Combinatorial Optimization of Engineering Systems based on

Diagrammatic Design

VOLODYMYR RIZNYK

Lviv Polytechnic National University,

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

UKRAINE

Abstract: - The objectives of the combinatorial optimization of engineering systems based on diagrammatic

design are enhancing technical indices of the systems with spatially or temporally distributed elements (e.g.,

radio-antenna arrays) concerning resolving ability, positioning precision, transmission speed, and performance

reliability, using the graphical performance of appropriate algebraic models of the system, such as cyclic

difference sets, Galois fields and “Ideal Ring Bundles”. The diagrammatic design provides configuring systems

with a smaller number of elements than at present, while upholding or improving on the other significant

operating quality indices of the system.

Key-Words: - non-uniform structure, spiral diagram, Galois field, Ideal Ring Bundle, combinatorial

optimization, design technique, quality indices, high-performance cyclic code, optimized self-

correcting code.

Received: October 23, 2023. Revised: April 9, 2024. Accepted: May 24, 2024. Published: July 1, 2024.

1 Introduction

In the last few years modern circuit and system

engineering have wide world gratitude for finding

optimal technological solutions using combinatorial

optimization of devices or systems based on

mathematical principles of the system design. This

article devotes solving technological problems

referring to structural optimization of engineering

devices or systems, using theory of combinatorial

configurations, [1], [2], [3], [4], including such

algebraic constructions as cyclic difference sets [5],

[6], [7], [8], projective planes [9], [10], finite field

theory [11], [12], [13], and concept of the Ideal

Ring Bundles (IRBs), [14].

2 Problem Formulation

The very important problem for innovative system

engineering is finding the optimal placement of

structural elements in spatially or temporally

distributed systems, including two- and

multidimensional structures of the system.

3 Problem Solution

Finding optimal solutions for wide classes of

technological problems requires the application of

combinatorial design techniques, using profitable

cyclic ordered numerical or vector diagrams of

Galois fields [11], [12], [13] and Ideal Ring

Bundles [14]. The task is to increase transmission

speed, positioning accuracy, resolving ability,

vector signal coding, data compressing, and

functionality of engineering devices or systems.

3.1 Ideal Ring Sequences

To the study of combinatorial properties of ring

sequences of positive integers let us calculate all

sums of connected sub-sequences of the sequence.

Sums of connected elements on ring ordered n-

stage sequence K = {k1, k2, …, kn } are given in

Table 1.

Table 1. Sums of connected elements of ring

ordered n-stage sequence K = { k1, k2, …, kn }

K = { k1, k2, …, kn }

...

n-1

i



...













...











...

n-1

i





k k

i n

i  

 



1 1

...

kn-1

i n

 



kn+k1

k k

n i







...





WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

149

Volume 23, 2024

Coordinates (pj, qj) correspond to appropriate

sum of connected elements (pj, qj) of a ring-ordered

n-stage sequence.

If pj 



qj , a ring sum is equal to ki ; in case pj



qj it is

i p





; or

 



jkk

for pj



qj.

In Table 1 the maximum number of distinct

sums S of connected elements of a ring-ordered n-

stage sequence is given by:

S = n2 – n + 1 (1)

If a sum of any number of connected elements

in the sequence enumerate the set of integers from

1 to S, we call this “ideal” ring sequence, shortly,

Ideal Ring (IR) with parameters S and n.

Now we see an example of an IR with S=13,

n=4, namely {1, 3, 2, 7}. Here we observe:

1=1 5=3+2 9 = 2+7 13= 1+3+2+7

2=2 6=1+3+2 10= 2+7+1

3=3 7=7 11= 7+1+3

4=1+3 8=7+ 1 12=3+2+7

One more variant of IRs with the same

parameters S=13, n=4 is four-stage cyclic sequence

{1, 2, 6, 4}:

1=1 5=4+1 9 = 1+2+6 13= 1+2+6+4

2=2 6=6 10= 6+4

3=1+2 7=4+1+2 11= 6+4+1

4=4 8= 2 +6 12=2+6+4

Theoretically, there are numerous Ideal Rings.

4 Graphic Design of Ideal Rings

4.1 One-dimensional Ideal Rings

One-dimensional Ideal Rings (IRs) are numerical

cyclic sequences of arbitrary one-dimensional

elements (e.g., parts of a set), which can be

presented graphically as a wreath of the elements.

Here is a chart of the IR {1, 3, 2, 7} (Figure 1).

Fig. 1: A chart of the IR {1, 3, 2, 7}

Easy to see, that if summing over more than

one complete revolving around the ring, you can

obtain all integers as such sums:

14= 1+3+2+7+1, 15= 2+7+1+3+2, 16=

3+2+7+1+3, etc.

A numerical spiral diagram of the IR {1, 3, 2,

7} is given below (Figure 2).

Fig. 2: Numerical spiral diagram of the IR {1, 3, 2,

The numerical diagram (Figure 2) consists of

four-stage (n =4) ring sequences placed one inside

another, so that the inner sequence is the IRB {1, 3,

2, 7} (yellow beads), and it forms the rest of the

ring sequences by summing of its consecutive terms

from 2 to 4, namely: {4, 5, 9, 8} (blue beads), {6,

12, 10, 11} (orange beads), and finally {13, 13, 13,

13}. The arrows point the direction of the summing

and give a place for circular sum's arrangement in

the spiral-like diagram. We can find this each ring

sum from 1 to n (n – 1) = 12 occurring exactly once

on the diagram. Using equation (1) easy to calculate

a maximum number of distinct sums S of any IRB:

3, 7, 13, 21, 31, etc. We say this numerical set is a

generative IRBs row for configure numerical spiral

diagram of any ring ordered n-stage sequence of the

IRB.

A chart of the IR {1, 2 ,6, 4} is depicted in

Figure 3.

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

150

Volume 23, 2024

Fig. 3: A chart of the IR {1, 2, 6, 4}

Numerical spiral diagram of the IR {1, 2, 6, 4}

is presented in Figure 4.

Fig. 4: Numerical spiral diagram of the IR{1, 2, 6,

Spiral diagram (Figure 4) consists of four-stage

(n=4) ring sequences placed one inside another, so

that the inner sequence is the IRB {1, 2, 6, 4}

(yellow beads), and it forms the rest of the ring

sequences by summing of its consecutive terms

from 2 to 4, namely: {3, 8, 10, 5} (blue beads), and

{7, 9, 12, 11} (orange beads). The arrows point the

direction of the summing and give a place for

circular sum's arrangement in the spiral-like

diagram. We can find this each ring sum from 1 to

n (n – 1) = 12 occurring exactly once on the

diagram.

4.2 Ideal Ring Bundles in Extension of Galois

Fields

Researches into the underlying diagrammatic

design involve the graphic presentation of cyclic

groups in extensions of Galois fields, [11], [12]. A

graphic rotational scheme in GF(32) of the Ideal

Ring {1,2,6,4} with parameters S=13 and n = 4,

is given in Figure 5.

Let us regard model of optimum rotational device as (4,

13)-IRB in terms of Galois theory. In this case prime

element x of GF(32) satisfies equation f(x)=x3-x-1. Here

f(x) is polynomial irreducible over GF(32), where p=3, and

m=2 (Fig.3).

x + 1

x2 + x

x2 + x +1

x2 + 2x +1

2x2+2x +1

2x2 +2

x + 2

x2 + 2x

2x2 + x +1

x2 + 2

Fig. 5: A graphic rotational scheme in GF(32) of

the Ideal Ring {1,2,6,4} with parameters S=13 and

n = 4

Prime elements x of GF(32) satisfy equation

f(x)= x3 – x –1, where f(x) is 3-degree polynomial

irreducible over GF(32), p=3, m=2:

122

212

211

210













xxx











(modd 3, x3 – x – 1)

The graphic scheme (Figure 6) displays two

non-uniform polygons of four (n1=4), and nine

(n2=9) vertexes as the GF(32):

(

0,3

)

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

151

Volume 23, 2024

Fig. 6: Graphic scheme of two non-uniform

polygons of four (n1=4), and nine (n2=9) vertexes

as the GF(32)

To see this, we observe representation of two

complementary IRBs. The first of them placed in

the vertexes x0, x1, x3, x9 is the IRB {1, 2, 6, 4}

with S=13, n=4, while the second – in vertexes x2,

x4, x5, x6, x7, x8, x10, x11, x12 starting the IRB

{2,1,1,1,1,2,1,1,3} with n=9, where ring sums of

the IRB enumerate the set of integers [1, S –1]

exactly 6-times. Graphic scheme (Figure 6) allows

combinatorial optimization of engineering devices

or systems based on wonderful geometric and

pictorial properties of Galois fields in rotational

symmetry graphical presentation.

4.3 Diagrammatic of Two-dimensional Ideal

Ring Bundles

Next, we regard the n-stage ring sequence K2D=

{(k11, k12), (k21, k22),..., (ki1, ki2),...,(kn1, kn2)}, where

we require all terms in each circular vector-sum to

be consecutive 2-stage sequences as elements of the

sequence. Example: IRB containing four (n=4)

two-dimensional (2-D) elements: k1= (0,2), k2=

(1,3), k3= (0,1), k4=(2,3) is depicted in the diagram

(Figure 7).

Fig. 7: Diagram of the 2-D IRB

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

Easy to calculate all 2-D vector-sums of the 2-D IRB

{(0,2), (1,3), (0,1), (2,3)}, taking twice modulo

m1 = n –1 =3, and m2 = n = 4 correspondently:

(1,0)≡(0,1)+(1,3);

(1,1)≡(1,3)+(0,2); (2,1)≡(0,2)+(2,3);

(2,0)≡(2,3)+(0,1);

(1,2)≡(0,1)+(1,3)+(0,2);(2,2)≡(0,2)+(2,3)+(0,1);

(0,3)≡(2,3)+(0,1)+(1,3); (0,0)≡(1,3)+(0,2)+(2,3).

Vector-sums of the 2-D IRB {(0,2),(1,3),(0,1),(2,3)},

taking twice modulo m1 =n –1 =3, and m2 = n = 4

is given in Table 2.

Table 2. Vector-sums of the 2-D IRB {(0,2), (1,3),

(0,1), (2,3)}, taking twice modulo m1 =n –1 =3, m2

= n = 4

These basic vectors {(0,2), (1,3), (0,1), (2,3)}

of the ring sequence themselves are circular 2-D

vector-sums too, forming a coordinate system of

size 3 × 4 over the surface of the torus:

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

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

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

Here is an example of the 2-D IR {(0,2), (1,3),

(0,1), (2,3)} vector ring as colored diagram (Figure

8).

Vector-sums

Basic vectors

(0,0)

(0,2)

(1,3)

(2,3)

(0,1)

(0,2)

(0,3)

(1,3)

(0,1)

(2,3)

(1,0)

(1,3)

(0,1)

(1,1)

(0,2)

(1,3)

(1,2)

(0,2)

(1,3)

(0,1)

(1,3)

(2,0)

(0,1)

(2,3)

(2,1)

(0,2)

(2,3)

(2,2)

(0,2)

(0,1)

(2,3)

x + 1

x2 + x

x2 + x +1

x2 + 2x +1

2x2+2x +1

2x2 +2

x2 +

x + 2

x2 + 2x

2x2 + x

++++

(0,1)

(0,2)

(2,3)

(1,3)

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

152

Volume 23, 2024

Fig. 8: Vector ring diagram of the 2-D IRB {(0,2),

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

Diagram (Figure 8) consists of four (n=4) n-

stage ring sequences of 2-D vectors, placed one

inside another so that the inner sequence is the 2-D

IRB {(0,2), (1,3), (0,1), (2,3)} (yellow beads). It

forms the rest of the ring sequences by summing

modulo 3 and 4 of its consecutive 2-D vectors

{(1,1), (1,0), (2,0), (2,1)} (blue beads), and finally

{(1,2), (0,3), (2,2), (0,0)} (orange beads). The

arrows point direction of the summing giving

locations of all modular sums in the ring diagram.

We can observe that each modular sum from (0,0)

to (2,3) occurs exactly once on the diagram.

The result of the computation forms two-

dimensional grid over torus 3 × 4, where 2-D

modular coordinates of each node of the grid

enumerated exactly once. Hence, the ring vector

sequence {(0,2), (1,3), (0,1), (2,3)} is two-

dimensional (t=2) Ideal Ring Bundle with S=13,

n=4, m1=3, m2=4. The IRB forms two-dimensional

coordinate grid over surface of torus m1 × m2. Here

each node point of the coordinate grid occurs

exactly once, as well as it creates binary 4-digit

(n=4) two-dimensional code with the weighted bits

{(0,2), (1,3), (0,1), (2,3)} under intelligent torus

coordinate system 3 × 4, where m1=3, m2=4 (Table

3).

Table 3 contains 12 binary four- digit (n = 4)

combinations (n2 –n = 12) for coding two attributes

(t = 2) both with three (m1 = 3) categories of the

first, and four (m2=4) – the second attributes

concurrently.

Table 3. Binary 2-D vector code based on the IRB

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

The 2-D IRB code under the torus coordinate

system m1 × m2 = 3 × 4

Vector

Digit weights of the 2-D code

(0,2)

(1,3)

(0,1)

(2,3)

(0,0)

(0,1)

(0,2)

(0,3)

(1,0)

(1,1)

(1,2)

(1,3)

(2,0)

(2,1)

(2,2)

(2,3)

5 Discussion

As is evident from Table 1 and equation (1),

combinatorial properties of n -stage ring sequences

of positive integers allows the enumeration of all

sums of connected sub-sequences of the sequence

in a finite interval [1, n2 – n + 1]. We call this Ideal

Ring Bundles (IRBs). Chart of IRB {1, 3, 2, 7} is

given in Figure 1. Numerical spiral diagram of the

IRB (Figure 2) demonstrates the structural

perfection of the combinatorial configuration. It

consists of four-stage (n=4) ring sequences placed

one inside another so that the inner sequence is the

IRB. Chart of IRB {1, 2, 6, 4} is given in Figure 3,

and its numerical spiral diagram of the IRB (Figure

4) well evident existing isomorphic constructions

of IRBs, as well as combinatorial varieties of them.

These diagrammatic schemes emulate designing

engineering devices or systems with optimal

spatially or temporally distributed structural

elements of the system. A graphic rotational

scheme (Figure 5) allows the optimization of

engineering systems using diagrammatic design in

Galois fields. The graphic scheme (Figure 6)

displays two non-uniform polygons of four (n1=4),

and nine (n2=9) vertexes as the GF(32). The first of

them form IRB {1, 2, 6, 4}, where each rings the

second matches IRB 9-poligon, where each ring

sums from 1 to S – 1 =11 occurs exactly once,

while sums enumerate the same set of integers

exactly 9-times. The last types of IRBs make it

(2,2)

(0,3)

(0,1)

(2,3)

(0,2)

(2,0)

(0,0)

(

0,3)

(1,3)

(2,1)

(1,0)

(1,1)

(1,2)

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

153

Volume 23, 2024

possible to configure optimized coding systems,

which provide an opportunity to detect up to 50 %

or correct up to 25 % errors of code combination

length with rising n to ∞ asymptotically [14]. A

diagram (Figure 7) demonstrates forming a

coordinate system of size 3 × 4 over the surface of

a torus based on the 2-D IRB {(0,2), (1,3), (0,1),

(2,3)} (Table 2). Diagram (Figure 8) consists of

four 4-stage ring sequences of 2-D vectors, placed

one inside another, so that the inner four sequences

are these basic vectors. Table 3 presents an

example of binary two-dimensional code words

with the weighted bits {(0,2), (1,3), (0,1), (2,3)},

which consist of no more than one monolithic part

both of bits “1” or “0” in code combinations as

being cyclic. Theoretically, there are infinitely

many varieties of useful combinatorial

configurations for the optimization of engineering

systems based on diagrammatic design. Such an

approach to be profitable for combinatorial

optimization of a number of technological

problems, e.g., high-performance error-corrected

encoding systems, self-correcting monolothic

codes, and multidimensional information manifold

coordinate systems for data processing attribute

sets at the same time without of parallel

computation, [14].

6 Conclusion

The essence of the proposed project technology is

combinatorial optimization of engineering systems

based on diagrammatic design for enhancing

technical indices of the systems with spatially or

temporally distributed elements. Because diagrams

and graphs are clear and easy to read and

understand, the study of diagrammatic reasoning is

about the understanding of concepts and ideas,

visualized with the use of the

diagrams and imagery. Diagrammatic and graphic

presentation of data means visual representation of

the data. It shows a comparison between two or

more sets of data and helps in the presentation of

highly complex data, e.g., geometric relationships

of cyclic groups in extensions of Galois fields,

helping design of optimized engineering systems.

Diagrams and graphs are clear and easy to read and

understand. The modern circuit and system

engineering have wide world gratitude connecting

fast development in the field of system engineering,

information technologies, and radio-electronics.

Combinatorial optimization of engineering systems

based diagrammatic design make it possible to

configure structure of the systems with the smaller

number of elements than at present, whereas

upholding or improving on the other significant

operating quality indices of the system.

References:

[1] M.Jr. Hall, Combinatorial Theory: Wiley-

Interscience, 1998, p.464, DOI: 10.1002 /

978111803286.

[2] Mitchel T.Keller, William T.Trotter, Applied

Combinatorics, 2017, p.393.

[3] Lorenz J.Halbeisen, Combinatorial Set

Theory. Second Edition: Springer, 2017,

p.610.

[4] Edwin E. Beckenbach, Applied Combinatorial

Mathematics, John Wiley and Sons, Inc. New

York, London, Sidney, 1964.

[5] Baumert Leonard D. Cyclic difference sets.

Lecture Notes in Mathematics, Vol. 182

Springer - Verlag, Berlin-New York 1971.

[6] Moore, EH; Pollastek, HSK,”Difference

Sets: Connecting Algebra, Combinatorics,

and Geometry”, 2013. AMS. ISBN: 978-0-

8218-9176-6.

[7] M. Buratti, Hadamard partitioned difference

families and their descendants, Cryptogr.

Commun., 11 (2019), p.557-562.

doi: 10.1007/s12095-018-0308-3.

[8] Jungnickel, D., Pott, A., Smith, K.W:

Difference sets. In: Colbourn, C.J., Dinitz,

J.H. (eds.) Handbook of Combinatorial

Designs. 2nd edn., pp 419–435. Chapman &

Hall/CRC, Boca Raton (2006)

[9] Stevenson, Frederick W. (1972), Projective

Planes, San Francisco: W.H. Freeman and

Company, ISBN: 0-7167-0443-9.

[10] Weintraub, Steven H. (1978), "Group actions

on homology quaternionic projective

planes", Proceedings of the American

Mathematical Society, 70, p.75-82.

[11] Joseph Rotman, Galois Theory (Second ed.):

Springer, 1998, doi: 10.1007/978-1-4612-

0617-0, ISBN: 0-387-98541-7. MR 164558.

[12] Cunxi Yu, and Maciej Ciesielski,”Formal

Analysis of Galois Field Arithmetic Circuits

- Parallel Verification and Reverse

Engineering,” in EEE Transactions on

Computer-Aided Design of Integrated

Circuits and Systems, 2018, vol. 38, Issue 2,

pp.354-365, DOI:

10.1109/TCAD.2018.2808457T.

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

154

Volume 23, 2024

[13] Pruss, P. Kalla, and F. Enescu, "Efficient

symbolic computation for word-level

abstraction from combinational circuits for

verification over finite fields", IEEE Trans.

Comput. -Aided Design Integr. Circuits Syst.,

vol. 35, no. 7, pp. 1206-1218, Jul. 2016.

[14] V.Riznyk, Optimization of Encoding Design

Based on the Spatial Geometry Remarkable

Properties, WSEAS Transactions on

Computers, Vol.20, pp.270-276,

https://doi.org/10.37394/23205.2022.21.33.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at

all stages from the formulation of the problem to

the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.e

n_US

WSEAS TRANSACTIONS on COMPUTERS

DOI: 10.37394/23205.2024.23.13

Volodymyr Riznyk

E-ISSN: 2224-2872

155

Volume 23, 2024