Abstract: This paper involves techniques for optimization of encoding design based on the remarkable
geometric property of ring symmetry which contains two complementary asymmetries as the World harmony
law for improving the quality indices of one- and multidimensional cyclic codes with respect to performance
reliability, transmission speed, and transmission content, using vector data coding. These design techniques
make it possible to configure encoding system with minimized number of digit weights, while maintaining or
improving on error protection, security, and function of autocorrelation. Such sets are t-dimensional
vectors, each of them together with all their modular sums enumerate the set node points grid of the
coordinate system with the corresponding sizes and dimensionality. Systemic researches based on
remarkable geometric properties of multi-modular mathematical structures such as “Glory to Ukraine Star”
(GUS) combinatorial configurations demonstrated.
Keywords: Rotational symmetry, Harmony, Golomb ruler, GUS-combinatorial configuration, Vector data coding, Self-
correcting code, Manifold coordinate system.
Received: September 17, 2021. Revised: May 13, 2022. Accepted: June 11, 2022. Published: July 5, 2022.
1. Introduction
The major goals of the modern information
technology are the expansion of big data process
engineering design and use concept of optimize teaching
approach to the practical tasks and assessment methods.
Another goal of the systems is creation of unified
information space with intelligent components of upper
management levels such as large amount of data, high
computing amount, and data flow intensity. In this aspect
of very profitable is the development of intelligent
components for the practical tasks and lectures studies,
using novel interpretations of mathematical principles
for transformation content and the other operating
characteristics of the system. The torus accepted as the
“perfect” shape that useful to describe objects as
mathematical model of systems with spatially distributed
structural elements of the system. Surface topology is
superior to geometry relating such phenomenon because it
deals with much more sophisticated and profound spatial
and temporal relationships. The toroidal shape used in
harmonic resolution analysis is similar to a doughnut but
rather than having an empty central “hole”, the topology
of a torus folds in upon itself and all points along its
surface converge together into a zero-dimensional point at
the centre called the vertex [2]. A major branch of
geometry is the study of geometrical structures on
manifolds [3]. A manifold is a curved space of some
dimension. Proposed concept involves techniques based
on cyclic groups theory [4] and Ideal Ring Bundles
algebra [5]. These aspects of the matter the issue are
examined about multidimensional Ideal Ring Bundles [5],
[6] and properties of “elegant” rotational symmetry and
asymmetry relationships [7]-[9]. Next is given research
into the mathematical principles relating to optimal
placement of structural elements in spatially or temporally
distributed systems [5], [6], [10]. The development of
new directions in fundamental and applied research in
systems engineering [5], [6], [10]-[13], coded design of
signals [5], [6], [8]-[10], [14]-[15], advanced information
technologies [10], [16]-[19]. The topological model of the
coordinate systems regarded as both algebraic
constructions, based on cyclic groups in extensions of
Galois fields [4], and intelligent non-redundant
combinatorial configurations, generated from “elegant”
ensembles of rotational symmetry composed from
complementary asymmetries [9]. These design techniques
make it possible to configure information technology with
vector data indexing and processing under basis of two-
and multi-dimensional coordinate system, where basis is a
sub-set of general number indexed vector data "category-
attribute", which belong to mapping nodal coordinate
points set of the system. The basis generates indexed
vector data "category-attribute" set using modular
summing for complete a reference grid of the coordinate
system. Moreover, we require each indexed vector data
"category-attribute" mutually uniquely corresponds to the
point with the eponymous set of the coordinate system. In
practice, the set points obtained using optimized basis of
the system. This methodology working out harmonious
VOLODYMYR RIZNYK
Lviv Polytechnic National University
Lviv, S. Bandery str. 12, 79013
UKRAINE
Optimization of Encoding Design Based on the Spatial Geometry
Remarkable Properties
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mutual penetration of rotational symmetry and asymmetry
as the remarkable property of real space for configure
multi-attribute intelligent information management
technologies under the coordinate system. Besides, a
combination of binary code with vector weight bits of the
database allowed, and the set of all values of indexed
vector data sets one-to-one correspondents to nodal points
set of the t-dimensional coordinate system. The
underlying mathematical principle relates to the optimal
placement of structural elements in spatially or temporally
distributed systems, using novel designs based on t-
dimensional combinatorial configurations, including the
appropriate algebraic theory of cyclic groups, number
theory, modular arithmetic, and geometric
transformations. This information technology brought out
relationship between the “elegant” ensembles of rotational
symmetry and intelligent models of torus coordinate
systems.
The role of the models becomes evident if teacher
selects methodology obviously to state the physical
essence of a studied problem. The aim of the article
involves techniques for improving the quality indices of
integrating control functions of technological, business
processes, creating unified big vector data information
space. The main problem of designing big vector data
coding systems is development of an approach to
configure two- and multidimensional optimum model of
the systems. The multidimensional coding systems, is
known to be of very important in information technology,
for improving the quality indices of the systems with
optimum compressed structure (e.g. two-dimensional
torus coordinate system). The paper regards innovative
techniques for development of vector data coding design
based on the idea of “perfect” spatially or temporally
distributed systems, using the appropriate combinatorial
configurations as a basis of expanded information field
for big data coding and processing.
2. Review of Literature
Geometric optimization as known can be performed as
follows: optimize the geometry in internal coordinates, in
redundant internal coordinates, and in Cartesian
coordinates. Each step of the geometry
optimization, Gaussian written to the output file the
current structure of the system, the energy for this
structure, the derivative of the energy with respect to the
geometric variables, and a summary of the convergence
criteria [20]. In recent times, a great number of new
concepts, parallel algorithms, processing tools, platform,
and applications are suggested and developed to improve
the value of big vector data [21], [24]-[26]. The vector
data-sets often involve a number of attributes, such as
name, type, length, content, and other indexes, which
have led to difficulties in large-scale data processing. In
recent times, a great number of new concepts, parallel
algorithms, processing tools, platform, and applications
are suggested and developed to improve the value of big
vector data. The geometric computing algorithms are
always very complex and time-consuming [20], [21]. A
framework that couples cloud and high-performance
computing for the parallel map projection of vector-based
big spatial data regarded in [24]. The projection provides
large-scale spatial modeling of big vector data under a
common coordinate system. High-dimensional datasets
can be very difficult to visualize. While data in two or
three dimensions can be plotted to show the inherent
structure of the data, equivalent high-dimensional plots
are much less intuitive. To aid geometric visualization
and processing of a dataset, the processing must be
optimized in some way with respect to the underlying
criteria. In the paper [26] a theoretical foundation of the
combinatorial 2D vector field topology set forth. A
discrete Morse theory for general vector fields Forman
describes in [27]. This theory applied successfully to
scalar fields on triangulated manifolds [28]. Classification
of digital n-manifolds based on the notion of complexity and
homotopic equivalence presents in paper [29]. The another
theoretical approach founded on structural perfection of
toroidal and multidimensional manifolds, namely the
concept of “Glory to Ukraine Star” combinatorial
configurations (GUS-configurations) stated in [11], [30].
3. Perfect Golomb Rulers and
Ideal Ring Protractors
In mathematics, a Golomb ruler is a set of marks
at integer positions along a ruler such that no two pairs of
marks are the same distance apart. The number of marks
on the ruler is its order, and the largest distance between
two of its marks is its length. There is no requirement that
a Golomb ruler be able to measure all distances up to its
length, but if it does, it called a perfect Golomb ruler. It
has been proved that no perfect Golomb ruler exists for
five or more marks [31].
A protractor is an instrument used for measuring angles
in degrees. An ideal (or perfect) ring protractor is a set of
marks at integer positions along a ring such that no two
pairs of marks are the same angular distance apart. The
number of marks on the ring is its order, and the largest
angular distance between two of its marks is equal to αmax
= 360o - αmin. In mathematics, and system engineering an
ideal ring protractor is known as “Ideal Ring Bundle”
(IRB) [5]. IRBs are cyclic sequences of positive integers,
which form perfect partitions of a finite interval [1, S] of
integers. The sums of connected sub-sequences of an IRB
enumerate the set of integers [1, S-1] exactly R- times. For
example, the IRB{1,2,6,4} containing four (n=4)
elements allows an enumeration of all numbers 1, 2,
3=1+2, 4, 5=4+1, 6, 7=4+1+2,...12=2+6+4 exactly once
(R=1). While the IRB {1,1,2,3} of order n = 4, and S=7,
enumerates each “circular sum” exactly twice (R=2): 1, 1;
2, 2=1+1; 3, 3=1+2; 4=3+1, 4=1+1+2; 5=2+3, 5=3+1+1;
6=1+2+3, 6=2+3+1. The concept of Ideal Ring Bundles
can be used for finding optimal solutions for wide classes
of technological problems in applications which need to
partition sets with the smallest possible number of
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intersections, e.g. metering tanks [5]. Unlike perfect
Golomb rulers [31] there are exist a priory endless
number of IRBs of order n, and the more n the more
number P of the n-sequence IRBs invariants. For
example, there are only two (P=2) invariants IRBs of
order n=4, R=1: {1,2,6,4}, and {1,3,2,7}. While, exists
P=159630 invariants of IRBs of order n=984, R=1 [30,
p.100].
Let us regard S-fold rotational symmetry as an ability
to reproduce the maximum number of combinatorial
varieties, using two-part asymmetrical division over a
planar space relative to central point of the symmetry.
To extract meaningful information from the underlying
data let us apply to rotational S-fold (S=7) symmetry
penetrated by two complementary asymmetric sub-
systems completions of the planar symmetric system
(Fig.1).
Fig.1 The 7-fold (S=7) rotational symmetry penetrated by two
complementary asymmetric sub-systems completions of the planar
symmetric system
To see this, we observe that the first sub-system (solid
lines) of the complete model forms perfect two-
dimensional spatial partitions of a fixed angular interval
[, 6] by step = 360º/7 exactly once (R1=1), while
the second (thinner lines) – exactly twice (R2=2) by the
same step.
Our reasoning proceeds from the fact, that the minimal
and maximal angular distances relation initiated by S-fold
rotational symmetry to be of prime importance for
discovery of the S–fold “elegant” symmetry-asymmetry
ensemble (Fig.2).
=
Fig. 2 A chart for discovery of the S –fold
“elegant” symmetry- asymmetry ensemble
We require the set of all N angular distances [αmin , N -
αmin] of S-fold harmoniously quantized space divided by a
set of n straight lines diverged from a central point O
non-uniformly allows an enumeration of all integers [1,
S–1] exactly R-times. If these requires request, we call
this phenomenon the “perfect” rotational S-fold
symmetry. From Fig.2 follows integer relation between of
variables S, n, and R [ 5, p.13]:
S = n (n-1)/R + 1 (1)
As follows from equation (1), there are exist a priory
infinite number of the “perfect” symmetry-asymmetry
“elegant” ensembles, and this is a necessary, but not a
sufficient condition.
4. Design of Optimal Cyclic Error-
Correcting Codes
4.1 Cyclic Codes Design Based on the IRBs
From (1) follows relationships for synthesis of cyclic
correcting code. To construct of the code it is necessary to
pad “1” into n cells of S-stage array of cells numbered
from 1 to S by number xj in accordance with relation:
xj – 1 ≡
)(mod
1
Sk
j
ii
, j=1,2,…,n, (2)
where ki is the i-th element of an IRB with parameters S,
n, and R, and the rest (S – n) cells are padded with zeroes.
For example, if ring sequence is {1,1,2,3}, n=4, S=7, then
k1=1, x1= 2; k2=1, x2=3; k3=2, x3=5; k4=3, x4=1, and we
obtain the next binary code combination:1110100. This is
a basic code word of the cyclic 7-code (S=7), and you can
get the rest S-1=6 code words by cyclic shift of the basic
one: 1110100, 0111010, …1101001. The code size is
doubled if bit positions of code words to appear as
opposite bits: 0001011, 1000101, …0010110.
Number of td errors to be detected, and corrected tc are
a function of minimal Hamming code distance dmin
according to the next relationships [5, p.102]:
dmin = 2(n – R) (3)
td ≤ dmin – 1, tc ≤ (td – 1)/2 (4)
The minimal code distance d2 for the doubled code size
is [5, p.103]: d2 = S – 2(n-R) (5)
of code words to appear as opposite bits: 0001011,
1000101, …0010110.
One of the indicators of the error-correcting codes is the
ratio of Hemming code distance dmin to length S of the
code combinations, which characterizes the efficiency of
the corrective ability of the code: Kе = dmin /S.
Table 1 provides calculations of quality indices of
optimal cyclic IRB error-correcting codes with information
parameters n, R, S. Code size P for this coding system is
P = 2S.
TABLE I. INDEXES OF THE OPTIMAL IRB CYCLIC ERROR-CORRECTING
CODES
R
dmin
Ke
td
tc
P
10
20
0,513
19
9
78
31
62
0,504
61
30
246
31
64
0,504
63
31
247
αmin=360o/S
αmax = N·αmin
O
α
α
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126
63
0,502
125
62
502
127
63
0,502
127
63
510
254
127
0,501
253
126
1014
255
127
0,501
255
127
1022
510
255
0,5005
509
254
2038
511
255
0,5005
511
255
2046
1022
511
0,5002
1021
510
4086
1023
511
0,5002
1023
511
4094
From Table 1, you can see that the optimum cyclic code
with has a higher corrective efficiency of the code, which
initially falls from 0.513 to 0.504 within the growth of the
number of digits from 20 to 62, and approach to the value
of 0.5 nonlinearly with any large increase in the number of
digits of code sequences. At the same time, the maximum
possible number comes of detected errors tends to 50%,
and to 25 % corrected errors of the code word length, while
code size P for this coding system is double the S.
4.2 Application of optimal IRB codes in data
communications
To improve the reliability of communications and
cryptographic protection and data transmission systems
using code sequences, it is necessary to use the rule for
optimizing IRB codes [30, p.129]:
The highest noise resistance acquired by IRB-code
sequences, in which the number of different binary
characters differs from each other by no more than one
character.
Table 2 shows the characteristic of optimal IRB code
sequences with a length of 7 ≤ S ≤ 39 for signal
recognition capacity in relation to signal/noise less than 1.
TABLE II. CHARACTERISTICS OF OPTIMAL IRB CODE SEQUENCES
LENGHTS
The autocorrelation function is calculated by a set of
step-by-step offsets of this sequence based on the
summation of all +1 and -1 elements, after a full cycle of
step-by-step offsets. We see that optimized cyclic IRB-
codes of length S provide correcting to (S-3)/4, i.e. (25%)
errors, while detecting to 50% ones [30, p.124-125]. The
results of calculations do not change from reversing the
order or changing the signs of elements to the opposite in
any of the variants of IRB sequences.
4.3 Gloria to Ukraine Stars Encoding
Systems
Next, we consider a new type of optimized encoding
systems based on the remarkable properties of two- or
more dimensional vector Ideal Ring Bundles, which
properties hold for the same S-ordered rotational
symmetry in varieties permutations of the vectors in the
IRBs, namely the ensembles of “Gloria to Ukraine Stars”
(GUS)s combinatorial configurations [11]. For example,
43-fold (S=43) rotational symmetry creates a set of
doubled IRBs two-dimensional (t=2) seven-stages (n=7)
GUSs [30, p.68-69]. Here is one of them (Fig.3).
Fig. 3 A chart of two-dimensional optimized
encoding system based on the 7–fold doubled
IRBs
This we can see a set of two 2D seven-stages (n=7)
IRBs: {(1,1), (1,3), (1,5), (1,0), (1,2), (1,4), (1,6)} (ring
cycle), and {(1,1), (1,5), (1,2), (1,6), (1,3), (1,0), (1,4)}
(star cycle).
In the Table 3 shows forming 7- digit 2D GUS – code
created by ring cycle{(1,1), (1,3), (1,5), (1,0), (1,2), (1,4),
(1,6)} under torus coordinate system 6×7 with two (t=2)
orthogonal ring- axes and common reference point (0,0).
Parameters of ІRВs
Autocorrelation function
n
R
S
tc
+1
-1
∆
│S/∆│100
%
4
2
7
1
3
4
-1
14,286
5
2
11
2
5
6
-1
9,0909
6
3
11
2
5
6
-1
9,0909
7
3
15
3
7
8
-1
6,6667
8
4
15
3
7
8
-1
6,6667
9
4
19
4
9
10
-1
5,2631
10
5
19
4
9
10
-1
5,2631
11
5
23
5
11
12
-1
4,3478
12
6
23
5
11
12
-1
4,3478
13
6
27
6
13
14
-1
3,7037
14
7
27
6
13
14
-1
3,7037
15
7
31
7
15
16
-1
3,2258
16
8
31
7
15
16
-1
3,2258
17
8
35
8
17
18
-1
2,8571
18
9
35
8
17
18
-1
2,8571
19
9
39
9
19
20
-1
2,5641
20
10
39
9
19
20
-1
2,5641
(1,1)
(1,6)
(1,3)
(1,4)
(1,5)
(1,0)
(1,2)
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TABLE III. THE 7-DIGIT 2D GUS –CODE CREATED BY RING
CYCLEUNDER TORUS COORDINATE SYSTEM 6 × 7
We can see that 2D vector sequence {(1,1), (1,3), (1,5),
(1,0), (1,2), (1,4), (1,6)} forms complete set of ring code
combinations on 2D ignorable array 67, and each of
them occurs exactly once (R=1).
Table 4 forms 7- digit 2D GUS - code created by star
cycle {(1,1), (1,5), (1,2), (1,6), (1,3), (1,0), (1,4)} under
torus coordinate system 6×7 with two (t=2) orthogonal
ring- axes and common reference point (0,0).
TABLE IV. THE 7-DIGIT 2D GUS –CODE CREATED BY STAR CYCLE
UNDER TORUS COORDINATE SYSTEM 6 × 7
We can see that 2D vector sequence {(1,1), (1,5), (1,2),
(1,6), (1,3), (1,0), (1,4)} forms complete set of ring code
combinations on 2D ignorable array 67, and each of
them occurs exactly once (R=1).
Hence, each of the regarded GUS encoding system
provides minimizing basis as two-dimension binary
vector code of fixed sizes with the same 2D digit weights
but differ its cyclic ordering.
We have tabulated optimized monolithic code, which
forms massive arranged (solid parts of bits) both symbols
“1” and/or “0” for each code combinations as being
cyclic. This property makes such vector ring codes useful
in applications in high performance vector coding systems
and information technology with improving of noise
immunity, advanced vector data coded design of signals.
Next, we consider an example of non-redundant two-
dimensional binary GUS code under torus coordinate
system 3×5 with n=4, and R=1 (Table 5).
TABLE V. THE 4-DIGIT NON-REDUNDANT 2D GUS –CODE UNDER
TORUS COORDINATE SYSTEM 3×5
Table 5 illustrates forming 4- digit (n=4) 2D GUS-code
{(1,2), (2,4), (1,3), (2,1)} under torus coordinate system
3×5. The code size is P (n) = 2n – 1 = 15 with n = 4 and R
= 1. Note, the basis of the torus coordinate grid created
under the non-redundant 2D GUS {(1,2), (2,4), (1,3),
(2,1)}.
Clearly, a t- dimensional toroid coordinate system
designed for vector data coding t attributes and mi
categories of each of them (i= 1,2,…, t) requires t
concurrent disjointed axes m1, m2,…,mi, …,mt with
common reference point for forming t- dimensional
coordinate grid of the system with sizes m1×m2 ×…×mt.
So, the underlying multidimensional information
manifold coordinate system can be described by
parameters S, n, R, t, mi (i = 1, 2, …, t). Here t is
dimension of vector data array, number of attributes, and
number of significant digits of t - dimensional code, mi is
a number of categories of i- th attribute, and number of
reference points on i- th ring axis in a toroidal coordinate
system. Besides, information about vector data array
depends of geometric sizes m1×m2 ×…×mt, of the
coordinate system.
5. Conclusion and Outlook
The remarkable geometric property of ring symmetry
containing two complementary asymmetries is the World
Harmony law, which provides improving the quality
indices of one- and multidimensional cyclic codes with
respect to performance reliability, transmission speed, and
transmission content, using optimized vector data
encoding design, based on the idea of “perfect” vector
combinatorial constructions such as t-dimensional Ideal
Ring Bundles (IRB)s and Gloria to Ukraine Stars (GUS)s.
These design techniques make it possible to configure
encoding system with minimized basis, while maintaining
or improving on error protection, security, and function of
autocorrelation. Two main classes of optimized binary
vector codes regarded, including optimized cyclic error-
correcting IRB-sequences, and “Gloria to Ukraine Stars”
(GUS)s encoding ensembles. In this turn, GUS-codes
involve self-correcting, so-called “monolithic” cyclic
codes, and non-redundant t-dimensional vector data GUS-
№
Vector
Digit weights of the GUS- code
(1,1)
(1,3)
(1,5)
(1,0)
(1,2)
(1,4)
(1,6)
1
(0,0)
1
1
1
0
1
1
1
2
(0,1)
1
1
1
1
1
1
0
3
(0,2)
1
1
0
1
1
1
1
4
(0,3)
1
1
1
1
1
0
1
5
(0,4)
1
0
1
1
1
1
1
…
…
…
…
…
…
…
…
…
41
(5,5)
1
1
1
0
0
1
1
42
(5,6)
1
0
0
1
1
1
1
№
Vector
Digit weights of the GUS- code
(1,1)
(1,5)
(1,2)
(1,6)
(1,3)
(1,0)
(1,4)
1
(0,0)
1
1
1
1
1
0
1
2
(0,1)
1
1
1
0
1
1
1
3
(0,2)
1
0
1
1
1
1
1
4
(0,3)
1
1
1
1
1
1
0
5
(0,4)
1
1
1
1
0
1
1
…
…
…
…
…
…
…
…
…
41
(5,5)
1
1
1
0
0
1
1
42
(5,6)
1
1
0
0
1
1
1
№
The 4- digit 2D GUS- code under torus grid 3 × 5
Vector
Digit weights of the binary GUS -code
(1,2)
(2,4)
(1,3)
(2,1)
1
(0,0)
1
1
1
1
2
(0,1)
1
1
0
0
3
(0,2)
0
1
1
0
4
(0,3)
1
0
0
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
(2,3)
0
1
1
1
15
(2,4)
0
1
0
0
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codes for encoding sets of two- or more attribute
categories at the same time under t-dimensional manifold
coordinate systems of appropriate spatial array. Moreover,
we require each indexed vector data "category-attribute"
mutually uniquely corresponds to the point with the
eponymous set of the system. Study the properties allows
a better understanding of the role of geometric structure in
the behaviour of artificial and natural objects in different
dimensionalities. Besides, a combination of binary code
with vector weight discharges of the database is allowed,
and the set of all values of indexed vector data sets are the
same that a set of numerical values. These design
techniques make it possible to configure big vector data
coding systems with smaller number of code words than
at present. The underlying skills are useful at high schools
and universities for in-depth training of students, which
study computer sciences and information technologies,
involving contemporary combinatorial and algebraic
theory for increasing interest to scientific researches.
«... harmony that the human mind is ediable to
reveal in nature is the only objective reality, the
only truth we can achieve; and what I will add is
that the universal harmony of the world is the
source of all beauty, it will be clear how we
should appreciate those slow and difficult steps
forward that little by little open it to us..."
Jules Henri Poincaré
References
[1] P.Wigner, Symmetries and reflections. Bloomington-
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