Optimum Vector Information Technologies Based on the Multi-

dimensional Combinatorial Configurations

VOLODYMYR RIZNYK

Lviv Polytechnic National University

79013, Lviv-13, Stepan Bandera Str., 12

UKRAINE

Abstract: - Paper devoiced to optimum vector information technologies based on the multi-dimensional

combinatorial configurations, such as Ideal Ring Bundles (IRBs). One-dimensional IRBs are ring ordered

positive integers that form finite set of integers from 1 to S using both these numbers and all its consecutive

terms. Two- and multi-dimensional IRBs make available to configure intelligent information and

telecommunication systems providing generate the maximum number of distinct vector sums of consecutive

terms in the combinatorial configuration. Applications profiting from optimum vector information

technologies based on the multi-dimensional combinatorial theory provide for example data mining

technologies and big vector data processing, data analysis and system security, signal compression and

reconstruction, vector computing and telecommunications, and other branches of sciences and advanced

information technologies.

Key-Words: - IRB, torus coordinate system, manifold, star- code, big data, optimum vector data coding

system, encryption, security

1 Introduction

Combinatorial optimization is a subfield

of mathematical optimization that relates finding

optimal solutions of problems connected from

sets that have finite number of elements, where the

sets of feasible solutions are discrete. Classic

combinatorial optimization problems are, for

example, the travelling salesman problem,

the knapsack proble, and the minimum spanning

tree problem. Such problems, exhaustive search is

not high-quality, and so directed on algorithms that

quickly rule out large parts of the search space or

estimate algorithms must be resorted to instead.

Combinatorial optimization is related to algorithm

theory, operations research, and computational

complexity theory. It has significant applications

in several fields, including artificial intelligence

and machine learning, auction theory and software

engineering, very large-scale integration (VLSI)

and applied mathematics, big vector data

processing and theoretical computer science, data

mining technologies and big data, intelligent

systems and data analysis, computational

linguistics and computer-aided design (CAD), and

other engineering areas. A number of research

works consider combinatorial configurations using

optimization of integer programming together with

discrete optimization of big vector data coding and

data mining technologies, which is composed with

multidimensional combinatorial configurations.

In turn, all of these topics have closely intertwined

research literature. It time involves innovative

combinatorial methodologies to efficiently allocate

wherewithal for finding solutions of many

mathematical problems.

A number of research works consider

combinatorial optimization consisting integer

programming together with discrete optimization

of big vector data coding and data mining

technologies, which is composed of many

problems dealing with effective researches of

intelligent multi-dimensional spatial combinatorial

structures, using their remarkable propwrties. All

of these topics have closely intertwined research

literature. It time involves innovative

combinatorial methodologies to efficiently allocate

wherewithal for finding solutions to optimum

vector information technologies based on the

multi-dimensional combinatorial configurations.

2 Problem Formulation

The problem to be of very important for configure

optimum vector information technologies and

systems of information security with improved the

quality indices of the system with respect to

performance reliability, data protection, and speed

transformation big information content.

Received: July 12, 2022. Revised: September 7, 2023. Accepted: October 11, 2023. Published: November 6, 2023.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

104

Volume 3, 2023

Mathematical problem is reduced to establishing a

mutually unambiguous display of vector binary

code combinations sets according to vector data

attribute-categories sets on the coordinate grid of

the t-dimensional surface of the manifold. The task

is to increase the number of code combinations of

t- dimensional binary code for the formation of

information parameters of signals by the number

of attributes and categories in the basis of the

outlined t- dimensional coordinate system. We

require the code combinations enumerate fixed

number of times both vector data attribute-

categories sets and node points set of the

coordinate grid with sizes m1× m2 ×…× mi ×…×mt,

where mi – number markers for referencing

indexed categories on i-th axis, which corresponds

to one of t attributes in the manifold coordinate

system.

The main goal of modern intelligent systems

engineering and big data mining technologies is

expansion of advanced data processing for optimal

solution of wide classes of problems, including big

vector data information systems and data mining

technologies focused on international

academicians, scientists and practitioners to

exchange new ideas for future collaboration.

3 Rewiev of Literature

Big vector data information technology, is known,

be able to defined as a softw are-utility that is

designed to analysis process and extract the data

from extremely complex and large data sets which

the traditional data processing software could

never deal with. In global review [1] presented big

spatial vector data management. In this paper,

autors discuss and itemize this topic from three

aspects according to different information

technical levels of big spatial vector data

management. It aims to help interested readers to

learn about the latest research advances and choose

the most suitable big data technologies and

approaches depending on their system

architectures. To support them more fully, firstly,

authors identify new concepts and ideas from

numerous scholars about geographic information

system to focus on big spatial vector data scope.

They conclude systematically not only the most

recent published literatures but also a global view

of main spatial technologies of big spatial vector

data, including data storage and organization,

spatial index, processing methods, and spatial

analysis. Finally, based on the above commentary

and related work, several opportunities and

challenges are listed as the future research interests

and directions for reference. This review paper

mainly focuses on big spatial vector data

management in the era of big data. The big spatial

data, data storage and organization, data

processing, and spatial analysis are discussed,

respectively. In the context of big spatial vector

data management, this study categories the

existing techniques and technologies, as well as

highlighting the mainstream academic views to

help the readers to better understand and handle

the problems from big spatial vector data

management efficiently. The existing work in big

spatial vector data management has mostly

emphasized on some characteristics (volume,

variety, or velocity) of big spatial data, and solved

certain problems in the technical level or

applications. Although there already have several

studies related to big spatial data. In addition, this

review summarizes the characteristics and domains

of big spatial data, and also overviews the big

spatial vector data management. Moreover, a

broad of literatures on the vector data model, data

storage, spatial index, pre-processing, spatial

query, visualization, and spatial analysis of big

spatial vector data are provided and classified.

Future research interests and directions are

contributed as a guide for researchers. The key–

value model is now the mainstream of the storage

model in a large number of NoSQL databases. In

the key–value model, each record consists of two

parts, also known as “Key/Value Pair”, which

supports simple data operation. Driven by the

wave of big data technology, big spatial vector

data has been affected and changed, especially for

the data management. This paper starts a

ddiscussion for the existing work of big spatial

vvector data (BSVD) management and

summarizes three main aspects, namely big spatial

data, data storage and organization, data

processing, and analysis, which are carried out a

detail description from the theoretical and

technical levels. The overview of BSVD

management is discussed firstly, and then, the big

spatial vector data model, storage mode and spatial

index are described in the layer of data storage and

organization. Furthermore, authors discussed the

data pre-processing, spatial query, visualization,

and spatial analysis. Finally, three future research

interests adirections are presented in the work.

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

105

Volume 3, 2023

In book [2] we can see a polynomial or a rational

function (matrix), characterizing a single-input

single-output (multi-input multi-output) system,

has the coefficients for parameters. The number of

such free parameters defines the dimension of the

space and a system with fixed parameters may be

represented by a point in parameter space. If at

least configuration for the one coefficient varies

about its nominal value, a region in parameter

space is generated. This region characterizes a

family of systems instead of one fixed system.

When the coefficients vary independently of each

other within specified compact intervals,

an interval system is generated. A well explored

case when the coefficients do not vary

independently occurs when the region in parameter

space is a bounded polyhedral set. A polyhedral

set is formed from the intersection of a finite

number of closed half-spaces and could be

unbounded. A bounded polyhedral set is a convex

polytope and vice versa. For an interval system,

the polytope degenerates into a boxed domain or a

hyper-rectangle. Extensive documentation of

research results concerned with the extraction of

information about the complete polytope from a

very small subset of the polytope with respect to

the property of stability for both continuous-time

and discrete-time systems is available in several

recent texts. The goal is to obtain tests for

invariance of useful properties of sets of

distinguished classes of functions from tests on a

small subset of such functions.

Many number of original models, conceptions,

parallel algorithms, platforms, applications and

processing gears, relate to improve the assessment

of big data technology [3 - 10]. The big data sets

again have a lot of factors of infrastructure,

including trade and industry, defense and

economic and other indexes, which have many

difficult problems. The papers [3-5] deal with

forecast of big vector data for remote sensing. The

techniques for create of a map procedure, which

make filtering, sorting, reconstruction and a

summary of big data operations presented at [6], at

the IEEE International Conferences on Data

Engineering. The paper [7] contains prompt vector

data ensemble experimental method

decomposition for the analysis of big spatial and

temporal datasets.

From the work [8] we can see that geometric

computing algorithms are time-consuming and

very complex, which make big spatial data

processing extremely slow. In the paper [9] a

configuration for the parallel map projection of

vector big spatial data which involves cloud and

high-performance computing regarded. Large

payer data amassed to explore big vector data for

advancing information in nursing methodologies,

clinical trials and lab research are described in

[11]. The suggestion of torus topological

coordinates for chemical structures is in concord

with relating the physics of torus confined plasmas

[12]. The research works provide spatial modeling

of large-scale vector data under a common

coordinate system. Still algorithmic complexity of

the display projections represents pressing

computational challenge.

Present theory of combinatorial configurations

are such spatial structures as perfect difference sets

[13] algebraic constructions based on cyclic

groups in extensions of Galois fields [14]

manifolds [15], structures connecting algebra

through geometry [16] In general case it was

possible to take in consideration a new conceptual

model of the data processing based on the laws of

worldwide harmony, such as Golden ratio [17] and

Perfect Distribution Phenomenon as relationships

“parts-whole” of complementary asymmetries

joined harmonically in the rotational symmetry,

forming optimum t- dimensional coordinate

system over closed manifold shape [18].

4 Problem Solution

Problem solution involves research into techniques

for finding technologically optimum into the

underlying mathematical principles relating to the

optimal structural and information parameters of

multi-dimensional combinatorial configurations

for development optimum vector information

technologies based on the configurations. These

design techniques will make it possible to

configure information technologies and systems

with fewer structural elements and bonds than at

present, while maintaining or improving on

computer power, data protection and the other

operating characteristics of the system, using

optimum vector information technologies based on

the multi-dimensional combinatorial

configurations.

4.1 Two-dimensional Vector Ideal Ring

Bundles

Two-dimensional Vector Ideal Ring Bundles (2D

vector IRBs) follow from n-stage two-dimensional

cyclic sequences of non-negative 2-stage (t=2)

integer sub-sequences of the sequence. A set of all

two-modular vector-sums (mod m1, mod m2) of the

sub-sequences creates grid over surface of torus m1

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DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

106

Volume 3, 2023

× m2. A set of all node coordinates of the grid

occurs exactly R-times, where m1 and m2 are ring

reference axes.

From the underlying definition follows formula

(1): R∙(S – 1) = n∙ (n – 1), (1)

where S, n, R, and m1 × m2 - parameters of

two- dimensional

Example. Two-dimensional (t=2) vector IRB

containing four (n=4) vectors {(0,2), (2,3), (0,1),

(1,3)} with parameters S= 13, n = 4, R = 1, and m1

= 3, m2 = 4 depicted in Fig. 1.

Fig. 1. Two-dimensional (t=2) vector IRB

containing four (n=4) {(0,2), (2,3), (0,1), (1,3)}

with parameters S= 13, n = 4, R = 1, and m1 = 3, m2

= 4

Creation of a torus coordinate system based on

the Ideal Ring Bundle {(1,3), (0,2), (2,3), (0,1)}

with parameters S =13, n = 4, R = 1, and m1=3, m2

= 4 showed in the Table 1.

Table 1

Node points of reference coordinate grid 34

created by modular vector sums (m1=3, m2=4)

using 2D vector Ideal Ring Bundle {(1,3), (0,2),

(2,3), (0,1)}

Node

points

Modular vector sums (m1=3, m2=4)

(1,3)

(0,2)

(2,3)

(0,1)

(0,0)

(1,3)

(0,2)

(2,3)

-

(0,1)

-

(0,1)

(0,2)

-

(0,2)

-

(0,3)

(1,3)

-

(2,3)

(0,1)

(1,0)

(1,3)

-

(0,1)

(1,1)

(1,3)

(0,2)

-

(1,2)

(1,3)

(0,2)

-

(0,1)

(1,3)

-

(2,0)

-

(2,3)

(0,1)

(2,1)

-

(0,2)

(2,3)

-

(2,2)

-

(0,2)

(2,3)

(0,1)

Node

points

Modular vector sums (m1=3, m2=4)

(1,3)

(0,2)

(2,3)

(0,1)

(2,3)

-

(2,3)

-

Table 1 contains n∙(n−1)=12 node points of

reference coordinate grid created by modular

vector sums (m1=3, m2=4) using 2D Ideal Ring

Bundle {(1,3), (0,2), (2,3), (0,1)}.

The sequence of the basic vectors {(1,3), (0,2),

(2,3), (0,1)} form two-dimensional (t =2) grid over

torus surface grid 3 × 4.

In the Fig. 2 we can see a symbolic presentation

of two (t =2) annular axes over surface of usual

torus with coordinate grid m1 × m2 = 3 × 4 and

reference point (0,0).

Fig.2. Symbolic presentation of two (t =2)

annular axes over surface of usual torus with grid

m1 × m2 = coordinate m1 × m2 = 3 × 4 and reference

point (0,0)

Symbolic presentation of torus coordinate system

is the simplest and well useful presentation of

combinatorial optimization of vector data coding

and processing based on remarkable properties of

two-dimensional IRBs.

Binary 2D vector code based on the IRB {(1,3),

(0,2), (2,3), (0,1)} in torus coordinate system

m1×m2=3×4 presented in Table 2.

Table 2

Binary 2D vector code based on the IRB {(1,3),

(0,2), (2,3), (0,1)} in torus coordinate system

m1×m2=3×4

Node

points

Digit weights

(1,3)

(0,2)

(2,3)

(0,1)

(0,0)

1

0

(0,1)

0

1

(0,2)

0

1

0

(0,3)

1

0

1

(1,0)

1

0

1

(1,1)

1

0

(0,2)

(1,3)

(2,3)

(0,1)

m1 =3 m2 =4

(0,0)

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

107

Volume 3, 2023

Node

points

Digit weights

(1,3)

(0,2)

(2,3)

(0,1)

(1,2)

1

0

1

(1,3)

1

0

(2,0)

0

1

(2,1)

0

1

0

(2,2)

0

1

(2,3)

0

1

0

Table 2 contains n∙ (n−1) =12 binary four- digit

(n = 4) combinations for coding two categories (t =

2) with three (m1 = 3) attributes of the first, and

four (m2 = 4) – the second category. Each of the

combinations allows coding two (t=2) indexed

data simultaneously.

4.2 Optimum Vector Codes

We can extend set of categories and increase

number of data code size, using optimum star-

codes. The optimum vector star-codes follow from

the self named “Gloria to Ukraine Stars” codes

[18]. The optimum vector star-codes may be

separating as self-correcting (redundant vector

codes), and the non-redundant optimum codes with

merits and limitations for each of them.

For example, two-dimensional (t=2) self-

correcting optimum vector code with digit weights

{(1,3), (0,2), (2,3), (0,1)} forms encoding design in

torus coordinate system m1×m2=3×4 the maximum

number Pmax= m1×m2=12 allowed vector code

words, each of them is no more one block of the

same binary symbol. We identify this class of star-

codes as the optimum vector annular monolithic-

group self-correcting codes (Table 2).

The non-redundant optimum vector star-code

with digit weights {(1,2),(2,4),(1,3),(2,1)} forms

encoding design in torus coordinate system

m1×m2=3×5. Unlike self-correcting star-codes, the

non-redundant vector star-code provides growing

code size for the same number of code positions.

maximum number Pmax= m1×m2=12 allowed vector

code words, each of them is no more one block of

the same binary symbol. We identify this class of

the optimum vector star-codes as the non-

redundant optimum vector codes (Table 3).

The code size of n- digit binary code cannot be

greater than the number of nonzero binary code

combinations formed by it.

Table 3 illustrates forming binary 4-digit 2D

optimum vector star-code {(1,2),(2,4),(1,3),(2,1)}

in torus coordinate system m1×m2=3× 5.

Table 3

Binary 4-digit 2D optimum vector star-code

{(1,2), (2,4),(1,3),(2,1)} in torus coordinate

system m1×m2=3×5

Node

points

Digit weights

(1,2)

(2,4)

(1,3)

(2,1)

(0,0)

1

(0,1)

1

0

(0,2)

0

1

0

(0,3)

1

0

1

(0,4)

0

1

(1,0)

0

1

0

1

(1,1)

1

0

1

(1,2)

1

0

(1,3)

0

1

0

(1,4)

1

0

(2,0)

1

0

1

0

(2,1)

0

1

(2,2)

1

0

1

(2,3)

0

1

(2,4)

0

1

0

Table 3 consists 4-digit (n=4) binary 2D (t=2)

star-code {(1,2),(2,4),(1,3),(2,1)} as non-redundant

code with parameters n=4, R=1, and code size P

(n) = 15.

Theorem. The power of the method of converting

the form of information with t–measurable

optimum star- code is greater than in classical

binary codes.

Proof. With the increase in the number of t

measurements of vector weight digits of the star-

code, the total number of transformations on the

set of basic vectors of weight digits as

multiplicative groups increases accordingly,

supplemented by options for mutual

rearrangements of digits in the structure of

"stellar" ensembles and corresponding

permutations of numbers within the base vectors,

which makes it possible to obtain more invariants

of code combinations than standard code.

Theorem is proved.

A more general model of the t-dimensional

toroidal coordinate system for vector data coding

and processing made from multidimensional

combinatorial configurations that provide an

ability to reproduce the maximum number of

vectors in the system [18], [19].

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DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

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Volume 3, 2023

5. Vector Data Processing in Spatial

Coordinate Systems

The basic ideas of vector data processing in spatial

coordinate system are as following:

- determine sizes of intelligent spatial coordinate

system and its dimensionality accordingly to

category- attribute of vector data list;

- make digital indexing category- attribute list in

the intelligent coordinate system;

- fetch from an information base applicable vector

code with respect to computer power and

processing program;

- make vector data processing in the intelligent

coordinate system.

The underlying methods provide opportunities

to apply them to configure suitable relation big

vector data models, using arbitrary number of

characteristics, and its associations.

For example, vector data coding by two (t = 2)

categories of three (n=3) attributes under torus

coordinate system with reference grid m1 × m2 = 2

× 3 based on the 2D Ideal Ring Bundle {(1,0),

(0,1), (0,2)} presented in Table 4.

Table 4

Vector data coding by two (t=2) categories of

three (n=3) attributes under torus coordinate

system with reference grid m1 × m2 = 2 × 3 based

on the 2D Ideal Ring Bundle {(1,0), (0,1), (0,2)}

Index of

category

Index of

attribute

Digit weights

(1,0)

(0,1)

(0,2)

1

0

1

0

1

0

1

2

1

0

1

0

2

0

1

0

1

0

1

Table 4 contains a set of six (m1× m2 = 2 × 3 = 6)

3-digit (n=3) binary code words for data

processing (storage, encryption, transmission etc.),

each of them have information about two (t = 2)

indexed data. For example, category with digital

index “1” and attribute with index “1” respond to

code word “1 1 0”, because {(1,0) + (0,1)} ≡

{1,1}(mod m1,mod m2), where m1 = 2, m2 = 3;

attribute with digital index “0” and category with

index “1” - to code word “0 1 0”.

Category with digital index “1” and attribute with

index “2” respond to code word “1 0 1” , because

{(1,0) + (0,2)} ≡ {1,2}(mod m1,mod m2). Category

with digital index “0” and attribute with index “2”

respond to code word “0 0 1”; category with

digital index “1” and attribute with index “0” - to

code word “1 0 0” or “111”. Finally, category

with digital index “0” and attribute with index “0”

respond to code word “0 1 1”. So, vector code

based on the IRB {(1,0), (0,1), (0,2)} make it

possible to organize vector information technology

for data processing over combinatory two-digital

sets of indexed arbitrary categories and attributes,

taking in account two (t = 2) categories and three

(n=3) attributes. Such technique provides storage,

sorting, and translation two indexed information

categories of the vector data concurrently. At the

same manner provides encoding big datasets of

arbitrary content at any level of indexing by a

priory infinitely large number of the datasets

providing big data processing without parallel

computing. .

Information encoded in vector signals with a

limited number of transient energy levels helps to

reduce energy costs and increases the level of

protection against external interference. In

addition, it is possible to encrypt the processed

data, for example, by periodic renaming of

coordinate axis numbers, rearranging vector

weight digits of the loop code, etc. Surface

topology is superior to geometry relating manifold

as “perfect” shape that useful to visualize objects

as mathematical model of multidimensional

systems for big vector data processing under

manifold reference systems based on suitable IRB

combinatorial configurations. The underlying

techniques can be used for easy-to-grasp

representation of big vector data processing under

the systems. Remarkable combinatorial properties

and structural perfection of the combinatorial

configurations provide high performance vector

computing technologies for effective big vector

data processing. Advanced big vector data

information technologies based on concept of IRB

combinatorial configurations provide competitive

advantages of the information technologies with

respect to processing speed and storage capacity

due to vector coding of compound attributes for

two and more their categories simultaneously.

Theoretically, there are infinitely many intelligent

IRB ensembles with increasing bit depth and

dimension of optimized t-dimensional vector

codes, which can be processed and sent by

communication channels for the same period of

time a greater amount of information compared to

the capabilities of classical analogues.

Vector IRB- and star-codes open up new

prospects for the use of combinatorial methods of

optimized encoding of multidimensional signals

for big data revolution in data storage and

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E-ISSN: 2769-2477

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Volume 3, 2023

processing, information and communication

technologies and vector computer engineering.

As it evident, combinatorial optimization of

big data processing under manifold coordinate

systems based on the concept of Ideal Ring

Bundles (IRB)s can be used for finding optimal

solutions for wide classes of problems related to

intelligent systems, including optimum vector data

coding and decoding design, data content

transformation, storage and security. One of them

presents mutually unambiguous compliance with a

set of indexed data “category-attributes” of a set of

binary vector code combinations formed by this

database have been achieved in the system. In turn,

it was possible due to reducing the natural

redundancy in the system.

6 Discussion

As it evident, the Table 2 demonstrates the

advantages of two- dimensional (t=2) binary

vector data coding in the minimized 3- digit (n =

3) 2D database of the torus coordinate system a

reference grid with sizes m1 × m2= 3 × 4. The

mutually unambiguous compliance with a set of

indexed data “category-attribute” of a set of binary

vector code combinations formed by this database

have been achieved in the system. The optimal t-

dimensional codes are based on the encoding of t–

measurable signals by annular monolithic-group

code, where any allowed ring code combination

allows the presence of no more than one block of

characters of the same name. This allows you to

instantly detect false combinations on the basis of

group distribution, and the code acquires self-

corrective properties. As it evident, the Table 3

demonstrates the advantages of binary vector data

coding in the minimized 4- digit (n = 4) 2D

database of the toroid coordinate system a

reference grid with sizes m1 × m2= 3 × 5. The

mutually unambiguous compliance with a set of

indexed data “category- attribute” of a set of

binary vector code combinations formed by this

database have been achieved in the system. In turn,

it was possible due to reducing the information

redundancy in the system.

Reasoning along similarly, Table 4 shows

method of vector data coding design in the

minimized database of the torus coordinate system

a reference grid with sizes m1 × m2 = 2 × 3.

In the formula (1), the underlying rule

displayed by the number R of methods of covering

all node points of t- dimensional outlined toroid

coordinate system for encoding of indexed vector

data “category- attribute” sets with t categories and

appropriated number of attributes in the set.

Besides, using n- digit base reconstruction of t-

dimensional binary vector code we provide the

rule: repetition is a loss of information in order to

increase so-many times’ security of vector data

coding under the coordinate system. Increasing the

number of allowed code combinations of the

coding method and improving the characteristics

of vector data coding and processing by

performance and interference, as well as the star-

code information capacity increases faster than

code size of the code with growing dimensionality

and number of its binary digits. Growing vector

code combination by one bit doubles the code size

of the encoding method in the outlined toroid

coordinate system with the corresponding

dimensions and dimensions. The number of

indexed attributes and categories in the form of

any long a priori of integer t- tuple allows one

code word to encode, forward and process in the

basis of the system simultaneously as many signs

of vector data as symbols contained in the t- tuples

without parallel computing. It respectively

provides increasing the performance of the system

at t times, where t is the number of annular axes of

the t –dimensional coordinate system.

Formalization allows you to reach a balanced

compromise on contradictory goals related to the

power and reliability of the information method. It

outlines theoretically a large- scale information

model of harmoniously built of multidimensional

geometric space as a hypothetical system of a

perfectly structured source of information as t-

dimensional locked spheres. This system has a

priori infinitely large number of sets of coordinate

sub- systems of quantum "density", generated by a

minimized basis of t- dimensional binary n- bit

code with compression coefficient value

approaching to 2n/n.

7 Conclusion

The essence of the proposed project technology is

processing multidimensional data in the database

without loss of information, processing data arrays

in streaming mode by lists of attribute sets at the

same time, ability to change the number of

categories and attributes when processing

information. Application profiting from optimized

vector information technologies under the torus

coordinate systems are the ability to update

completed tables with indexing of names,

packages, procedures, etc. These design techniques

International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

110

Volume 3, 2023

provides optimum vector information technologies

based on the underlying combinatorial

configurations, using formalization of

interdependence between information parameters

of vector data coding systems on a single

mathematical platform. It reflects the essence of

the proposed concept of converting

multidimensional form of information with binary

code in a structured field of toroid coordinate

systems of corresponding dimensions and

dimensionalities for provide optimum coding and

processing vector data arrays with numbering

categories and attributes under minimized basis of

outlined t- dimensional toroid coordinate system,

using the underlying formalization (1), (2).

The scientific novelty of obtained results is the

formalization of interdependence between

information parameters of vector data coding

systems on a single mathematical platform. This

approach allows forming the outlined t–

dimensional toroidal coordinate system, using

smaller as all number of coordinates set basis of t-

tuples. This, in turn, provides optimum coding and

processing vector data arrays with numbering

attributes and categories under minimized basis of

outlined t- dimensional toroidal coordinate system

by holding criterion and limitations to achieve a

favorable compromise between contradictory goals

in changing of computational values within of

theoretically defined by formulas (1) and (2)

boundaries. The upper limit of the information

capacity encoding method for given optimum

vector code size with numbering categories and

attributes of vector data sets, as well as R of

various ways of coding the same "category-

attribute" sets defined.

Physical results - a better understanding the role

of geometric structure in the behavior of natural

and man-made objects. The existence of an a priori

of an infinitely large number of minimized bases,

which give rise to numerous varieties of

multidimensional "star" coordinate systems, opens

up new possibilities for solving a wide range of

mathematical and applied problems of computer

science, cybernetics, and management on the

platform of system mathematics. We take into

account the developments of modern theory of

systems as a set of philosophical, methodological

and scientific, and applied problems of analysis

and synthesis of multidimensional systems.

offering ample scope for progress in systems

engineering, cybernetics, and industrial

informatics.

Prospect for further research are the

development of new direction in multidimensional

information technologies, computing,

telecommunications and systems engineering for

improving such quality indices as information

capacity, reliability, transmission speed,

positioning precision, resolving and ability to

reproduce the maximum number of combinatorial

varieties in the system with a limited number of

elements and bonds, using remarkable properties

and structural perfection of the underlying

multidimensional combinatorial configurations.

The experimental results allow recommending

the proposed methodology for direct applications

to information and computational technologies,

telecommunications, radio- and electronic

engineering, radio- physics, and other engineering

areas, as well as in education. These design

techniques allows configure optimum two- and

multidimensional vector data coding system, using

innovative methods based on the underlying

combinatorial structures, offering ample scope for

progress in systems engineering, cybernetics, and

industrial informatics.

Urgent problem of optimum vector information

technologies and processing large data content

based on the multidimensional combinatorial

configurations solved.

Acknowledgement:

The basic results of the research presented in

completed work on the R&S project “Designing

Software for Vector Data Processing and

Information Protection Based on Combinatorial

Optimization” (State registration 0113U001360).

State account number 0218U000988.

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International Journal of Computational and Applied Mathematics & Computer Science

DOI: 10.37394/232028.2023.3.12

Volodymyr Riznyk

E-ISSN: 2769-2477

112

Volume 3, 2023

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