Vladimir I. Levenshtein

Association schemes and coding theory

This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense). It is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code. Special emphasis is put on the linear programming method, inspired by the MacWilliams transform. This produces upper bounds for the size of a code with a given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain well-defined polynomial properties; this leads one into the realm of orthogonal polynomial theory. In particular, some universal bounds are derived for codes and designs in polynomial type association schemes. Throughout the paper, the main concepts, methods, and results are illustrated by two examples that are of major significance in classical coding theory, namely, the Hamming scheme and the Johnson scheme. Other topics that receive special attention are spherical codes and designs, and additive codes in translation schemes, including Z/sub 4/-additive binary codes.

Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Universal bounds for the cardinality of codes in the Hamming space F/sub r//sup n/ with a given minimum distance d and/or dual distance d are stated. A self-contained proof of optimality of these bounds in the framework of the linear programming method is given. The necessary and sufficient conditions for attainability of the bounds are found. The parameters of codes satisfying these conditions are presented in a table. A new upper bound for the minimum distance of self-dual codes and a new lower bound for the crosscorrelation of half-linear codes are obtained. >

Efficient Reconstruction of Sequences from Their Subsequences or Supersequences

In the paper two combinatorial problems for the set Fnq of sequences of length n over the alphabet Fq={0, 1, ?, q?1} are considered. The maximum size N?q(n, t) of the set of common subsequences of length n?t and the maximum size N+q(n, t) of the set of common supersequences of length n+t of two different sequences of Fnq are found for any nonnegative integers n and t. The number N?q(n, t)+1 (respectively, N+q(n, t)+1) is equal to the minimum number N of different subsequences of length n?t (supersequences of length n+t) of an unknown sequence X?Fnq which are sufficient for its reconstruction. Simple algorithms to recover X?Fnq from N?q(n, t)+1 of its subsequences of length n?t and from N+q(n, t)+1 of its supersequences of length n+t are given.

Efficient reconstruction of sequences

We introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either a combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest N such that N erroneous versions always suffice to reconstruct a sequence of length n, either exactly or with a preset accuracy and/or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within Hamming distance d with error probability /spl epsiv/ is found when d/n and /spl epsiv/ tend to 0 as n/spl rarr//spl infin/. A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained.

Perfect (d,k)-codes capable of correcting single peak-shifts

Codes for the multibit peak-shift recording channel, called (d,k)-codes of reduced length N, are considered. Arbitrary (d,k)- and perfect (d,k)-codes capable of correcting single peak-shifts of given size t are defined. For the construction of perfect codes, a general combinatorial method connected with finding good weight sequences in Abelian groups is used, and the concept of perfect t-shift N-designs is introduced. Explicit constructions of such designs for t=1, t=2, and t=(p-1)/2 are given, where p is a prime. This construction is universal in that it does not depend on the (d,k)-constraints. It also allows automatic correction of those peak-shifts that violate (d,k)-constraints. The construction is extended to (d,k)-codes of fixed binary length and allows the beginning of the next codeword to be determined. The question whether the designed codes can be represented as systematic codes with minimal redundancy is considered as well. >

Bounds on the minimum support weights

The minimum support weight, d/sub r/(C), of a linear code C over GF(q) is the minimal size of the support of an r-dimensional subcode of C. A number of bounds on d/sub r/(C) are derived, generalizing the Plotkin bound and the Griesmer bound, as well as giving two new existential bounds. As the main result, it is shown that there exist codes of any given rate R whose ratio d/sub rd/sub 1/ is lower bounded by a number ranging from (q/sup r/-1)/(q/sup r/-q/sup r-1/) to r, depending on R. >

New lower bounds on aperiodic crosscorrelation of binary codes

For the minimum aperiodic crosscorrelation /spl theta/(n,M) of binary codes of size M and length n over the alphabet {1,-1} there exists the celebrated Welch bound /spl theta//sup 2/(n,M)/spl ges/(M-1)n/sup 2//2Mn-N-1 which was published in 1974 and remained in this form up to now. In the article this bound is strengthened for all M/spl ges/4 and n/spl ges/2. In particular, it is proved that /spl theta//sup 2/(n,M)/spl ges/n-2n//spl radic/3M, M/spl ges/3 and /spl theta//sup 2/(n,M)/spl ges/n-[/spl pi/n//spl radic/8M], M/spl ges/5. In the asymptotic process when M tends to infinity as n/spl rarr//spl infin/, these bounds are twice as large as the Welch bound and coincide with the corresponding asymptotic bound on the square of the minimum periodic crosscorrelation of binary codes. The main idea of the proof is a new sufficient condition for the mean value of a nonnegative definite matrix over the code to be greater than or equal to the average over the whole space. This allows one to take into account weights of cyclic shifts of code vectors and solve the problem of their optimal choice.

Optimal conflict-avoiding codes for three active users

We consider the problem to construct a code of the maximum cardinality which consists of binary vectors of length n with three ones and has the following property: a matrix of size 3 times n from any cyclic shifts of any three different code vectors contains the identity matrix of size 3 times 3 (with accuracy up to a permutation of columns). This property (in more general form) was considered in the connection with the problem to avoid conflicts in the channels of multiple access under a restriction to the number of active users (see L. A. Bassalygo and M. S. Pinsker, Problemy Peredachi Informatsii, 1983; J. L. Massey and P. Mathys, IEEE Trans. Inform. Theory , 1985; B.S. Tsybakov and A.R. Rubinov, Problems of Information Transmission , 2002). The cardinality of such a code corresponds to the number of all users, and this property means that each from any three active users can successfully transmit a packet of information in one of three attempts to do it during n slots of time without a collision with other active users. In particular, cyclic Steiner triple systems give examples of such conflict-avoiding codes if we choose representatives of the cyclic classes as code vectors. In the paper we present some constructions of conflict-avoiding codes of triples which are better as compared with those obtained from the cyclic Steiner triple systems

Bounds for deletion/insertion correcting codes

Bounds for codes capable of correcting deletions and insertions are presented.

On designs in compact metric spaces and a universal bound on their size

Abstract For finite and compact infinite metric spaces, a concept of a (weighted) τ-design is introduced which depends on a choice of a substitution function. To estimate the minimum size of a τ-design a system of orthogonal polynomials is defined using the average measure of metric balls and the substitution function. A universal lower bound on the size of τ-designs is obtained with the help of the solution of the known extremum problem for systems of orthogonal polynomials. The concept of a τ-design and the bound considered coincide with those in the case of polynomial association schemes of Delsarte and the Euclidean sphere for the proper choices of the substitution functions. This bound is also calculated for some other spaces.