Vyacheslav V. Rykov

New constructions of superimposed codes

Kautz-Singleton (1964) suggested a class of binary superimposed codes which are based on the q-ary Reed-Solomon codes (RS codes). Applying a concatenation of the binary constant-weight error-correcting codes and the shortened RS codes, we obtain new constructions of superimposed codes. Tables of their parameters are given. From the tables it follows that the rate of obtained codes exceeds the corresponding random coding bound.

New Applications and Results of Superimposed Code Theory Arising from the Potentialities of Molecular Biology

Superimposed codes (SC) were introduced by Kautz-Singleton (1964) [1], who worked out the important constructive methods. Dyachkov-Rykov [2, 3, 4, 5, 6] and Erdos-Frankl-Furedi [7] obtained upper and lower bounds on the rate of SC. Dyachkov-Macula-Rykov [8, 9, 10, 11] investigated the development of constructions for SC (nonadaptive pooling designs) intended for the clone-library screening problem. (See Balding-Torney [12] and Knill-Bruno-Torney [13]). In this paper, we give an introduction to the problem and a detailed survey of our recent results on constructive methods of SC. We discuss superimposed distance codes and list-decoding superimposed codes.

Trivial two-stage group testing for complexes using almost disjunct matrices

Abstract Let [t] represent a finite population with t elements. Suppose we have an unknown d-family of k-subsets Γ of [t]. We refer to Γ as the set of positive k-complexes. In the group testing for complexes problem, Γ must be identified by performing 0, 1 tests on subsets or pools of [t]. A pool is said to be positive if it completely contains a complex; otherwise the pool is said to be negative. In classical group testing, each member of Γ is a singleton. In this paper, we exhibit and analyze a probabilistic trivial two-stage algorithm that identifies the positive complexes.

DNA sequences and quaternary cyclic codes

There is a need to find generalized or universal tags for multiplex genome analysis. We use sets of reverse complement cyclic codes to generate DNA codes that can be used for DNA tags. These tags are placed on DNA chips and connect with the capture tags floating in a solution. The capture tags attach themselves to these generalized codes and give us much information about an individual strand. We show that the reverse complement cyclic and shortening cyclic codes are suitable for this application. We found theoretical bounds of parameters for these codes and the parameters and the generator polynomials of these codes for length up to 41.

New t-Gap Insertion-Deletion-Like Metrics for DNA Hybridization Thermodynamic Modeling

We discuss the concept of t-gap block isomorphic subsequences and use it to describe new abstract string metrics that are similar to the Levenshtein insertion-deletion metric. Some of the metrics that we define can be used to model a thermodynamic distance function on single-stranded DNA sequences. Our model captures a key aspect of the nearest neighbor thermodynamic model for hybridized DNA duplexes. One version of our metric gives the maximum number of stacked pairs of hydrogen bonded nucleotide base pairs that can be present in any secondary structure in a hybridized DNA duplex without pseudoknots. Thermodynamic distance functions are important components in the construction of DNA codes, and DNA codes are important components in biomolecular computing, nanotechnology, and other biotechnical applications that employ DNA hybridization assays. We show how our new distances can be calculated by using a dynamic programming method, and we derive a Varshamov-Gilbert-like lower bound on the size of some of codes using these distance functions as constraints. We also discuss software implementation of our DNA code design methods.

Superimposed codes for multiple accessing of the OR-channel

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

On associative nets

We consider the Boolean model of associative memory using neural nets. The previous results from the superimposed code theory are applied to obtain exchange relations between the model parameters and the size of stored information.