Newfamily of error-correcting codes based on genetic algorithms

IAES International Journal of Artificial Intelligence (IJ-AI)
Vol. 14, No. 2, April 2025, pp. 1077∼1086
ISSN: 2252-8938, DOI: 10.11591/ijai.v14.i2.pp1077-1086 ❒ 1077
New family of error-correcting codes based on genetic
algorithms
El Mehdi Bellfkih1
, Said Nouh1
, Imrane Chemseddine Idrissi1
, Khalid Louartiti2
, Jamal Mouline1
1Department of Mathematics and Computer Science, Faculty of Science Ben M’sick, University Hassan II, Casablanca, Morocco
2Department of Mathematical Sciences and Decision Support, ENSA, Abdelmalek Essaâdi University, Tetouan, Morocco
Article Info
Article history:
Received Nov 28, 2023
Revised Nov 17, 2024
Accepted Nov 24, 2024
Keywords:
Decoding
Design
Error-correcting codes
Generator vector
Genetic algorithm
Minimum distance
ABSTRACT
This paper introduces a novel error-correcting code (ECC) construction and
decoding approach utilizing genetic algorithms (GAs). Classical ECCs often
struggle with efficiency in correcting multiple errors due to time-consuming
matrix-based encoding and decoding processes. Our GA-based method opti-
mizes generator vectors to maximize the minimum distance between codewords,
enhancing error correction capabilities. Specifically, we construct a new family
of ECCs with code length 31, dimension 12, and minimum distance 7, reducing
complexity from O(kn) to O(k(n − k)) by encoding message blocks with vec-
tors instead of matrices. In the decoding phase, the GA effectively corrects errors
in received codewords. Experimental results show that at a signal-to-noise ratio
(SNR) of 7.7 dB, our method achieves a bit error rate (BER) of 10−5
after only
9 generations of the GA. These results demonstrate improved error correction
and decoding performance compared to traditional methods. This study con-
tributes an innovative approach using GAs for error correction, offering simpler
encoding and robust performance in coding schemes.
This is an open access article under the CC BY-SA license.
Corresponding Author:
El Mehdi Bellfkih
Department of Mathematics and Computer Science, Faculty of Science Ben M’sick, University Hassan II
Casablanca, Morocco
Email: elmehdi.bellfkih@gmail.com
1. INTRODUCTION
The transmission and storage of information are susceptible to corruption due to various physical or
logical faults, which can result in system-wide failures. To mitigate such risks, robust testing and fault tolerance
mechanisms are essential for ensuring secure and stable communication flows. Error-correcting codes (ECCs)
play a pivotal role in safeguarding data integrity and reliability by incorporating redundant information into
transmitted messages.
The efficacy of ECCs lies in their ability to detect and/or correct errors that may arise during data
transmission or storage. This error-correction capability is crucial for maintaining data integrity under adverse
conditions. While linear block codes, such as Hamming codes, offer decent error-correction capability, they
are inherently limited in their scope. In contrast, nonlinear block codes, exemplified by turbo codes, exhibit
superior error-correction capabilities but are accompanied by higher decoding complexities [1], [2]. As shown
in the Figure 1, the minimum distance of a code is directly related to its error detection and correction capability.
A code with a larger minimum distance can detect and correct more errors compared to a code with a smaller
minimum distance.
Journal homepage: http://ijai.iaescore.com

1078 ❒ ISSN: 2252-8938
Figure 1. Correlating minimum Hamming distance with error detection and correction capabilities
In the realm of ECC construction, linear block codes, grounded in linear algebra, are renowned for
their simplicity of implementation, analysis, and comprehension. They excel at detecting and correcting errors
within a confined bit range. Examples include Hamming codes, Reed-Solomon codes, and Bose-Chaudhuri-
Hocquenghem (BCH) codes. Conversely, nonlinear block codes present a more intricate landscape, demanding
deeper analytical understanding and implementation efforts. Yet, they boast broader error-correcting capabili-
ties, effectively managing errors across a larger bit spectrum. Notable examples encompass Reed-Muller codes,
Golay codes, and BCH codes.
However, despite the advancements in ECC design, the process of decoding remains a challenging
task. Traditional decoding methods often encounter computational bottlenecks, particularly when dealing with
complex codes. Herein lies the potential for employing metaheuristic approaches to decode ECCs efficiently.
Metaheuristic algorithms, renowned for their adaptability and problem-solving prowess, offer a promising
avenue for tackling the intricacies of ECC decoding. By leveraging metaheuristic techniques, such as genetic
algorithms (GAs), simulated annealing, or particle swarm optimization, researchers can explore novel decoding
strategies capable of surmounting the complexities associated with ECCs.
There is various classes of codes in coding theory, and various method to construct them aiming to
achieve the best results e.g., the reliable communication, better complexity, easy construction of code. Let F2
be a field of order 2 and Fk
2 be a vector space of length k. Here we present our new k-dimensional binary linear
code C over Fn
2 whose G is its generator matrix, or g(x) is its polynomial generator (the rows of G form a basis
for C). [n, k, d]2 denotes a 2-ary linear code with length n, dimenion k and minimum distance d. An element
of C is called a codeword, its weight is the number of nonzero coordinate. The minimum distance of C is the
smallest Hamming distance between distinct codewords (is also the smallest weight in case of binary linear
codes) denoted by d(C). The Singleton bound as in (1) states that a (n, k, d)-code or [n, k, d]-code satisfy.
d(C) ≤ n − k + 1 (1)
A code with linearity condition and achieves the equality in the Singleton bound is called maximum
distance separable (MDS) code. To achieve the goal of finding high-performing ECC, our approach takes
advantage of the optimization nature of the problem. By formulating the problem as an optimization problem,
we can leverage the power of optimization algorithms, such as GAs [3], [4], to search for the best possible
solutions. The GA framework allows us to efficiently explore the vast solution space and find good ECC with
high minimum distances, making it an ideal approach for this type of problem. GAs are a type of optimization
algorithm that is inspired by the process of natural selection and evolution [5]. GAs are used to solve complex
problems by simulating the process of evolution, where a population of potential solutions evolves over time
towards an optimal solution [6]–[9]. They works by representing a problem as a set of candidate solutions, also
known as a population. Each candidate solution is encoded as a string of parameters, called a chromosome.
The chromosomes in the population are then evaluated using a fitness function that assigns a numerical score to
each chromosome based on how well it solves the problem. The chromosomes with the highest fitness scores
are selected as parents and used to produce offspring, which are new candidate solutions, through a process
called crossover. In this process, the genetic information from the parent chromosomes is combined to produce
Int J Artif Intell, Vol. 14, No. 2, April 2025: 1077–1086

Int J Artif Intell ISSN: 2252-8938 ❒ 1079
a new chromosome (child). This process is repeated over several generations, leading to the evolution of the
population towards better solutions. GAs also include a mechanism for introducing random variations into
the population, called mutation. This allows the algorithm to explore new regions of the solution space and
helps to prevent getting stuck in local optima. They are well-suited for problems that have multiple solutions
or where the solution space is complex and difficult to explore using traditional optimization methods. In the
next section, we will consider related research findings to contextualize and augment our study’s conclusions.
2. RELATED WORKS
The field of ECCs plays a crucial role in ensuring data integrity and reliability in various applications.
Despite the widespread use of existing ECCs, there are challenges that hinder their efficiency, particularly in
terms of the time-consuming encoding and decoding processes. To address these limitations, researchers have
turned to innovative approaches such as GA for code design and decoding. GA offer a promising avenue for
generating codes with high minimum distances, thus enhancing error detection and correction capabilities. A
range of studies have explored the design and decoding of ECC [10]–[12]. Natarajan et al. [13] developed
algebraic ECC for informed receivers, while Elkelesh et al. [11] proposed a GA-based low density parity check
(LDPC) code design scheme. Das and Touba [14] introduced a new class of single burst ECC with parallel
decoding, and Zhang et al. [15] presented a decoding algorithm for five-error-correcting binary quadratic
residue codes. These studies collectively contribute to the advancement of ECC, with a focus on informed
receivers, LDPC codes, burst error correction, and decoding algorithms.
McGuire and Sabin [16] have employed GA to search for linear binary codes with optimal minimum
distance. In another paper, Maini et al. [17] developed suboptimal soft decision decoders for linear block
codes. GA have also been utilized to tackle the problem of finding ECCs that correct a maximum number of
errors [18]. These studies highlight the effectiveness of GA in addressing various aspects of error correction
code design and decoding, and recognized as one of the most powerful optimization methods due to its ver-
satility and ease of implementation across various problem domains. One of the key strengths of GA lies in
their diverse set of operators and options, which allow for flexible exploration and exploitation of the search
space [19]–[21]. These operators, including selection, crossover, and mutation, provide a rich toolbox that
can be tailored to specific optimization problems. Moreover, the GA’s inherent parallelism and population-
based nature enable it to effectively handle complex and multimodal optimization landscapes. In fact, it can
be viewed as a variant of the minimum distance problem, which is known to be NP-hard. The minimum
distance of a code plays a crucial role in its error detection and correction capabilities. However, determin-
ing the exact minimum distance of a code is computationally complex and requires exhaustive search over
all codewords. This computational hardness motivates the exploration of heuristic approaches, such as GA,
to efficiently search for codes with large minimum distances. The design of ECCs has traditionally relied on
coding-theoretic principles, aiming to optimize code properties such as minimum Hamming distance and de-
coding threshold. However, recent advancements have explored the application of artificial intelligence (AI)
techniques, particularly GA, for ECC design. Huang et al. [10] investigate an AI-driven approach using GA
to design optimal codes within specific families, showcasing comparable performance to existing codes and
even superior performance in certain cases. Amirzadeh et al. [22] focus on joint GA and linear program-
ming optimization for LDPC codes, striving for low complexity, high coding threshold, and decoding stability.
Mahran [23] explores the optimization of turbo product codes (TPC) parameters using GA, finding a balance
between error performance and code complexity. Joundan et al. [24] present a GA approach for designing linear
codes with large minimum weight and small dual minimum distance, demonstrating effective error correction
performance. These studies collectively highlight the potential of GA in ECC design, offering opportunities for
improving code performance, complexity, and error correction capabilities in various communication systems.
GAs have emerged as a powerful tool for ECCs decoding. Chaibi et al. [25] present a GA-based
decoder for LDPC codes, demonstrating its superior performance compared to the sum-product decoder.
Azouaoui et al. [26] propose hard-decision and soft-decision decoding techniques based on GAs for general
ECC, showcasing their effectiveness over various transmission channels. Broulı́m et al. [27] explore the appli-
cation of GA optimization algorithms to design parity-check matrices for LDPC codes, enabling the correction
of burst errors. Nouh et al. [28] focus on decoding block codes using GAs and permutations set, showing
comparable error correcting performances to existing methods. Elkelesh et al. [11] present a decoder-tailored
polar code design using GAs, achieving the same error-rate performance as existing decoding algorithms while
New family of error-correcting codes based on genetic algorithms (El Mehdi Bellfkih)

1080 ❒ ISSN: 2252-8938
reducing the decoding complexity. Berkani et al. [29] propose compact GAs with larger tournament size for
improved decoding of linear block codes, demonstrating the effectiveness of larger tournament sizes in soft
decision decoding. These studies collectively highlight the potential of GAs in ECC decoding and code design,
offering enhanced error correction performance, reduced complexity, and improved decoding capabilities in
various communication.
3. PROPOSED METHODS
In this section, our GA-based methods are proposed using the principal factors (fitness function,
crossover, and mutation factors). we will delve into the application of GAs based methods in the encoding
and decoding phases of ECCs. Specifically, we will explore how GAs can be utilized to optimize these cru-
cial stages of the coding process. For the encoding phase, we will discuss the use of GAs based methods
to determine optimal generator vectors, considering factors such as code properties and encoding complexity.
In the decoding phase, we will examine how GA based method can aid in finding the corrected corrupted
received words, focusing on factors such as decoding performance, and error correction capability. Through a
detailed analysis, we aim to shed light on the main factors and considerations when employing GA for efficient
encoding and decoding of ECCs.
3.1. Construction phase
Our primary objective is to identify a generator vector that maximizes the distance between encoded
messages. By employing GA in the encoding phase, we aim to find the most suitable generator vector that
enhances error correction capabilities. However, we will rely on encoding through multiplying by generator
vector and conversion based on binary and decimal.
The Figure 2 showcases the sequential steps involved in encoding a message using a generator vector.
The process begins with the division of the message into blocks, represented in decimal form. These blocks
are then converted into their corresponding binary forms, ensuring that the message is represented using binary
digits. The next stage focuses on the encoding process itself. The binary message, consisting of k bits, under-
goes multiplication with a generator vector of n-k bits. This multiplication results in a binary message of length
n bits, which represents the encoded message with added redundancy for error correction or detection. Finally,
the encoded message is converted back to its original decimal form. This figure provides a clear visualization
of the encoding process, emphasizing the transformation from decimal to binary representation, the application
of the generator vector for encoding, and the subsequent conversion back to decimal form.
Figure 2. Encoding process using generator vector
The diagram in Figure 3 illustrates a GA-based method for finding an optimal generator vector.
The GA operates on a population of candidate generator vectors, with the number of generations and initial
population size specified as input parameters. Elitism is employed as the selection strategy, ensuring that the
fittest individuals are preserved in each generation. The fitness function, defined as the minimum distance
achieved by a generator vector, guides the evaluation and selection process. Crossover and mutation operators
are applied to introduce diversity and explore new solutions within the population. The initial population con-
sists of generator vectors with n − k bits, where n is the total number of code-word bits and k is the number of
message bits. The GA iteratively evolves the population to converge towards an optimal generator vector that
maximizes the minimum distance.
fgen = min{d(C) : C = {bwi × gen, ∀i < 2k
}} (2)
Where bwi are messages of k bits and gen is a generator vector of n-k bits.

Figure 3. Diagram of the method based on the GA to find the generator vector for an ECC
The provided Figure 4 demonstrates the crossover operation in our GA-based method. If a randomly
generated probability is less than or equal to a predefined value (pm = 0.97), the crossover is applied. Two
parent individuals, each represented by a binary sequence of n bits, are selected based on fitness function value
as mentioned in (2). A random position, denoted as p, is chosen within the length of the sequence. The first
child is created by combining the section from the first parent starting from position 0 up to position p, with the
section from the second parent starting from position p up to position n. Similarly, the second child is formed
by combining the section from the second parent from position 0 to p, and from the first parent from position
p to n. Additionally, the figure indicates that the mutation operation follows a similar principle. If a randomly
generated number between 0 and 1 is less than or equal to a predefined value (pc = 0.02), the mutation occurs.
It involves flipping the value at a specific position in the child’s binary sequence.
Figure 4. Crossover and mutation factors
3.2. Decoding phase
We present a GA-based method for correcting corrupted received code-words in ECCs. Our objec-
tive is to accurately recover the original information from the received word, even in the presence of errors.
The proposed method utilizes GA to iteratively search for the optimal solution that converge to the correct
code-word.
fcodeword = d(receivedword, codeword) (3)

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The diagram in Figure 5 illustrates a GA-based method for decoding received words in ECCs. The
algorithm takes several inputs, including the length n and dimension k of the ECC, the number of corrections
allowed t, the number of generations for the algorithm to iterate, and the initial population consisting of code-
words generated using the available generator vectors. Elitism is employed as the selection strategy, and the
fitness function is defined as the minimum distance between the received word and the code-words as in (3)
in the population. The crossover and mutation operations are applied with specific rates and with the same
strategy as shown in the Figure 4, aiming to explore and exploit the solution space. The initial population is
initialized with generator vectors of size n-k bits. Through the iterations of the GA, the method aims to decode
the received word and recover the original information accurately.
Figure 5. Diagram of the method based on the GA for decoding new ECCs
4. RESULTS AND DISCUSSION
In this section, we present the results obtained from our study on the construction and decoding
of ECCs. The subsections below detail the outcomes of our investigations into both the construction and
decoding phases, highlighting the performance and efficacy of our proposed methodologies. Through rigorous
experimentation and analysis, we assess the effectiveness of our approach in achieving robust error correction
capabilities and efficient decoding processes.
4.1. Construction of error-correcting codes
The provided Table 1 outlines the default parameters used in running the GA-based method for finding
the generator vector of the ECC with a length of 31 and a dimension of 12. These parameters, which include
settings such as population size, crossover rate, and mutation probability, serve as the initial configurations for
the GA, providing a starting point for the optimization process. By carefully selecting these default parameters,
the algorithm efficiently navigates the search space to identify generator vectors that maximize the minimum
distance between codewords, thereby enhancing the ECC’s error-correcting capabilities.
After running the GA-based method with the default parameters mentioned in Table 1, we obtained a
set of generator vectors for an ECC of length 31 and dimension 26. The Table 2 results include the minimum
distance achieved by these generator vectors, which is equal to the known lower bound. This suggests that the
GA effectively identified generator vectors that offer optimal error correction capabilities for the given ECC
dimensions.

Table 1. The default parameters for GA based method for codes of moderate lengths
Parameter value
Initial population size 20000
Selection elitism
Crossover rate 0.93
Mutation rate 0.02
Number of generations 50
Table 2. Set of ECCs of parameters (31,12)
n k d Generator
31 12 7 1110001001011000001
31 12 7 1110000111101100101
31 12 7 1101001101011000101
31 12 7 1010000110111010111
31 12 7 1110010110100100101
31 12 7 1011100111111000111
31 12 7 1111011100111010011
31 12 7 1111010010101000111
31 12 7 1111100110001111011
31 12 7 1111111010010011001
31 12 7 1111001011000011001
31 12 7 1110101001111100111
31 12 7 1111110111011010011
31 12 7 1110100010101001111
The application of the GA-based approach resulted in the discovery of ECCs with dimension 12 and
length 31, showcasing minimum distances that equal to the known lower bound. This significant achievement
holds promising implications for error detection and correction in practical scenarios. These codes exhibit an
exceptional capability to detect and correct errors, surpassing the performance of previously known codes. The
listed codes in Table 2 exemplify superior error-correcting properties, indicating their potential for enhancing
data integrity and ensuring reliable information transmission and storage. Also, Our GA based method has
successfully identified optimal generator vectors, enabling a more efficient encoding process. Instead of multi-
plying message blocks of length k by a matrix of dimension (k,n), we now multiply them by a vector of length
n-k. This reduction in dimensionality results in significant complexity gains, leading to improved efficiency in
the encoding process. The results are summarized in the Table 3.
Table 3. Encoding complexity
Encoding process Complexity
Encoding via generator matrix O(kn)
Encoding via generator vector O(k(n − k))
4.2. Decoding
After successfully finding a set of generator vectors that maximize the error-correcting capabilities
of our ECCs, we proceed to the decoding phase, where we introduce a GA-based method for decoding these
new codes. This method leverages GA to efficiently correct errors in the received codewords by exploring
possible solutions and selecting the most optimal one based on a fitness function. The focus of this section is
on evaluating the bit error rate (BER) performance of the decoding process, demonstrating how effectively our
GA-based decoder restores the original messages under various levels of noise.
The Table 4, presents the chosen default parameters for the GA-based decoding method include a
relatively small population size and a limited number of generations. This decision was made to ensure a
manageable computational complexity during the decoding process. Our algorithm is designed to create a
population of candidate words derived from a received word. Specifically, the algorithm generates a set of
InitPop words closely related to the input received word. Additionally, we implement an adjustment by
increasing the minimum allowable distance between generated words. These two strategic steps collectively
serve to reduce algorithmic complexity and enhance computational efficiency in terms of speed. Furthermore,
in instances where correction of the received word is not feasible due to an error count surpassing the predefined

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threshold value (t), the algorithm provides a set of proximate words. This information proves valuable in
scenarios where understanding the proximity of the received data is of significance.
Table 4. The default parameters for GA based method for decoding ECCs
Parameter Value
Initial population size 500
Selection elitism
Crossover rate 0.93
Mutation rate 0.07
Number of generations 1000
The Figure 6 illustrates the exceptional decoding performances of our method for our found code
with a length of 31, dimension 12, and a minimum distance of 7. Notably, at an signal-to-noise ratio (SNR) of
7.7 dB, the BER stands at 10−5
, highlighting the decoder’s initial performance. As the SNR increases to 8.5 dB,
the BER decreases to 10−6
, underscoring the decoder’s enhanced error-correcting capabilities with improved
SNR. This progression signifies the decoder’s effectiveness in achieving higher levels of data accuracy under
varying signal conditions.
Figure 6. BER performance of GA-based decoder
In spite of the substantial increase in the number of generations as indicated in Table 4, intended to
ensure the successful decoding of received words irrespective of the number of errors, the achieved outcomes
remain below the values specified in Table 4. This observation is substantiated by the statistical summary
presented in Table 5, which provides insights into the average and standard deviation. Notably, the low values
of both parameters in Table 5 signify the commendable efficiency and effectiveness of our algorithm in the
decoding process across varying SNRs.
Table 5. Statistical summary of algorithm performance
Number of words Max number of generations Avg number of generations Std number of generations
100000 1000 ≈ 8.7 ≈ 8.2
5. CONCLUSION
This research article has demonstrated the effectiveness of utilizing GA-based methods for both the
construction and decoding of ECCs. By employing these methods, we have successfully identified generator
vectors with high minimum Hamming distances, thereby streamlining the encoding process and enhancing the
BER performance of the codes. However, we acknowledge the limitation of achieving relatively low rates.
Moving forward, our future objectives entail addressing this limitation by optimizing the generation of gener-
ator vectors for specified parameters of length n and dimension k, as well as refining our decoder to further

improve the BER performances of the ECCs. Through these endeavors, we aim to bolster the efficiency and
efficacy of ECCs in real-world communication and storage systems.
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1086 ❒ ISSN: 2252-8938
BIOGRAPHIES OF AUTHORS
El Mehdi Bellfkih holds a Ph.D. in applied mathematics from Hassan II University,
specializing in coding theory, error-correcting codes, artificial intelligence, and machine learning.
His research explores innovative solutions in these fields to address complex computational problems.
He can be contacted at email: elmehdi.bellfkih@gmail.com.
Said Nouh holds a Ph.D. in computer sciences at National School of Computer Science
and Systems Analysis (ENSIAS), Rabat, Morocco in 2014. He is currently professor (higher degree
research (HDR)) at Faculty of sciences Ben M’Sick, Hassan II University, Casablanca, Morocco. His
current research interests artificial intelligence, machine learning, deep learning, telecommunications,
information, and coding theory. He can be contacted at email: said.nouh@univh2m.ma.
Imrane Chemseddine Idrissi is a Ph.D. in computer science at Faculty of Sciences Ben
M’Sik (FSBM), Hassan II University, Casablanca, Morocco. He received a master’s thesis in data
science and big data at ENSIAS Mohammed V university in 2019. His current research interests
include networks and systems, telecommunications, information, coding theory, machine learning,
and deep learning. He can be contacted at email: imrane.chemseddine-etu@etu.univh2c.ma or im-
ran.chems@gmail.com.
Khalid Louartiti originally hailing from Taounate, Morocco, he earned his Ph.D. from
Sidi Mohamed Ben Abdellah University in Fes, Morocco. Presently serving as a Professor at the
National School of Applied Sciences (ENSA) in Tetouan, Morocco. His research focuses on graph
theory, modules, ideals, commutative algebra, and amalgamated algebra. He can be contacted at
email: lokha2000@hotmail.com.
Jamal Mouline originally from Ouazzane, Morocco, he earned his Ph.D. from Provence
University in France. Presently, he holds the position of a Professor in the Department of Mathematics
and Informatics at Hassan II University in Morocco. His research focuses on fixed point theory and
combinatorial theory. He can be contacted at email: mouline61@gmail.com.

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