PROBLEMS OF INFORMATION TRANSMISSION
A translation of Problemy Peredachi Informatsii

CONTENTS                  

Weakly Resolvable Block Designs and Nonbinary Codes Meeting the Johnson Bound
L. A. Bassalygo, V. A. Zinoviev, and V. S. Lebedev
pp. 1–12

Abstract—We present two new families of resolvable block designs. We introduce the notion of a weakly resolvable block design and prove the equivalence of such designs and nonbinary codes meeting the Johnson bound. We construct a new family of such codes.

Reduction of Recursive Filters to Representations by Sparse Matrices
A. Yu. Barinov
pp. 13–31

Abstract—A recursive filter as a part of a recursive convolutional code is of practical importance in composite interleaved code circuits. We consider a matrix description of recursive filters in the time domain over the finite field $\mathbb F_2$. We analyze and formalize the reduction of matrices describing recursive filters (with puncturing) to sparse matrices of a special form. We mainly address the analysis of binary sequences of recursive filters with puncturing every second bit. We describe the application of the obtained sparse matrices to finding punctured transfer functions for such filters. We propose an approach to the minimal circuit realization of the punctured transfer functions. We give examples of circuit realizations of punctured turbo codes as duo-binary turbo codes.

Multi-twisted Additive Codes with Complementary Duals over Finite Fields
S. Sharma and A. Sharma
pp. 32–57

Abstract—Multi-twisted (MT) additive codes over finite fields form an important class of additive codes and are generalizations of constacyclic additive codes. In this paper, we study a special class of MT additive codes over finite fields, namely complementary-dual MT additive codes (or MT additive codes with complementary duals) by placing ordinary, Hermitian, and $\ast$ trace bilinear forms. We also derive a necessary and sufficient condition for an MT additive code over a finite field to have a complementary dual. We further provide explicit enumeration formulae for all complementary-dual MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We also illustrate our results with some examples.

On $q$-ary Propelinear Perfect Codes Based on Regular Subgroups of the General Affine Group
I. Yu. Mogilnykh
pp. 58–71

Abstract—A code is said to be propelinear if its automorphism group contains a subgroup acting on its codewords regularly. A subgroup of the group $G\kern-.3ptA(r,q)$ of affine transformations is said to be regular if it acts regularly on vectors of $\mathbb{F}_q^r$. Every automorphism of a regular subgroup of the general affine group $G\kern-.3ptA(r,q)$ induces a permutation on the cosets of the Hamming code of length $\dfrac{q^r-1}{q-1}$. Based on this permutation, we propose a construction of $q$-ary propelinear perfect codes of length $\dfrac{q^{r+1}-1}{q-1}$. In particular, for any prime $q$ we obtain an infinite series of almost full rank $q$-ary propelinear perfect codes.

Bounds on Threshold Probabilities for Coloring Properties of Random Hypergraphs
A. S. Semenov and D. A. Shabanov
pp. 72–101

Abstract—We study the threshold probability for the property of existence of a special-form $r$-coloring for a random $k$-uniform hypergraph in the $H(n,k,p)$ binomial model. A parametric set of $j$-chromatic numbers of a random hypergraph is considered. A coloring of hypergraph vertices is said to be $j$-proper if every edge in it contains no more than $j$ vertices of each color. We analyze the question of finding the sharp threshold probability of existence of a $j$-proper $r$-coloring for $H(n,k,p)$. Using the second moment method, we obtain rather tight bounds for this probability provided that $k$ and $j$ are large as compared to $r$.