1 rounds.