=Paper=
{{Paper
|id=Vol-3702/paper32
|storemode=property
|title=A Method of Control and Operational Diagnostics of Data Errors Presented in a Non-positional Number System in Residual Classes
|pdfUrl=https://ceur-ws.org/Vol-3702/paper32.pdf
|volume=Vol-3702
|authors=Alina Yanko,Victor Krasnobayev,Oleg Kruk
|dblpUrl=https://dblp.org/rec/conf/cmis/YankoKK24
}}
==A Method of Control and Operational Diagnostics of Data Errors Presented in a Non-positional Number System in Residual Classes==
https://ceur-ws.org/Vol-3702/paper32.pdf
A Method of Control and Operational Diagnostics of Data
Errors Presented in a Non-positional Number System in
Residual Classes
Alina Yanko1, Victor Krasnobayev2 and Oleg Kruk 1
1 National University «Yuri Kondratyuk Poltava Polytechnic», Pershotravneva Avenue 24, Poltava,36011, Ukraine
2 N. Karazin Kharkiv National University, Svobody sq., 4, Kharkiv, 61022, Ukraine
Abstract
The basis of modern infocommunication systems is computer means of data transmission and
processing. Therefore, one of the alternative means of achieving maximum efficiency in the functioning
of the infocommunication systems when processing data in real time is to improve, first of all, such
characteristics of computer data processing systems (CDPS) as the reliability and performance of
information processing, as well as the fault tolerance of its functioning.
Currently, the quality of implementation of information processing procedures is largely determined
by the selected mathematical model for organizing the information processing process in the CDPS.
Therefore, research and finding ways to solve the problem of increasing the reliability of real-time
CDPS, without reducing the productivity of processing large data arrays based on new mathematical
models, is an urgent task. One of the possible innovative ways of solving the formulated problem is the
usage of a non-positional number system in residual classes (NPNSRC) to create CDPS. The versatility
of NPNSRC codes is explained not only by their high correcting abilities, arithmetic and the ability to
fight against error packets, but also by their adaptability to flexible changes in correcting properties,
without changing the coding method. The article discusses issues related to the control and diagnosis
of data errors in the CDPS operating in the NPNSRC. The main attention is paid to the consideration of
a method for quickly diagnosing data errors in the NPNSRC. Reducing diagnostic time increases the
efficiency of diagnosing solitary errors in a non-positional code structure in the NPNSR. A specific
example of the implementation of a method for diagnosing data errors in the NPNSRC is given.
Keywords
Computer data processing system, diagnosing data errors, non-positional code structure, non-
positional number system in residual classes, number projection, orthogonal basis. 1
1. Introduction
An analysis of existing methods for increasing fault tolerance has shown that the best and most
widely used in practice today are two methods: redundancy and control, diagnostics with
further restoration of the CDPS functionality. When choosing control and diagnostics methods,
the main attention should be paid to the ability of this control and diagnostics method to detect
errors, as well as the amount of equipment and time spent on control [1-3].
The principles of control of non-positional code structures in the NPNSRC are the same as
the principles of control in the positional number system (PNS), while taking into account the
principles of formation of the NPNSRC and the influence of properties of the NPNSRC on the
structure of the CDPS [4], let us note, in a general way, the principles of data control in non-
positional code structures: the principle of reliability of control; the principle of continuity of
control; the principle of operational control.
The most effectiveness from the usage of the NPNSRC is accomplished in cases when the
realized algorithms comprise arithmetic operations such as addition, multiplication and
subtraction [5]. However, in a CDPS operating in the NPNSRC, in addition to the above
arithmetic operations, it is necessary to carry out so-called non-modular (positional) operations
[6-8]. Such operations include control operations and correction (diagnosis and correction) of
CMIS-2024: Seventh International Workshop on Computer Modeling and Intelligent Systems, May 3, 2024,
Zaporizhzhia, Ukraine
al9_yanko@ukr.net (A. Yanko); v.a.krasnobaev@gmail.com (V. Krasnobayev); olegkruk1975@gmail.com (O. Kruk)
0000-0003-2876-9316 (A. Yanko); 0000-0001-5192-9918 (V. Krasnobayev); 0009-0004-4241-2676 (O. Kruk)
© 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
data errors. Data control, diagnosis and correction operations, compared to arithmetic
operations in the NPNSRC, require significant time for their implementation. The need to
implement control (monitoring) operations, diagnosis and correction of data reduce the overall
effectiveness of the usage of the NPNSRC in real-time CDPS. When processing data in real time,
the considerable time required for the implementation of control, diagnostic, and error
correction procedures calls into question the feasibility of using NPNSRC as a general-purpose
CDPS. The need to ensure the high efficiency of the functioning of the CDPS in the NPNSRC
requires the development and implementation of methods of operational data control,
diagnostics and correction, other than methods employed in ordinary binary PNS [9].
Within the framework of the concept of development of fast-acting and reliable CDPS
presented in the NPNSRC, the urgent task is the development and application of methods and
means of operational data control, diagnosis and error correction. In the article, the main focus
is devoted to data diagnostics process in the NPNSRC.
The purpose of the research is to develop a method for controlling (monitoring) and
operational diagnostics errors in data presented in the NPNSRC, using the orthogonal basis of
partial sets of bases (modules).
2. Data correction process in the non-positional number system in
residual classes
The correction process (detection and correction) of errors in the information code structure D
of data consists of the following main stages [4, 9]:
data control (monitoring) (the process of detecting the presence of an error in
D (d1 ||d 2 || ...|| d j || ...|| d k ) , presented in the NPNSRC);
data diagnostics (localization of error locations with a given diagnostic depth);
error correction in the non-positional code structure (recovery of distorted residues
{d j } ( j 1, k ) of the incorrect number D and obtaining the correct number D ).
The number D (d 1 ||d 2 || ...|| d j 1 || d j || d j 1 || ...|| d k ) in without redundant NPNSRC is
represented by a set of residues {d j } ( j 1, k ) according to the selected system of information
k
bases (modules) {f j } in the numerical interval 0, L , where L f j is overall amount of
j 1
information code words [9]. In this case, the greatest common divisor of any two NPNSRC bases
is equal to (d i ,d j ) 1 ; i , j 1, k (i j ) .
In order for the non-positional code structure in the NPNSRC to have the necessary
corrective abilities, it is required that it contain sufficient information redundancy. First, the
extant information redundancy in the original structure of the non-positional code should be
determined and quantified [10]. Secondly, when tasked with providing data with additional
corrective capabilities, introduce additional (artificial) information redundancy (apply the
information redundancy method) by introducing additional (control) bases {f c } NPNSRCS [11].
Without loss of generality of reasoning, when tasked with providing data in the NPNSRCS
with additional corrective capabilities, we will assume that only one additional control bases
f c f k 1 is added to the k information bases, which is coprime with any of the k existing
information bases [4, 9]. In this case, the non-positional code structure
D (d 1 ||d 2 || ...|| d j || ...|| d k || d k 1 ) in the NPNSRC is represented by a set of {f i } (i 1, k 1)
bases in the full (working) numerical interval 0, L1 , where L1 L f k 1 is the overall amount
of code words for this NPNSRC with one control base [9].
It is known [4, 12] that for non-positional code structure in the NPNSRC the minimum code
distance is defined by the expression V min c 1 , where c is the number of control bases used
in the non-positional code structure in the NPNSRC, i.e. minimum code distance depends both
on the number of control bases and on the size of each of it.
g
If for the control bases {f z j } the condition f z j f c is satisfied, then the introduction into
j 1
the system of the NPNSRC bases of one control base f c f k 1 is equivalent to the presence of g
control bases f z 1 , f z 2 ,..., f z g . Taking into account the fact that all numbers taking part in data
processing in the CDPS, along with outcome of the operation are in the interval 0, L , then it is
clearly that if as outcome of data processing the final outcome D is obtained and at the same
time D L , this means that the resulting number D is distorted (incorrect). Thus, if D L ,
then the conclusion is that the number D is correct, and if D L , then the number D is
incorrect. In this case, only solitary errors (only in one {d j } of the number D ) are assumed, or
a packet of errors no longer than s [log2 (f j 1)] 1 binary digits in one residue modulo f j .
All existing methods for monitoring data in the NPNSRC are based on this principle of
comparing the value of the number D with the value 0, L of the information numerical
interval. Note that the comparison principle is also used in the development of diagnostic and
error correction methods. In the future, in this article, we will consider the method of
operational (quick) diagnosis.
3. Control and diagnostics of data in the non-positional number
system in residual classes
In [4] there are a number of scientific statements, the outcomes of the proof of which underlie
for methods for controlling and diagnosing data errors presented in the NPNSRC. It should be
reminded that in what follows only a solitary error is assumed (in one residue d j ( j 1, k 1)
of the number D (d 1 ||d 2 || ...|| d j || ...|| d k || d k 1 ) presented in the NPNSRC).
Let the number D (d 1 ||d 2 || ...|| d j || ...|| d k || d k 1 ) being checked be given in the NPNSRC
with informational {f j } ( j 1, k ) and one control base f c f k 1 . It is necessary, firstly, to
control (determine the correctness) of the number D , and, secondly, to diagnose the residues
{d j } ( j 1, k 1) of the number D , i.e. determine distorted (or undistorted) residues.
Data controlling and diagnostics are carried out sequentially in two stages.
First stage. Method for controlling (monitoring) data of non-positional code structure
D (d 1 ||d 2 || ...|| d j 1 || d j || d j 1 || ...|| d k || d k 1 ) , which from the following algorithm of actions:
1. Determine the values of the orthogonal basis B j ( j 1, k 1) for the complete system of
bases (modules) {f j } NPNSRC:
e j L1
Bj , (1)
fj
where e j is weight of the orthogonal basis B j .
2. Using the system of orthogonal basis B j , the original number D in the NPNSRC is
represented in the PNS [13]:
k 1
DPNS ( d j B j ) mod L1 . (2)
j 1
3. Carry out positional comparison operations between the values of DPNS and L . If the
comparison result showed that DPNS L , then number D is correct. If DPNS L , then the
number D is considered incorrect if only one of the residues {d j } of the number D is
distorted.
Second stage. Method for diagnosing the residues {d j } ( j 1, k 1) of the code structure D
of data, based on the use of the obtained results of the following statement.
Statement. Let in an ordered (f j f j 1 ) NPNSRC with k information and one control base
f c f k 1 the number D (d 1 ||d 2 || ...|| d j || ...|| d k || d k 1 ) satisfy the following condition:
L1
L Lk 1 D Li , (3)
f k 1
k
where L f j is overall amount of information code words in the NPNSRC (including only
j 1
information bases);
k 1
L1 f j is overall amount of all code words in the NPNSRC (including information bases and
j 1
control base);
Lk 1 is overall amount code words with one control base f c f k 1 ;
k 1
Li f p is overall amount of code words for excluding the base f i , that is
p 1,
p i .
Li f 1f 2... f i 1f i 1... f k 1 .
Then the residues of the number D are not distorted (correct) if only a solitary error (in one
residue d j ) is possible. The second stage of the developed method will be considered in more
detail using a specific example.
4. An example of the application of the control and operational
diagnostics method
Let's consider an example of using the control and diagnostic method in the NPNSRC for a one-
byte (l = 1) machine word (8 binary digits) CDPS. In this case, a complete NPNSRC with one
control base is specified by information f 1 3, f 2 4, f 3 5, f 4 7 and control
f c f k 1 f 5 11 bases. At the same time, the requirements for unambiguous representation
of code words in a given information numeric 0, L range are ensured.
k 1
For a given NPNSRC we can calculate: L1 f j f 1 f 2 f 3 f 4 f k 1 3 4 5 7 11 4620
j 1
k
– overall amount of code words in this NPNSRC; L f j f 1 f 2 f 3 f 4 3 4 5 7 420 –
j 1
overall amount of information code words in this NPNSRC. In this case, the full (working)
0, L1 and informational 0, L numerical ranges of numbers are defined, respectively, as
0, 4620 and 0, 420 . All possible partial sets of the NPNSRC bases for a one-byte (l = 1)
L1
CDPS are presented in Table 1, where Li , f i is a base that is not included in the given
fi
complete system of the NPNSRC bases.
For example, consider the process of finding Li , since the complete system of bases of the
considered NPNSRC consists of five bases: f 1 3, f 2 4, f 3 5, f 4 7 and f 5 11, the base that
is not included in the first line (i 1) of Table 1 is f i 3, so we divide overall amount of code
4620
words in this NPNSRC L1 4620 by this base: L1 1540 or can get the same result by
3
multiplying all sets of bases NPNSRC in the first line (i 1) of Table 1: L1 4 5 7 11 1540.
Table 1
Set of partial operating bases NPNSRC (l = 1) [14]
j
f1 f2 f3 f4 Li
i
1 4 5 7 11 1540
2 3 5 7 11 1155
3 3 4 7 11 924
4 3 4 5 11 660
5 3 4 5 7 420
Let, in the process of data processing, in lieu the correct D (1 ||0 || 0 || 2 || 1)
(DPNS 100 L 420) result of the operation a number of the form D (0 ||0 || 0 || 2 || 1), where
DPNS 3180 L 420. It is necessary to verify the correctness of the number D and diagnose its
residues {d j } ( j 1, 5).
First stage.
1. Let's define all possible orthogonal basis B j ( j 1, 5) using formula (1) for the complete
system of bases f 1 3, f 2 4, f 3 5, f 4 7 and f 5 11 NPNSRC [14]:
B1 (1, 0, 0, 0, 0) 1540, e1 1,
B 2 (0, 1, 0, 0, 0) 3465, e2 3,
B3 (0, 0, 1, 0, 0) 3696, e3 4,
B 4 (0, 0, 0, 1, 0) 2640, e 4 4,
B5 (0, 0, 0, 0, 1) 2520, e5 6.
2. Using the values of orthogonal basis B j ( j 1, 5), let's determine the value of number D
in the NPNSRC is represented in the PNS according to formula (2):
DPNS (0 1540 0 3465 0 3696 2 2640 1 2520)mod L1 7800 mod 4620 3180.
3. Let's compare the obtained number of DPNS and the value L 420. Since
DPNS 3180 L 420 , we make conclusion that the received D is incorrect by {d j } of the
correct number D (1 ||0 || 0 || 2 || 1).
Second stage.
1. Let's determine the values of partial orthogonal basis B ji for each of the 5 possible sets
of the NPNSRC bases [15]. In general, the value of partial orthogonal basis B ji is determined
based on the following comparison [16, 17]:
Li e ji
B ji 1(mod f j ), (4)
fj
where e ji 1, f j 1 – weight of the orthogonal basis B ji .
So, for j 4 and i 5 we have:
B1i (1, 0, 0, 0),
B 2i (0, 1, 0, 0 ),
B (0, 0, 1, 0),
B3i (0, 0, 0, 1).
4i
Let's determine the values of B j 1 for the first (i 1) set of bases: f 1 4, f 2 5, f 3 7 and
4
f 4 11 (see Table 1). In this case L1 f j 4 5 7 11 1540 ( L 420 ). We determine the
j 1
values of the partial orthogonal basis based on the known relationship [18-20].
L e
Determine the value of B11 1 11 . In this case, we use f 1 4 . In this case,
f1
L1 1540
385, e11 1, f 1 1 1, 4 1 1, 3. Let's compose possible values of B 11
f1 4
depending on possible e11 :
1 385 1(mod 4),
2 385 2(mod 4),
3 385 3(mod 4).
In this case, to fulfill the condition (4) we have that B11 1 385 385.
L1 e21 L1 1540
Let's determine the value of B 21 . In this case f 2 5, 308,
f2 f2 5
e21 1, f 2 1 1, 5 1 1, 4. Let's make a set of comparisons:
12 308 3(mod 5),
308 1(mod 5), 34308 4(mod 5),
308 2(mod 5).
In this case, B 21 2 308 616.
L1 e31 L1 1540
Let's determine the value of B31 . In this case f 3 7, 220,
f3 f3 7
e31 1, f 3 1 1, 7 1 1, 6. Let's make a set of comparisons:
1 220 3(mod 7), 4 220 5(mod 7),
2 220 6 (mod 7 ), 220 1(mod 7),
5
3 220 2(mod 7), 6 220 4(mod 7).
In this case, B31 5 220 1100.
L1 e 41 L1 1540
Let's determine the value of B 41 . In this case f 4 11, 140,
f4 f4 11
e41 1, f 4 1 1, 11 1 1, 10. Let's make a set of comparisons:
1 140 8(mod 11), 6 140 4(mod 11),
2 140 5(mod 11), 7 140 1(mod 11),
3 140 2(mod 11), 8 140 9(mod 11),
4 140 10(mod 11), 9 140 6(mod 11),
5 140 7(mod 11),
10 140 3(mod 11).
In this case, B 41 7 140 980.
Let's determine the values of B j 2 for the second (i 2) set of bases: f 1 3, f 2 5, f 3 7 and
4
f 4 11 (see Table 1). In this case L2 f j 3 5 7 11 1155 ( L 420 ).
j 1
L2 e12 L2 1155
Let's determine the value of B12 . In this case f 1 3, 385,
f1 f1 3
e12 1, f 1 1 1, 3 1 1, 2. Let's make a set of comparisons:
12 385 1(mod 3),
385 2(mod 3).
In this case, B12 1 385 385.
L2 e22 L2 1155
Let's determine the value of B 22 . In this case f 2 5, 231,
f2 f2 5
e22 1, f 2 1 1, 5 1 1, 4. Let's make a set of comparisons:
1 231 1(mod 5),
2 231 2(mod 5),
3 231 3(mod 5),
4 231 4(mod 5).
In this case, B 22 1 231 231.
L2 e32 L2 1155
Let's determine the value of B32 . In this case f 3 7, 165,
f3 f3 7
e32 1, f 3 1 1, 7 1 1, 6. Let's make a set of comparisons:
1 165 4(mod 7), 4 165 2(mod 7),
2 165 1(mod 7), 5 165 6(mod 7),
3 165 5(mod 7),
6 165 3(mod 7).
In this case, B32 2 165 330.
L2 e 42 L2 1155
Let's determine the value of B 42 . In this case f 4 11, 105,
f4 f4 11
e42 1, f 4 1 1, 11 1 1, 10. Let's make a set of comparisons:
1 105 6(mod 11), 6 105 3(mod 11),
2 105 1(mod 11), 7 105 9(mod 11),
3 105 7(mod 11), 8 105 4(mod 11),
4 105 2(mod 11), 9 105 10(mod 11),
5 105 8(mod 11), 10 105 2(mod 11).
In this case, B 42 2 105 210.
Let's determine the values of B j 3 for the third (i 3) set of bases: f 1 3, f 2 4, f 3 7 and
4
f 4 11 (see Table 1). In this case L3 f j 3 4 7 11 924 ( L 420 ).
j 1
L e L3 924
Let's determine the value of B13 2 13 . In this case f 1 3, 308,
f1 f1 3
e13 1, f 1 1 1, 3 1 1, 2. Let's make a set of comparisons:
1 308 2(mod 3),
2 308 1(mod 3).
In this case, B13 2 308 616.
L2 e23 L3 924
Let's determine the value of B 23 . In this case f 2 4, 231,
f2 f2 4
e23 1, f 2 1 1, 4 1 1, 3. Let's make a set of comparisons:
1 231 3(mod 4),
2 231 2(mod 4),
3 231 1(mod 4).
In this case, B 23 3 231 693.
L e L3 924
Let's determine the value of B33 2 33 . In this case f 3 7, 132,
f3 f3 7
e33 1, f 3 1 1, 7 1 1, 6. Let's make a set of comparisons:
1 132 6(mod 7), 4 132 3(mod 7),
2 132 5(mod 7), 5 132 2(mod 7),
3 132 4(mod 7),
6 132 1(mod 7).
In this case, B33 6 132 792.
L2 e 43 L3 924
Let's determine the value of B 43 . In this case f 4 11, 84,
f4 f 4 11
e43 1, f 4 1 1, 11 1 1, 10. Let's make a set of comparisons:
1 84 7(mod 11), 6 84 9(mod 11),
2 84 3(mod 11), 7 84 5(mod 11),
3 84 10 mod 11), 8 84 1(mod 11),
4 84 6(mod 11), 9 84 8(mod 11),
5 84 3(mod 11), 10 84 4(mod 11).
In this case, B 43 8 84 672.
Let's determine the values of B j 4 for the fourth (i 4) set of bases: f 1 3, f 2 4, f 3 5 and
4
f 4 11 (see Table 1). In this case L4 f j 3 4 5 11 660 ( L 420 ).
j 1
L4 e14 L4 660
Let's determine the value of B14 . In this case f 1 3, 220,
f1 f1 3
e14 1, f 1 1 1, 3 1 1, 2. Let's make a set of comparisons:
1 220 1(mod 3),
2 220 2(mod 3).
In this case, B14 1 220 220.
L4 e24 L4 660
Let's determine the value of B24 . In this case f 2 4, 165,
f2 f2 4
e24 1, f 2 1 1, 4 1 1, 3. Let's make a set of comparisons:
1 165 1(mod 4),
2 165 2(mod 4),
3 165 3(mod 4).
In this case, B 24 1 165 165.
L4 e34 L4 660
Let's determine the value of B34 . In this case f 3 5, 132,
f3 f3 5
e34 1, f 3 1 1, 5 1 1, 4. Let's make a set of comparisons:
12 132 2(mod 5),
132 4(mod 5), 34132 1(mod 5),
132 3(mod 5).
In this case, B34 3 132 396.
L4 e44 L4 660
Let's determine the value of B 44 . In this case f 4 11, 60,
f4 f 4 11
e44 1, f 4 1 1, 11 1 1, 10. Let's make a set of comparisons:
1 60 5(mod 11), 6 60 8(mod 11),
2 60 10(mod 11), 7 60 2(mod 11),
3 60 4(mod 11), 8 60 7(mod 11),
4 60 9(mod 11), 9 60 1(mod 11),
5 60 3(mod 11), 10 60 6(mod 11).
In this case, B 44 9 60 540.
Let's determine the values of B j 5 for the fifth (i 5) set of bases: f 1 3, f 2 4, f 3 5 and
4
f 4 7 (see Table 1). In this case L5 f j 3 4 5 7 420 ( L 420 ).
j 1
L e L5 420
Let's determine the value of B15 5 15 . In this case f 1 3, 140,
f1 f1 3
e15 1, f 1 1 1, 3 1 1, 2. Let's make a set of comparisons:
1 140 2(mod 3),
2 140 1(mod 3).
In this case, B15 2 140 280.
L5 e25 L5 420
Let's determine the value of B 25 . In this case f 2 4, 105,
f2 f2 4
e25 1, f 2 1 1, 4 1 1, 3. Let's make a set of comparisons:
1 105 1(mod 4),
2 105 2(mod 4),
3 105 3(mod 4).
In this case, B 25 1 105 105.
L e L5 420
Let's determine the value of B35 5 35 . In this case f 3 5, 84,
f3 f3 5
e35 1, f 3 1 1, 5 1 1, 4. Let's make a set of comparisons:
1 84 4(mod 5),
2 84 3(mod 5),
3 84 2(mod 5),
4 84 1(mod 5).
In this case, B35 4 84 336.
L5 e 45 L5 420
Let's determine the value of B 45 . In this case f 4 7, 60,
f4 f4 7
e45 1, f 4 1 1, 7 1 1, 6. Let's make a set of comparisons:
1 60 4(mod 7), 4 60 2(mod 7),
2 60 1(mod 7), 5 60 6(mod 7),
3 60 5(mod 7),
6 60 3(mod 7).
In this case, B 45 2 60 120.
The set of calculated partial orthogonal basis B ji is given in Table 2.
Table 2
Partial orthogonal basis Bji for l = 1 [14]
j
1 2 3 4
i
1 385 616 1100 980
2 385 231 330 210
3 616 693 792 672
4 220 165 396 540
5 280 105 336 120
2. Let's determine the correctness of the residues of the number D . First, let’s compose all
possible projections Di of the number D (0 ||0 || 0 || 2 || 1) :
D (0, 0, 2, 1),
1
D2 (0, 0, 2, 1),
D3 (0, 0, 2, 1),
D4 (0, 0, 0, 1),
D5 (0, 0, 0, 2).
3. Let's represent the values of the projections Di in the PNS [14]:
D1PNS (d 1 B11 d 2 B 21 d 3 B31 d 4 B 41 ) mod L1
(0 385 0 616 2 1100 1 980) mod 1540 100 420.
D2PNS (d 1 B12 d 2 B 22 d 3 B32 d 4 B 42 ) mod L2
(0 385 0 231 2 330 1 210) mod 1155 870 420.
D3PNS (d 1 B13 d 2 B 23 d 3 B33 d 4 B 43 ) mod L3
(0 616 0 693 2 792 1 672) mod 924 418 420.
D4PNS (d 1 B14 d 2 B 24 d 3 B34 d 4 B 44 ) mod L4
(0 220 0 165 2 396 1 540) mod 660 540 420.
D5PNS (d 1 B15 d 2 B 25 d 3 B35 d 4 B 45 ) mod L5
(0 280 0 105 2 336 1 120) mod 420 240 420.
Thus, of all the received projections Di of the number D (0 ||0 || 0 || 2 || 1) the projections
D1 , D3 , D5 L 420 and the projections D2 , D4 L 420. Therefore, the results of controlling
and diagnosing the incorrect D number state that among all five residues of the number it is
the residues d 1 , d 3 and d 5 may be erroneous, but the residues d 2 and d 4 are definitely not
distorted.
5. Conclusions
This article improves the method of control and operational diagnostics of data errors
presented in the NPNSRC, using the orthogonal basis B ji of partial sets of bases (modules)
NPNSRC. The values of orthogonal basis B ji are formed from the complete system of bases
{f i } (i 1, k 1) , and it usage provides an opportunity to organize the parallel processing of
projections Di of numbers D of a non-positional code structure in the NPNSRC. This
circumstance makes it possible to enhance efficiency of controlling and diagnosing data in the
CDPS operating in the NPNSRC.
Examples of specific realization of the process of controlling (monitoring) and diagnosing
errors are given. It is established that the use of the proposed method will improve the
efficiency of controlling and diagnosing data errors in the data processing system operating in
the NPNSRC, and will also technically simplify the procedure for processing non-positional code
structures.
References
[1] R. Wilmhof, F. Taylor, On hard errors in RNS architectures, IEEE Transactions on Acoustics,
Speech, and Signal Processing 31 (1983) 772–774. doi:10.1109/TASSP.1983.1164105.
[2] H. Krishna, K.-Y. Lin, J.-D. Sun, A coding theory approach to error control in redundant
residue number systems. I. Theory and single error correction, IEEE Transactions on
Circuits and Systems II: Analog and Digital Signal Processing 39 (1992) 8–17.
doi:10.1109/82.204106.
[3] Lie-Liang Yang, L. Hanzo, Redundant residue number system based error correction codes,
in: IEEE 54th Vehicular Technology Conference, Atlantic City, NJ, 2001, pp. 1472–1476.
doi:10.1109/VTC.2001.956442.
[4] I.Ya. Akushskii, D.I. Yuditskii, Machine Arithmetic in Residual Classes, Sov. Radio, Moscow,
1968.
[5] V. Krasnobayev, A. Yanko, D. Kovalchuk, Control, An Improved Method for Performing the
Arithmetic Operations of Modulo Addition of the Remainders of Numbers, in: 2023 13th
International Conference on Dependable Systems, Services and Technologies (DESSERT),
Athens, Greece, 2023, pp. 1–6. doi:10.1109/DESSERT61349.2023.10416508.
[6] G. Sobczyk, Modular Number Systems, New Foundations in Mathematics, Birkhäuser,
Boston, MA, 2013, pp. 1–21. doi:10.1007/978-0-8176-8385-6_1.
[7] V. Yatskiv, T. Tsavolyk, Two-dimensional corrective codes based on modular arithmetic, in:
The Experience of Designing and Application of CAD Systems in Microelectronics, Lviv,
2015, pp. 291–294. doi:10.1109/CADSM.2015.7230860.
[8] N.I. Chervyakov, A.V. Veligosha, K.T. Tyncherov, S. A. Velikikh, Use of modular coding for
high-speed digital filter design, Cybernetics and Systems Analysis 34 (1998) 254–260.
doi:10.1007/BF02742075.
[9] V. Krasnobayev, A. Yanko, D. Kovalchuk, Control, Diagnostics and Error Correction in the
Modular Number System, in: The Sixth International Workshop on Computer Modeling and
Intelligent Systems (CMIS 2023), Zaporizhzhia, Ukraine, 2023, pp. 199–213.
doi:10.32782/cmis/3392-17.
[10] P. V. A. Mohan, Residue Number Systems: Theory and Applications, Birkhäuser Basel,
Switzerland, 2016.
[11] R. Liu, L. Li, Y. Yang, Performance Residual Based Fault Detection for Feedback Control
Systems, IEEE Transactions on Circuits and Systems II: Express Briefs 68 (2021) 3291–
3295. doi:10.1109/TCSII.2021.3062718.
[12] S. Onyshchenko, A. Yanko, A. Hlushko, Improving the efficiency of diagnosing errors in
computer devices for processing economic data functioning in the class of residuals,
Eastern-European Journal of Enterprise Technologies 5, 4 (125) 63–73.
doi:https://doi.org/10.15587/1729-4061.2023.289185.
[13] M. Zaharchenko, M. Hadzhyiev, A. Nazarenko, N. Salmanov, N. Shvets, Comparison of
Positional and Timer Coding in the System of Residual Classes, in: 2022 IEEE 16th
International Conference on Advanced Trends in Radioelectronics, Telecommunications
and Computer Engineering (TCSET), pp. 414–418.
doi:10.1109/TCSET55632.2022.9766866.
[14] A. Moskalenko, V. Krasnobayev, S. Koshman, The concept of diagnostic data errors of
computing systems witch functioning in the system of residue classes, Computer science
and cybersecurity (CS&CS) 2 (6) (2017) 39–46. URL:
https://periodicals.karazin.ua/cscs/article/download/8733/9891.
[15] V. Krasnobayev, A. Yanko, A. Hlushko, Information Security of the National Economy Based
on an Effective Data Control Method, Journal of International Commerce, Economics and
Policy 14 (3) (2023). doi:10.1142/S1793993323500217.
[16] J. James, A. Pe, A novel method for error correction using Redundant Residue Number
System in digital communication systems, in: 2015 International Conference on Advances
in Computing, Communications and Informatics (ICACCI), Kochi, 2015, pp. 1793–1798.
doi:10.1109/ICACCI.2015.7275875.
[17] Z. Wang, E. Zheng, J. Liu, T. Guo, Adaptive Orthogonal Basis Function Detection Method for
Unknown Magnetic Target Motion State, Applied Sciences 14 (2):902 (2024).
doi:10.3390/app14020902.
[18] O. Laktionov, L. Lievi, A. Tretiak, M. Movin, Investigation of combined ensemble methods for
diagnostics of the quality of interaction of human-machine systems, Naukovyi Visnyk
Natsionalnoho Hirnychoho Universytetu 4 (2023) 138–143. doi:10.33271/nvngu/2023-
4/138.
[19] L. Mu, X. Liu and L.-L. Yang, Redundant Residue Number System Coded Diffusive Molecular
Communications, in: 10th International Conference on Wireless Communications and
Signal Processing (WCSP), Hangzhou, China, 2018, pp. 1–6,
doi:10.1109/WCSP.2018.8555582.
[20] Y. Jia, S. Wang, J. Jin, H. Long, Hang, Adaptive Orthogonal Basis Scheme for OTFS, in:
Proceedings of the 17th International Conference on Communications and Networking,
volume 500 of Lecture Notes of the Institute for Computer Sciences, Springer, Switzerland,
2023. doi:10.1007/978-3-031-34790-0_5.