Nikita Polyanskii

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.

Separable Codes for the Symmetric Multiple-Access Channel

A binary matrix is called an s-separable code for the disjunctive multiple-access channel (disj-MAC) if Boolean sums of sets of

On multistage learning a hidden hypergraph

s

On a hypergraph approach to multistage group testing problems

columns are all distinct. The well-known issue of the combinatorial coding theory is to obtain upper and lower bounds on the rate of s-separable codes for the disj-MAC. In our paper, we generalize the problem and discuss upper and lower bounds on the rate of q-ary s-separable codes for models of noiseless symmetric MAC, i.e., at each time instant the output signal of MAC is a symmetric function of its

DNA codes for generalized stem similarity

s

Bounds on the rate of superimposed codes

input signals.

Symmetric disjunctive list-decoding codes

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.

Almost cover-free codes and designs

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.

On capacities of the two-user union channel with complete feedback.

The concept of a generalized stem similarity function and the corresponding DNA codes are introduced. We give parameters for some optimal constructions called maximum distance separable DNA codes and obtain bounds on the maximum size of DNA codes.