# Application of Discrete Structures and Numerical Sequences in Block Codes

The purpose to compress information using composition of universal codes with the recursive algorithm of original data recovery was achieved in this work. It obtains compression coefficient higher than in symbolic coding. Proposed method of time coding has reasonable values of compression coefficient and its purpose is coding with compression. For this purpose, entirely new kind of universal coding with the polybasic numeral system was created.

# Minimal Systems of Generators and Relations and Properties of Wreath Products of Perfect Groups

Generators and defining relations for wreath products of perfect group which is two generating and alternating groups (m  2 times) are given. System of generators of metaperfect groups are found. Generators and defining relations for wreath products of 2-generating perfect groups were found, including alternating groups, i.e. (m  2 time). Systems of generators for metaperfect groups were investigated. A constructive proof of the minimality found system of generators was presented. It is shown that metaperfect group is not locally finite group.

# Modified Algorithm of Shanks with Ordered Blocks

Majority of cryptanalytic methods can be modified due to parallel algorithms. One of them is the method of Shanks solving discrete logarithms problem. The main goal of this article is to construct algorithm, which allows parallel calculating all values from low and high pitch tables, to make this search more directed and to put in order all values of table elements. It will allow applying method of blocks searching, separation on the ordered sub-blocks, accelerate applying method of value indexing (or value hash).

# Generators and Relations of Syllows p-Subgroups of Group Sn

In this article, we investigate generators and relations of syllows subgroups of some symmetric group. It will enable studying syllows subgroups of other groups since every finite subgroup is isomorphicaly embedded in the syllows subgroup of some symmetric group. We find a set of relations for a fixed system of generators and prove that this set of relations is minimal between sets of relations. Research methods are the method of Shreier’s canonical words and rewriting process.

# Fractal Recursive Function Posses-sing Hyperexponential Growth

This article considers the research methods of deterministic fractal sets. Specifically, it develops constructive methods for studying recursive functions. The obtained results allow consider deterministic fractal functions from the novel theoretical standpoint.

# Fast Timer Encryption Method

A statistical-oriented method of data compression which using non traditional timer encryption is elaborated. The method uses special weight function. The necessary estimates of compression are developed. Proposed method of multiple system of labels for data compression and decompression method of collection of texts in one area of knowledge. It also was proposed to use a system of multiple labels for a set of texts, which are combined in solid archive. It was proved, that joint unzipping is more effective on the number of operations as a result – on time.

# Problem of Finiteness Conjecture and Joint Spectral Radius

This paper studies the constraint of a vector norm under periodic and aperiodic action of matrices from a finite set of matrices with rational elements as well as the presence of AZR and PZR. We consider a complex case when every matrix from has its own numbers that are both bigger and smaller units.