Arkadii G. D'yachkov

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.

Families of Finite Sets in which No Intersection of ℓ Sets Is Covered by the Union of s Others

Abstract In 1964, Kautz and Singleton ( IEEE Trans. Inform. Theory 10 (1964), 363–377) introduced the superimposed code concept. A binary superimposed code of strength s is identified by the incidence matrix of a family of finite sets in which no set is covered by the union of s others ( J. Combin. Theory Ser. A 33 (1982), 158–166 and Israel J. Math. 51 (1985), 75–89). In the present paper, we consider a generalization called a binary superimposed ( s,l )-code which is identified by the incidence matrix of a family defined in the title. We discuss the constructions based on MDS-codes (The Theory of Error-correcting Codes, North-Holland, Amsterdam, The Netherlands, 1983) and derive upper and lower bounds on the rate of these codes.

Exordium for DNA Codes

We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNA-sequence codewords. This enhancement involves abstractions of the pertinent aspects of DNA; it nevertheless ensures specificity of binding for all pairs of single strands derived from its codewords—the key desideratum of DNA codes– i.e. with binding feasible only between reverse complementary strands. We defer discussing the combinatorial-optimization superincumbencies of code construction. Generalization of deletion similarity to an optimal sequence-alignment score could readily effect advantageous improvements (Kaderali, Masters Thesis, Informatics, U. Köln, 2001) but would render the combinatorics opaque. We mention motivating applications of DNA codes.

A weighted insertion-deletion stacked pair thermodynamic metric for DNA codes

Thermodynamic distance functions are important components in the construction of DNA codes and DNA codewords are structural and information building blocks in biomolecular computing and other biotechnical applications that employ DNA hybridization assays. We introduce new metrics for DNA code design that capture key aspects 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. We introduce the concept of (t-gap) block isomorphic subsequences to describe new string metrics that are similar to the weighted Levenshtein insertion-deletion metric. We show how our new distances can be calculated by a generalization of the folklore longest common subsequence dynamic programming algorithm. We give a Varshamov-Gilbert like lower bound on the size of some of codes using our distance functions as constraints. We also discuss software implementation of our DNA code design methods.

On DNA Codes

We develop and study the concept of similarity functions for q-ary sequences. For the case q = 4, these functions can be used for a mathematical model of the DNA duplex energy [1,2], which has a number of applications in molecular biology. Based on these similarity functions, we define a concept of DNA codes [1]. We give brief proofs for some of our unpublished results [3] connected with the well-known deletion similarity function [4–6]. This function is the length of the longest common subsequence; it is used in the theory of codes that correct insertions and deletions [5]. Principal results of the present paper concern another function, called the similarity of blocks. The difference between this function and the deletion similarity is that the common subsequences under consideration should satisfy an additional biologically motivated [2] block condition, so that not all common subsequences are admissible. We prove some lower bounds on the size of an optimal DNA code for the block similarity function. We also consider a construction of close-to-optimal DNA codes which are subcodes of the parity-check one-error-detecting code in the Hamming metric [7].

Cover-free families and superimposed codes: constructions, bounds and applications to cryptography and group testing

This paper deals with (s,l)-cover-free families or superimposed (s,l)-codes. They generalize the concept of superimposed s-codes and have several applications for cryptography and group testing. We present a new asymptotic bound on the rate of optimal codes and develop some constructions.

DNA codes for nonadditive stem similarity

We study two new concepts of combinatorial coding theory: additive stem similarity and additive stem distance between q-ary sequences. For q = 4, the additive stem similarity is applied to describe a mathematical model of thermodynamic similarity, which reflects the “hybridization potential” of two DNA sequences. Codes based on the additive stem distance are called DNA codes. We develop methods to prove upper and lower bounds on the rate of DNA codes analogous to the well-known Plotkin upper bound and random coding lower bound (the Gilbert-Varshamov bound). These methods take into account both the “Markovian” character of the additive stem distance and the structure of a DNA code specified by its invariance under the Watson-Crick transformation. In particular, our lower bound is established with the help of an ensemble of random codes where distribution of independent codewords is defined by a stationary Markov chain.

New results on DNA codes

For q-ary n-sequences, we develop the concept of similarity functions that can be used (for q = 4) to model a thermodynamic similarity on DNA sequences. A similarity function is identified by the length of a longest common subsequence between two q-ary n-sequences. Codes based on similarity functions are called DNA codes. DNA codes are important components in biomolecular computing and other biotechnical applications that employ DNA hybridization assays. We present our unpublished results connected with the conventional deletion similarity function used in the theory of error-correcting codes. The main aim of this paper - to obtain lower bounds on the rate of optimal DNA codes for a biologically motivated similarity function called a similarity of blocks. We also present constructions of suboptimal DNA codes based on the parity-check code detecting one error in the Hamming metric

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.

Random Coding Bounds for DNA codes based on Fibonacci ensembles of DNA sequences

We consider DNA codes based on the concept of a weighted 2-stem similarity measure which reflects the ldquohybridization potentialrdquo of two DNA sequences. A random coding bound on the rate of DNA codes with respect to a thermodynamic motivated similarity measure is proved. Ensembles of DNA strands whose sequence composition is restricted in a manner similar to the restrictions in binary Fibonacci sequences are introduced to obtain the bound.