Pavel A. Vilenkin

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.

Two Models of Nonadaptive Group Testing for Designing Screening Experiments

We discuss two non-standard models of nonadaptive combinatorial search which develop the conventional disjunct search model of Du and Hwang (1993) for a small number of defective elements contained in a finite ground set or a population. The first model called asearch of defective supersets (complexes)was suggested in D’yachkovetal. (2000c,d). The second model which can be called asearch of defective subsets in the presence of inhibitorswas introduced for the case of an adaptive search by Farachetal. (1997) and De Bonis and Vaccaro (1998). For these models, we study the constructive search methods based on the known constructions for the disjunct model from Kautz and Singleton (1964) and from D’yachkovetal. (2000a,b).

Nonadaptive and Trivial Two-Stage Group Testing with Error-Correcting d e-Disjunct Inclusion Matrices

We discuss three types of inclusion matrices (subset, subspace and sequence). We exhibit their disjunct properties and their applications to error-correcting nonadaptive group testing. Under some limited conditions, these structures are optimal for their disjunct properties.

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.

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

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.

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.