Sergey V. Avgustinovich

A UNIQUE DECOMPOSITION THEOREM FOR FACTORIAL LANGUAGES

We study decompositions of a factorial language to catenations of factorial languages and introduce the notion of a canonical decomposition. Then we prove that for each factorial language, a canonical decomposition exists and is unique.

On the Ranks and Kernels Problem for Perfect Codes

A construction is proposed which, for n large enough, allows one to build perfect binary codes of length n and rank r, with kernel of dimension k, for any admissible pair (r, k) within the limits of known bounds.

On the Structure of Symmetry Groups of Vasil'ev Codes

The structure of symmetry groups of Vasil’ev codes is studied. It is proved that the symmetry group of an arbitrary perfect binary non-full-rank Vasil’ev code of length n is always nontrivial; for codes of rank n − log(n + 1) +1, an attainable upper bound on the order of the symmetry group is obtained.

Avoidance of boxed mesh patterns on permutations

We introduce the notion of a boxed mesh pattern and study avoidance of these patterns on permutations. We prove that the celebrated former Stanley-Wilf conjecture is not true for all but eleven boxed mesh patterns; for seven out of the eleven patterns the former conjecture is true, while we do not know the answer for the remaining four (length-four) patterns. Moreover, we prove that an analogue of a well-known theorem of Erdos and Szekeres does not hold for boxed mesh patterns of lengths larger than 2. Finally, we discuss enumeration of permutations avoiding simultaneously two or more length-three boxed mesh patterns, where we meet generalized Catalan numbers.

On abelian versions of critical factorization theorem

In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.

Intersection matrices for partitions by binary perfect codes

We investigate the following problem: given two partitions of the Hamming space, their intersection matrix provides the cardinalities of the pairwise intersections of the subsets of these partitions. If we consider partitions by extended perfect codes, how many intersection matrices can we construct?.

Embedding in a perfect code

Abstract. A binary 1-error-correcting code can always be embedded in a 1-perfect code ofsome larger length. For any 1-error-correcting binary code C of length m we will construct a 1-perfect binary codeP(C) of length n = 2 m −1 such that fixing the last n−m coordinates by zeroes in P(C) gives C.In particular, any complete or partial Steiner triple system (or any other system that forms a1-code) can always be embedded in a 1-perfect code of some length (compare with [5]). Since theweight-3 words of a 1-perfect code P with 0 n ∈ P form a Steiner triple system, and the weight-4words of an extended 1-perfect code P with 0 n ∈ P form a Steiner quadruple system, we have, ascorollaries, the following well-known facts: a patrial Steiner triple (quadruple) system can alwaysbe embedded in a Steiner triple (quadruple) system [6] ([3]) (these results, as well as many otherembedding theorems for Steiner systems, can be found in [4, 1]).Notation:• F m denotes the set of binary m-tuples, or binary m-words.• F˙

To the Metrical Rigidity of Binary Codes

A code C in the n-dimensional metric space En over GF(2) is called metrically rigid if each isometry I : C → En can be extended to an isometry of the whole space En. For n large enough, metrical rigidity of any length-n binary code that contains a 2-(n, k, λ)-design is proved. The class of such codes includes, for instance, all families of uniformly packed codes of large enough lengths that satisfy the condition d − ρ ≥ 2, where d is the code distance and ρ is the covering radius.

On the Number of

We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.

1

A two-dimensional word is a function on Z^2 with finite number of values. The main problem we are interested in is the periodicity of two-dimensional words satisfying some local conditions. In this paper we prove that every bounded centered function on the infinite rectangular grid is periodic. A function is called centered if the sum of its values in every ball is equal to 0. Similar results are obtained for the infinite triangular and hexagonal grids.