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THE IMPACT OF BLOCK PROGRAMMING ENVIRONMENTS ON STUDENTS'
ERROR ANALYSIS SKILLS
Kayumova Shahnoza
Senior lecturer, Department of "Exact Sciences",
University of Science and Technology
Abstract:
This paper discusses the comparative error-correcting ability of block codes such as
Hamming, Golay, BCH, and Reed-Solomon codes. The introduction emphasizes the importance
of ensuring reliable data transmission under real-world interference conditions such as thermal
noise, multipath propagation, and impulse interference. Methods for improving noise immunity
are discussed, including convolutional and block coding. The paper presents analytical formulas
for calculating the probability of errors in decoded messages using various codes. The
calculation results confirm that the error-correcting ability depends on the code parameters and
the error probability in the channel. A study is conducted on the efficiency of codes in the
presence of burst errors of different lengths, including conditions under which decoding
remains successful. The paper emphasizes that the choice of the optimal code is determined by
the specifics of the telecommunication system and the characteristics of the communication
channel.
Introduction
Information plays an increasingly important role in all types of human activity. Recently, the
requirements for information transmission systems have increased dramatically. It is necessary
to transmit ever greater volumes of information over ever greater distances at ever greater
speeds. At the same time, the transmitter's energy resources are usually limited. The
requirements for the reliability of data transmission are also growing.
In recent years, methods of digital processing and transmission of information in various
telecommunication systems have significantly developed. One of the most important functions
in the operation of such systems is to ensure reliable protection of data from interference. The
radio channel is a weak link in data transmission systems, since it is in it that transmitted signals
are subject to distortion and attenuation due to the negative impact of numerous factors.
Interference and fading reduce the reliability of information transmission. Increasing the
reliability of information transmitted over a communication channel can be organized in various
ways, for example, by increasing the transmitter power, improving the sensitivity of the
receiver, increasing the gain of antennas. The implementation of these methods usually requires
significant material costs, and most importantly, does not ensure an increase in the reliability of
transmitted information with frequency-selective fading. Increasing the noise immunity of
information is achieved in various ways, but many of them are ineffective for one reason or
another. For example, increasing the transmitter power is limited by strict requirements for
electromagnetic compatibility of radiation sources, and multiple repetition of transmitted blocks
leads to a significant increase in channel occupancy and a corresponding increase in the
information processing time [1].
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At present, the task of ensuring the reliability of information transmission is solved in most
cases by using noise-resistant coding, which is a class of signal transformations performed to
improve the quality of communication. Modern people are surrounded by devices that use
noise-resistant coding algorithms in their work. Noise-resistant coding technologies have
become a mandatory element of data storage and transmission systems. The basis of modern
coding theory is the work of V.A. Kotelnikov and K.E. Shannon. Subsequently, the theory of
noise-resistant coding was developed by many researchers. However, the problem of an
unambiguous choice of the type of coding for a specific information transmission channel has
not yet been solved.
Currently, there are a significant number of options for constructing and decoding methods
for error-correcting codes that are capable of correcting both single and group errors. At the
same time, the characteristics of practical implementations of error-correcting codes lag
significantly behind the theoretical limits. Significant difficulties arise in satisfying the
requirements for code efficiency in order to achieve the constantly growing requirements for
data transmission and storage systems.
Reasons for deterioration of signal transmission quality
The source of interference in an ideal channel is thermal noise generated in the receiver [2, 3,
4]. Thermal noise typically has a constant spectral power density over the entire signal band and
a Gaussian voltage probability density function with zero mean. The signal in an ideal channel
attenuates with distance in exactly the same way as when propagating in ideal free space. The
signal power decreases proportionally to the square of the distance. With such ideal propagation,
the signal power is quite predictable.
Additional sources of losses in a real radio channel are natural and artificial sources of noise
and interference, the negative impact of which is often more significant than the thermal noise
of the receiver [5, 6, 7].
In radio communication, signal propagation occurs in the atmosphere and near the earth's
surface. A radio signal can travel from a transmitter to a receiver along multiple paths. This
phenomenon, called multipath propagation, can cause fluctuations in the amplitude, phase, and
angle of arrival of the received signal, which is called multipath fading. Fading causes random
fluctuations in the signal.
For a typical radio channel, the received signal consists of several discrete multipath
components, resulting in a spreading of the signal over time (or signal dispersion).
In the case of signal dispersion, the types of degradations due to fading are divided into
frequency-selective and frequency-non-selective. In the case of non-stationary channel behavior,
the types of degradations due to fading are divided into fast and slow.
In addition to independent errors, grouped errors may occur in the channel. They are formed
in channels with memory. One of the main causes of such errors are interruptions that occur due
to a smooth decrease in the signal level below the receiver's sensitivity threshold, when signal
reception practically ceases. Interruptions can be caused by various activities, and some of them
can even cause termination of a communication session. In addition, interruptions can be
caused by equipment malfunctions, imperfect operation, measurement, etc. Interruptions and
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impulse interference are the main cause of grouped errors when transmitting discrete messages
over various types of communication channels. Impulse interference is interference
concentrated in time. It is a random sequence of pulses with random amplitudes and following
each other at random time intervals, and the transient processes caused by them do not overlap
in time. The most common causes of such interference are: switching connections in electronic
equipment, interference from high-voltage lines, lightning discharges, and reception of reflected
signals.
Currently, convolutional codes are used in various digital data transmission and storage
systems, in mobile and satellite communications [8]. Noise-immune codes are quite diverse and
differ in the encoding-decoding method, the number of coded information bits, the introduced
redundancy, and the number of errors corrected. The parameters of noise-immune codes are
selected based on the characteristics of a specific digital communications system.
Currently, there are three main understandings of efficiency: efficiency in the sense of
effectiveness - this is the ability to produce an effect (result) of some actions, which cannot
always be measured using quantitative indicators; efficiency in the sense of productivity,
performance, economy - this is an indicator of the effectiveness of activities, reflecting the
amount of output per unit of costs (the fewer resources spent on achieving the planned results,
the higher the productivity); efficiency in the sense of effectiveness, optimality - this is the
ability to produce the planned result in the desired volume, can be expressed by a measure
(percentage ratio) of the actually produced result to the standard/planned one. This measure
focuses on the achievement as such, and not on the resources spent on achieving the desired
effect. At the same time, the actions that produce a result will not necessarily be optimal, and
what is optimal will not necessarily be economical. Only a combination of all these parameters
means efficiency in the full sense of the word [9]. Quantitatively, efficiency is assessed using
an efficiency indicator [10]. Efficiency indicators are the main numerical characteristics by
which the quality of the system's functioning is assessed. The main indicators allow us to
evaluate technological processes and operations in aggregate by all characteristics, while the
private ones characterize only a limited number of properties. Determining the composition and
content of the system of indicators necessary for conducting an efficiency assessment is a
classic task of systems analysis [11, 12].
Based on the analysis of modern educational literature [1, 9, 11], in the part concerning
error-correcting coding, the main indicators are:
- code rate;
- probability of a bit error in a decoded information message;
- energy gain from the use of error-correcting coding;
- spent computing resources;
- complexity of hardware implementation.
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The efficiency of error-correcting coding is assessed according to certain evaluation criteria.
Criterion - 1) means for making a judgment; standard for comparison; rule for evaluation; 2)
measure of the degree of closeness to the goal.
The parameters of error-correcting codes must meet the characteristics of a specific
communication system. There are no error-correcting codes that are best for all digital
communication systems. Approaches to assessing the effectiveness of error-correcting codes
may vary.
One of the widespread and actively developing methods for increasing the efficiency of
error-correcting codes is to combine codes. Convolutional codes are one of the large classes of
error-correcting codes.
Hamming codes form one of the best-known families of linear block codes [13, 14, 15, 16,
17]. For every natural number m ≥ 3, there exists a binary Hamming code with the following
parameters:
- length of code words
2
1
m
n
=
-
;
- number of information bits
2
1
m
k
m
=
- -
;
- number of verification digits
m n k
= -
;
- corrective ability
1
t
=
,
min
3
d
=
.
The Hamming code requires minimal redundancy for a given block length to correct one
error. The Hamming code is a perfect code.
The advantage of this code is its simplicity and, as a result, high encoding and decoding
speeds. The disadvantage is the ability to correct only single errors.
Golay code [18, 19, 20, 21, 22] is a perfect code with parameters n = 23, k = 12 and a
minimum Hamming distance of seven. This code guarantees the correction of all three-bit
errors. The advantage of this code is a relatively simple decoding algorithm and the ability to
withstand three-bit errors. The Golay code (23,12) has a generating polynomial
11
10
6
5
4
2
( )
g x
x
x
x
x
x
x
x
=
+
+
+
+
+
+
. The Golay code is encoded by implementing polynomial
division. Decoding the Golay code is usually done using the Meggitt decoder.
Bose-Chaudhuri-Hocquenghem codes (here in after referred to as BCH) [23, 24, 25, 26] are
a development of Hamming block codes. This type of code provides greater freedom in
choosing the block length, degree of coding, alphabet size, and error correction capabilities.
Theoretically, BCH codes can correct an arbitrary number of errors. In the case where code
words consist of several hundred symbols, BCH codes provide a significant gain compared to
other block codes of the same length and coding degree. Most often, BCH codes use code
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words of length
2 1
h
n
=
-
, where h = 3, 4, 5… For BCH codes, the maximum coding efficiency
is achieved with coding degrees between 1/3 and 3/4. From a mathematical point of view, the
construction of BCH codes is based on the operation of calculating the remainder from dividing
the vector of the information word by a generating polynomial.
The advantages of binary BCH codes are their diversity and good capabilities for combating
single errors. The disadvantages are rather complex decoding algorithms (especially for long
codes) and the inability to resist error bursts.
The symbols of the BCH code are taken from a finite Galois field [27, 28, 29]. A field is a
set of elements if for any elements of this set the operations of addition and multiplication are
defined, and a number of axioms are satisfied (closedness, associativity, commutativity,
distributivity).
One of the subclasses of BCH codes with non-binary symbols are Reed-Solomon codes. The
Reed-Solomon code [30, 31, 32] is a non-binary case of the BCH code. The symbols of non-
binary codes are multi-bit (m-bit) sequences. Reed-Solomon codes have a minimum distance
min
1
d
n k
= - +
and are capable of correcting
(
)
]
/ 2[
t
n k
=
-
errors. In communication channels,
the set of transmitted signals is always finite. Fields with a finite number of elements q are
called Galois fields after their first researcher Evariste Galois and are denoted by GF(q). The
number of elements of the field q is called the order of the field. Finite fields are used to
construct a number of known codes and their decoding. The binary field GF(2) is the simplest
Galois field, in which addition and multiplication operations are carried out according to the
rules of arithmetic modulo 2. The binary field is used to construct binary BCH codes. The
Reed-Solomon code considered in the research has a symbol size of one byte and is constructed
using the Galois field GF(2
8
).
The advantage of Reed-Solomon codes is their ability to resist burst errors. The
disadvantages are complex decoding algorithms.
Comparison of the Correcting Capability of Block Codes
The probability of occurrence of a bit error in a decoded information message is one of the
main values characterizing the correcting ability of noise-resistant codes. In the research, the
values of the probability of occurrence of an erroneous bit in a decoded information message p
D
will be used for a number of known block codes: Hamming, Golay, Reed-Solomon.
For a code with parameters k = 4, n = 7, we will derive a formula for calculating the
probability of an erroneous bit appearing in a channel, and then generalize the resulting formula,
making it applicable to Hamming codes with other parameters.
Let us denote the probability of having two erroneous bits out of n as P
2out of n
. According to
Bernoulli's formula [33, 34, 35, 36], it is equal to:
2
2
2
!
(1
)
2!(
2)
n
atn
B
B
n
P
p
p
n
-
=
-
-
1)
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For different values of the probability of occurrence of an erroneous bit in the channel, using
formula (1), we obtain the following results:
at p
B
= 10
-2
:
2at7
= 2∙10
-3
and
3at7
= 3.36∙10
-5
;
at p
B
= 10
-3
:
2at7
= 2.1∙10
-5
and
3at7
= 3.49∙10
-8
;
at p
B
= 10
-4
:
2at7
= 2.1∙10
-7
and
3at7
= 3.05∙10
-11
.
Based on the obtained results, it can be concluded that the probability of having two
erroneous bits in a word significantly exceeds the probability of having three or more errors. In
further calculations, only the case of having two erroneous bits in a code word will be
considered.
Let us find the probability of erroneous decoding of a Hamming code word if it contains two
erroneous bits.
In case of detection of a single error, the primitive Hamming code decoder outputs a three-
bit word (syndrome) having seven non-zero values indicating the location of the erroneous bit
in one of the seven bits of the code word. A zero-syndrome value indicates the absence of errors.
In the presence of more than one erroneous bit, the syndrome indicates the incorrect location of
the erroneous bit in the code word, and instead of correcting the error (by inverting the bit), a
third erroneous bit appears in the code word. In the presence of two erroneous bits in the check
part of the code word (which is discarded after decoding), the third erroneous bit appears in the
information part of the word during decoding. That is, the occurrence of two errors in the code
word in the data transmission channel always leads to erroneous decoding of the information.
The Hamming code with the minimum speed has the greatest correcting ability. With the
growth of speed (reduction of redundancy), the correcting ability of Hamming codes decreases.
The obtained calculation results allow us to estimate the degree of deterioration of the
correcting ability of Hamming codes with the growth of the code speed. In this case, the code
speed with the growth of m tends to 1, i.e., it can vary from 0.57 to 1 (by 1.75 times).
Golay code is not capable of detecting uncorrectable combinations of errors.
An erroneous decoding of a code word with a small degree of approximation is equal to the
probability of 4 errors out of 23 bits:
4
23 4
4
19
(23!/ (4! (23 4)!))
(1
)
8855
(1
)
BK
B
B
B
B
P
p
p
p
p
-
=
-
-
=
-
2)
If a decoding error occurs, the code word is transferred to another word, six bits away. Then
the probability of an erroneous bit appearing in the decoded information message is:
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4
19
(6 / 23)
2310
(1
)
D
W
B
B
P
p
p
p
=
=
-
3)
As a result of calculation using formula (3.14), the following results were obtained:
at p
B
= 10
-1
: P
D
=3.11∙10
-2
;
at p
B
= 10
-2
: P
D
= 1.90∙10
-5
;
at p
B
= 10
-3
: P
D
= 2.26∙10
-9
;
at p
B
= 10
-4
: P
D
= 2.30∙10
-13
.
Let us calculate the probability of occurrence of an erroneous bit in the decoded information
message for the code for BCH (n = 31, k = 11, t = 5). Erroneous decoding of the code word is
equal to the probability of occurrence of 6 errors out of 31 bits:
6
31 6
6
25
(31!/ (6! (31 6)!))
(1
)
736281
(1
)
BK
B
B
B
B
P
p
p
p
p
-
=
-
-
=
-
4)
The distance between code words is ten, then if there is a decoding error, the code word goes
into another, ten bits away. Then the probability of an erroneous bit appearing in the decoded
information message is:
6
25
(10 / 31)
237510
(1
)
D
W
B
B
P
p
p
p
=
=
-
5)
Let us conduct a study of the efficiency of the Reed-Solomon code with different
probabilities of occurrence of an erroneous bit in the channel. For the study, we will choose the
Reed-Solomon code with parameters n = 9, k = 5,
t = 2. Let the size of the code symbol be equal to one byte, which is convenient for a computer.
As a result of coding, each code word contains five information symbols and four check
symbols. The structure of such a code word containing nine symbols (72 bits).
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This code is capable of correcting any two erroneous symbols. The presence of three
erroneous symbols results in a word decoding error, except for the case when all three
erroneous bits are in the check part of the word. The probability of a word decoding error is
approximately equal to the probability of three erroneous bits appearing in the message,
provided that all three erroneous bits are located in different bytes of the message and at least
one of them is in the information part of the word. Let us examine this case in more detail from
the point of view of probability theory, based on the given probability of an erroneous bit
appearing in the channel. The probability of this RRS event is equal to the sum of the
probabilities of the following three events:
1) in the verification part there are two “erroneous” bytes, in the information part – one;
2) in the verification part there is one “erroneous” byte, in the information part – two;
3) there are no “erroneous” bytes in the verification part, and three in the information part.
Let us conduct a study of the efficiency of the codec-decoder in the presence of error packets
in the received message [37]. Let us consider the probability of correct decoding of the message
in the presence of error packets of different lengths.
discussions
The code under consideration is able to correct any two corrupted symbols in the code word.
Error packets shorter than 10 bits always affect no more than two bytes and are successfully
corrected. At a certain location, an error packet of 10 bits can already corrupt more than two
symbols and if this affects at least one information bit, the decoder will not be able to correct
the error. The probability of this event for a message of one code word is 0.11. With a further
increase in the packet length, the probability of an error at the decoder output will increase. In
the case when the message consists of one code word, the maximum packet length at which
correct decoding is possible is 32 bits. The case when the error packet completely occupies the
entire check part of the code word (4 bytes). The probability of an error at the decoder output in
this case is 0.975 (Figure 1).
FIGURE 1.
Error rates of different codes
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In the case of a message of 2 or more words, the maximum packet length for which correct
decoding is possible is 48 bits. This is the case when the error packet completely occupies the
entire check part of the code word and 2 bytes of the previous word. The error probability at the
decoder output is 0.988 for the case when the message consists of 3 code words. Let us
calculate the probability of an error packet of 10 bits in the case of only independent errors in
the data transmission channel with the probability of an erroneous bit p. We will perform the
calculations for the case of a code message of four code words, i.e. 288 bits in size.
Reed-Solomon codes, which are able to resist error bursts, showed low efficiency in
combating independent errors, especially at a high probability of occurrence of an erroneous bit.
At the probability of occurrence of an erroneous bit in the data transmission channel equal to 1
10-2, Hamming codes demonstrated a higher correcting ability.
One of the disadvantages of block codes is the fixed block length. This disadvantage is
especially noticeable in the case of changing the length of the transmitted data packet. If the
packet length is not divisible by the block length without remainder, then symbols that do not
carry information have to be added to the information sequence, which leads to a decrease in
coding efficiency [38].
CONCLUSION
The Article calculates the correcting ability of common block codes. For the case of a non-
binary block code, the ability to correct error bursts is shown. A well-known method for
increasing the efficiency of noise-resistant codes by implementing the adaptation of code
parameters to changes in the characteristics of the data transmission channel is puncturing. Here
we provide some basic advice for formatting your mathematics, but we do not attempt to define
detailed styles or specifications for mathematical typesetting. You should use the standard
styles, symbols, and conventions for the field/discipline you are writing about.
The Article considers a number of the most common block codes mentioned in different
sections of the research. The error-correcting ability is calculated. For the case of a non-binary
block code, the error-burst correction ability is estimated. The possibilities of block code
puncturing are considered. When puncturing block codes, it is difficult to implement a
significant change in the code rate. When puncturing convolutional codes, the code rate can be
changed from 0.5 to 0.83 (by 66%), and when puncturing block codes, from 0.555 to 0.625 (by
13%). Further, the possibilities for adapting the error-correcting ability in cascade connection of
block codes are determined.
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