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A Steiner triple system (STS) contains a transversal subdesign TD(3, w) if its point set has three pairwise disjoint subsets A, B, C of size w and w2 blocks of the STS intersect with each of A, B, C (those w2 blocks form a TD(3,w)). We prove several structural properties of Steiner triple systems of order 3w + 3 that contain one or more transversal subdesigns TD(3, w). Using exhaustive search, we find that there are 2 004 720 isomorphism classes of STS(21) containing a subdesign TD(3, 6) (or, equivalently, a 6 × 6 Latin square). [ABSTRACT FROM AUTHOR]

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs. [ABSTRACT FROM AUTHOR]

We study the symmetry group of a binary perfect Mollard code M( C,D) of length tm + t + m containing as its subcodes the codes C and D formed from perfect codes C and D of lengths t and m, respectively, by adding an appropriate number of zeros. For the Mollard codes, we generalize the result obtained in [1] for the symmetry group of Vasil'ev codes; namely, we describe the stabilizer Sym( M( C,D)) of the subcode D in the symmetry group of the code M( C,D) (with the trivial function). Thus we obtain a new lower bound on the order of the symmetry group of the Mollard code. A similar result is established for the automorphism group of Steiner triple systems obtained by the Mollard construction but not necessarily associated with perfect codes. To obtain this result, we essentially use the notions of 'linearity' of coordinate positions (points) of a nonlinear perfect code and a nonprojective Steiner triple system. [ABSTRACT FROM AUTHOR]

We consider generalized Preparata codes with a noncommutative group operation. These codes are shown to induce new partitions of Hamming codes into cosets of these Preparata codes. The constructed partitions induce 2-resolvable Steiner quadruple systems S( n, 4, 3) (i.e., systems S( n, 4, 3) that can be partitioned into disjoint Steiner systems S( n, 4, 2)). The obtained partitions of systems S( n, 4, 3) into systems S( n, 4, 2) are not equivalent to such partitions previously known. [ABSTRACT FROM AUTHOR]

All different Steiner systems S(2, 4, 3) of order 2 and rank 2 − m − 1 + s over $$\mathbb{F}_2$$, where 0 ≤ s ≤ m − 1, are constructed. The number of different systems S(2, 4, 3) whose incident matrices are orthogonal to a fixed code is obtained. A connection between the number of different Steiner systems and of different Latin cubes is described. [ABSTRACT FROM AUTHOR]

An algorithm for generating parity-check matrices of regular low-density paritycheck codes based on permutation matrices and Steiner triple systems S( v, 3, 2), v = 2 − 1, is proposed. Estimations of the rate, minimum distance, and girth for obtained code constructions are presented. Results of simulation of the obtained code constructions for an iterative 'belief propagation' (Sum-Product) decoding algorithm applied in the case of transmission over a binary channel with additive Gaussian white noise and BPSK modulation are presented. [ABSTRACT FROM AUTHOR]

The structure of all different Steiner triple systems S(2−1, 3, 2) of rank 2− m+2 over $\mathbb{F}_2 $ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order 2 − 1 and rank ≤ 2 − m + 2. The number of such different systems of order 2 − 1 and rank less than or equal to 2 − m + 2 which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order 2 − 1 and rank ≤ 2 − m + 2 are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank. [ABSTRACT FROM AUTHOR]

A letter to the editor is presented in response to the article "Steiner Triple Systems S(2m - 1, 3, 2) of Rank 2m -m + 1 over F2" published in a 2012 issue.

Steiner systems S(2 − 1, 3, 2) of rank 2 − m+1 over the field $$\mathbb{F}_2$$ are considered. A new recursive method for constructing Steiner triple systems of an arbitrary rank is proposed. The number of all Steiner systems of rank 2 − m+1 is obtained. Moreover, it is shown that all Steiner triple systems S(2 − 1, 3, 2) of rank r ≤ 2 − m+1 are derived, i.e., can be completed to Steiner quadruple systems S(2, 4, 3). It is also proved that all such Steiner triple systems are Hamming; i.e., any Steiner triple system S(2 − 1, 3, 2) of rank r ≤ 2 − m + 1 over the field $$\mathbb{F}_2$$ occurs as the set of words of weight 3 of a binary nonlinear perfect code of length 2−1. [ABSTRACT FROM AUTHOR]

For an arbitrary Steiner system S( v, k, t), we introduce the concept of a component as a subset of a system which can be transformed (changed by another subset) without losing the property for the resulting system to be a Steiner system S( v, k, t). Thus, a component allows one to build new Steiner systems with the same parameters as an initial system. For an arbitrary Steiner system S( v, k, k − 1), we provide two recursive construction methods for infinite families of components (for both a fixed and growing k). Examples of such components are considered for Steiner triple systems S( v, 3, 2) and Steiner quadruple systems S( v, 4, 3). For such systems and for a special type of so-called normal components, we find a necessary and sufficient condition for the 2-rank of a system (i.e., its rank over $\mathbb{F}_2$) to grow under switching of a component. It is proved that for k ≥ 5 arbitrary Steiner systems S( v, k, k − 1) and S( v, k, k − 2) have maximum possible 2-ranks. [ABSTRACT FROM AUTHOR]

Inaccuracies in computations in the paper of the authors on classification of perfect binary codes of lengths 15 and 16 and of rank 13 are fixed. An explicit construction of all extended perfect codes of length 16 and rank 13 with a given kernel size is presented. Perfect binary codes of length 15 and rank 14 obtained by the general doubling construction are classified. [ABSTRACT FROM AUTHOR]

Abstract  A transformation of Steiner quadruple systems S(υ, 4, 3) is introduced. For a given system, it allows to construct new systems of the same order, which can be nonisomorphic to the given one. The structure of Steiner systems S(υ, 4, 3) is considered. There are two different types of such systems, namely, induced and singular systems. Induced systems of 2-rank r can be constructed by the introduced transformation of Steiner systems of 2-rank r − 1 or less. A sufficient condition for a Steiner system S(υ, 4, 3) to be induced is obtained. [ABSTRACT FROM AUTHOR]

Abstract  We study modal logics of regions in a real space ordered by the inclusion and compact inclusion relations. For various systems of regions, we propose complete finite modal axiomatizations; the described logics are finitely approximable and PSPACE-complete. [ABSTRACT FROM AUTHOR]

Abstract  The paper deals with the problem of constructing a code of the maximum possible cardinality consisting of binary vectors of length n and Hamming weight 3 and having the following property: any 3 � n matrix whose rows are cyclic shifts of three different code vectors contains a 3 � 3 permutation matrix as a submatrix. This property (in the special case w = 3) characterizes conflict-avoiding codes of length n for w active users, introduced in [1]. Using such codes in channels with asynchronous multiple access allows each of w active users to transmit a data packet successfully in one of w attempts during n time slots without collisions with other active users. An upper bound on the maximum cardinality of a conflict-avoiding code of length n with w = 3 is proved, and constructions of optimal codes achieving this bound are given. In particular, there are found conflict-avoiding codes for w = 3 which have much more vectors than codes of the same length obtained from cyclic Steiner triple systems by choosing a representative in each cyclic class. [ABSTRACT FROM AUTHOR]

Abstract  A list decoding algorithm is designed for the first-order binary Reed-Muller codes of length n that reconstructs all codewords located within the ball of radius n/2(1 − ɛ) about the received vector and has the complexity of O(n ln2(min{ɛ −2, n})) binary operations. [ABSTRACT FROM AUTHOR]

Abstract  A Steiner triple system of order n (for short, STS(n)) is a system of three-element blocks (triples) of elements of an n-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple (i,j,k) ∊ STS(n) a topological triangle with vertices i, j, and k. Gluing together like sides of the triangles that correspond to a pair of disjoint STS(n) of a special form yields a black-and-white tiling of some closed surface. For each n ≡ 3 (mod 6) we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order n. We also show that for half of the values n ≡ 1 (mod 6) there are nonisomorphic tilings of nonorientable closed surfaces. [ABSTRACT FROM AUTHOR]

Abstract  Two new constructions of Steiner quadruple systems S(v, 4, 3) are given. Both preserve resolvability of the original Steiner system and make it possible to control the rank of the resulting system. It is proved that any Steiner system S(v = 2 m , 4, 3) of rank r ≤ v − m + 1 over F2 is resolvable and that all systems of this rank can be constructed in this way. Thus, we find the number of all different Steiner systems of rank r = v − m + 1. [ABSTRACT FROM AUTHOR]