EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6414 ISSN 1307-5543 – ejpam.com Published by New York Business Global Structural Insights and Decoding Strategies for BCH Codes over Quasi-Galois Rings Muhammad Sajjad1,∗, Muhammad Shoaib Abid2, Maha Alammari3, Robinson-Julian Serna4 1 NUTECH School of Applied Science and Humanities, National University of Technology, Islamabad, 44000, Pakistan 2 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 3 Department of Mathematics, College of Science, King Saud University, P.O. Box 22452 Riyadh 11495, Saudi Arabia 4 Escuela de Matemáticas y Estad́ıstica, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia Abstract. Robust data disclosure constitutes an essential problem for contemporary system com- munication, and the theory of coding becomes the key to maintaining data integrity. This paper discusses constructions and decoding of Bose–Chaudhuri–Hocquenghem (BCH) codes over Quasi- Galois Rings (QGRs) – generalization of classical Galois rings. The QGRs provide richer algebraic structures that lead to improved error correction capabilities, greater code rates, and more code- words than their Galois counterparts. We provide an all-round theory of construction for BCH codes over QGRs, describe them in their construction process, and walk through an efficient de- coding technique. Our findings demonstrate the ability of BCH codes under QGR to provide high reliability and performance for the communication systems, which will make them a candidate for future use in data transmission and storage. 2020 Mathematics Subject Classifications: 94B75, 11T71, 94A24, 68P30, 14G50, 94A05 Key Words and Phrases: Quasi-Galois Rings, BCH Codes, Error Correction, Code Rate, Ring- Based Coding Theory, Cyclic Subgroups 1. Introduction It relates to how information that is to be transmitted is put into another form that can easily be transmitted through a channel, which is usually a wire or broadcasting system. This entails the use of different codes to reduce errors that may be occasioned by the transmission medium, such as noise, interference or loss of signal strength. The ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6414 Email addresses: muhammad.sajjad@nutech.edu.pk (M. Sajjad), janjuashoaib814@gmail.com (M. S. Abid), malammari@ksu.edu.sa (M. Alammari), robinson.serna@uptc.edu.co (R. J. Serna) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 2 of 22 principles of error-detecting and error-correcting codes are very essential to ensure high data reliability, and this involves the Hamming codes, Reed-Solomon codes, and BCH codes. These techniques enhance the dependability of the original data in that the receiver checks and rectifies errors without having to ask for retransmission. As inferred from the above observations, this paper aims at showing that the selection of a particular scheme to be used can greatly affect the dependability and efficiency of the communication system with respect to the RB/SSS trade-off. Hence, the coding theory is a very enabling tool for a plethora of utilization opportunities in the digital world, such as in digital communication, data storage, deep-space signals and even in the new fashion wireless networks [1]. This section is devoted to the discussion of major achievements and the state of the progress as well as the results of recent decades in the study of BCH codes over different algebraic structures. The first step was defined by Assmus and Mattson in [1] who gave an axiomatic approach to error-correcting codes and stated fundamental theorems of modern coding theory. Their work provided what is now acknowledged as the first formal math- ematical definitions for making and deciphering codes that are still in use today. Augot, Betti, and Orsini [2] also elaborated on linear and cyclic codes and their importance in the real world, in fields like cryptography and data transmission. A thorough introduction is given to their basic algebraic properties and the encoding methods that are crucial in en- hancing the code’s performance. Blake [3] studied codes over certain rings, thus enriching and developing theoretical fundamentals of coding theory for various algebraic structures except for fields. His work laid the foundation for studying BCH codes over non-field rings such as the integer residue rings [4, 5] and modular rings [6, 7] which are vital in studying BCH codes over QGRs. Shah et al. [8] discussed constructions of codes by the semigroup rings, which discussed new way of code construction and encoding techniques. Their work helps to increase the area of algebraic applications of BCH codes and construct them over QGR, enhancing the usability and reliability of error-correcting codes. For achieving efficient error correction in finite field applications, Kim, Lee and Yoo [9] introduced an infinite family of Griesmer quasi-cyclic self-orthogonal codes. Their work shows that BCH codes are far better compared to QGRs in improving the coding efficiency and resilience in actual applications. Zullo [10] also discussed multi-orbit cyclic subspace codes and linear sets, which give more of the algebraic properties and the structures of cyclic codes. It is necessary to state that this research advances the knowledge of the theoretical background and real-world applications of BCH codes over various algebraic structures. Andrade and Palazzo [3] studied the construction and decoding of BCH codes over finite commutative rings that constituted the first step towards constructing the BCH code over rings other than the Galois fields. Their study stresses the ability to apply BCH codes in various algebraic settings, including possible over QGRs. Shankar [11] surveyed BCH codes over an arbitrary integer ring, extending the theory of BCH codes to other algebraic structures. The contribution of this research is useful in improving the reliability and usability of BCH codes in various communication and storage systems. Interlando and Palazzo [12] have given a structural description of cyclic code over Zm, and the decoding process in detail. Their research provides significant information regarding the enhancement of the BCH codes applied to the modular rings and other non-field structures, Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 3 of 22 as well as the possible use in the QGRs. Sajjad et al.’s [13–17] recent works have involved decoding algorithms that are designed to work on certain algebraic structures, including Gaussian fields, Eisenstein fields, quaternion integers, and octonion integers. These studies have substantially contributed to the improvement of the error correction capability of BCH codes as applicable to today’s communication systems. Lee, Li, Wu and Zeng [18] continued with the investigation of the hulls of primitive binary and ternary BCH codes with respect to particular structures that can improve code performance indicators. Their paper offers useful information about the enhancement of BCH codes in terms of performance for error correction and information reliability in implementation. In a work devoted to the description of specific properties and decoding of specific binary BCH codes of length n = 2m+1, Liu Li Fu Lu and Rao [19], this research with the collection of BCH codes which can be used for different communication schemes and data-storing methods. Further, new studies by different authors [20, 21] have explored details of BCH codes with the parameters and families of negacyclic and constacyclic BCH codes besides using them in cryptography and communication. These works present the continued research on the improvements of the additive BCH codes in terms of redundancy and code length in various algebraic structures. Several studies have contributed significantly to the advancement of BCH codes and their decoding strategies. Asif and Shah [22] explored a computational approach to BCH codes and demonstrated their effectiveness in image encryption applications, showcasing the practical utility of algebraic coding in data security. Shah and Andrade [23] proposed a decoding method aimed at enhancing both the code rate and error correction capabilities, thereby improving the overall performance of BCH codes. Further, Shah, Qamar, and de Andrade [24] investigated the construction and decoding of BCH codes over a chain of commutative rings, providing an algebraic foundation for extending classical code struc- tures. Additionally, Shah, Khan, and Andrade [25] introduced a novel decoding technique by embedding an n-length binary BCH code within a n(n+1) -length cyclic code, allowing for improved decoding accuracy. These foundational works underscore the importance of algebraic generalizations and decoding efficiency in enhancing the reliability of BCH codes for modern communication systems. Furthermore, cyclic codes were compared by Shah and De Andrade [26] with the help of B[X], B [ X, 1 pk Z0 ] , and B [ X, 1 kpZ0 ] pointing to further developments in constructing methods as well as decoding rules. Their work offers understanding on how to implement BCH code for certain algebraic structures, focusing on enhanced error correction and speed. Zhu, Li, and Zhu [27] identified parameters of two classes of negacyclic BCH codes, their structure, and decoding. Thus, the results of this research can be used to improve the efficiency and practical use of BCH codes in the field of cryptography and in digital communication systems. Wang, Sun and Ding [28] proposed two families of negacyclic BCH codes that were proved to be more powerful in correcting more errors and more reliable in data transmission. Citing their discoveries, they emphasize that BCH codes are applicable in any algebraic structure, which makes them useful in today’s communication networks. Moreover, the research of Pang, Zhu, Yang, Gao, Zhou, Kai, and others [29, 30] focuses on BCH codes with a larger hull dimen- Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 4 of 22 sionality and ternary LCD consta-cyclic BCH codes. These have included broadening of the theoretical as well as practical applications of the BCH codes and, in addition, the im- provements of the performance characteristics of BCH codes in extended communication networks. An ever-evolving communication technology means that there is a constant need for enhanced methods of error correction for the effective transfer of data. BCH (Bose- Chaudhuri-Hocquenghem) codes have always been acknowledged for their suitability for error correction because of the clear algebraic structure of the code and the availability of efficient decoding procedures. However, the desire to make coding more efficient has resulted in the consideration of what is called codes over other algebraic structures like the integer residue rings and finite commutative rings [1, 31]. Therefore, the theory of quasi-Galois rings (QGRs) is a natural and quite a powerful generalization of the theory of Galois rings that could be useful for improving the BCH codes [1]. Compared to the previous Galois rings, QGRs have additional algebraic characteristics that make it pos- sible to build BCH codes with higher error-correcting capability and code rate. These properties include an increased number of code words, which makes the BCH codes over QGRs more suitable for modern communication systems, and an improved code rate. New studies have established that it is possible to construct and decode BCH codes over QGRs. For instance, while applying Andrade and Palazzo’s work on BCH codes over finite com- mutative rings, there has been a marked enhancement in the error correction efficiency [1]. Likewise, Shankar’s work further extending the BCH codes over arbitrary integer rings has added more feathers in the hat of these codes in numerous applications [31]. These works form a base upon which further research can be conducted in order to better understand the characteristics of QGRs and the use of these groups in constructing BCH codes. This article has important contributions to the domain of coding theory in order to implement BCH codes over Quasi-Galois Rings (QGRs) for enhancing the actual commu- nication systems. From this, a comprehensive theoretical background for BCH codes over QGRs is presented to explain possible higher code rates and a larger number of code words compared to the use of Galois rings. Thus, the article demonstrates the practical applica- bility of the QGR-based BCH codes as a result of presenting novel construction methods and efficient decoding strategies. The enhanced understanding of these codes’ theoretical and practical characteristics underlines their capability to dramatically improve the effec- tiveness and stability of today’s communication networks in various applications. Thus, this investigation is expected to bring the BCH code over QGRs into light as potent and effective candidates to be incorporated in the advanced communication systems as well as contribute to the continuous improvement of coding methods in the modern world. 2. Preliminaries This section consists of basic results that are useful to understand the upcoming sec- tions [12–14, 18, 19, 26]. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 5 of 22 2.1. Quasi-Galois Ring Let a Quasi-Galois Ring (Q-GR) be a local, finite, commutative ring with cardinality prs and characteristic p (where p is a prime and n, s are any positive integers). Particularly from the perspective of applications in coding theory, Q-GRs are highly intriguing because they possess the desirable attribute of having prime characteristics. It is denoted by and defined as A(ps, n) = Fps [x] ⟨xn⟩ = { n−1∑ i=0 aiθ i : ai ∈ Fps } (1) where θ is a formal non-trivial root of the polynomial xn ∈ Fps [x], i.e., θ n = 0. 2.2. Units in Q-GR If a0 ̸= 0 in the expression, n−1∑ i=0 aiθ i, ai ∈ Fps , then the element is a unit in A(ps, n). 2.3. Nilpotent Elements in Q-GR If a0 = 0 in, A(ps, n) = Fps [x] ⟨xn⟩ = { n−1∑ i=0 aiθ i : ai ∈ Fps } , then the element is nilpotent. If a0 = 1, then it corresponds to a principal unit element. Proposition 2.1. Let A(ps, n) be a Q-GR. Then the group of units in A(ps, n) is isomor- phic to the direct product of two groups, U(A(ps, n)) ∼= G1 ×G2, where G1 is a cyclic group of order ps−1, and G2 is an Abelian p-group of order ps(n−1). Particularly, if s = 1 and n = 2, then G2 is a cyclic group of order p. While for p = 2, s = 1, n = 3, we have G2 ∼= C4. 2.4. Ideal Structure of Q-GR Since Q-GR is a local ring, it contains a unique maximal ideal, and the ring is given by: A(ps, n) = Fps [x] ⟨xn⟩ = { n−1∑ i=0 aiθ i : ai ∈ Fps } , θn = 0. Every proper ideal in Q-GR is of the form: Jk = θkA(ps, n), 1 ≤ k ≤ n− 1. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 6 of 22 3. BCH Code Construction over QGR Before designing BCH codes over Q-GRs, it is necessary to specify the Galois extension of Q-GRs in order to factorize xn − 1 over the group of units of the applicable extension ring of the known local ring (QGR) and then construct the generator polynomial for the BCH code. 3.1. Galois Extension of Q-GR A(ps, n) Let A(ps, n) be a class of local finite rings with a unique maximal ideal m(ps, n), and let the residue field be: K = A(ps, n)/m(ps, n) ∼= Fps . Consider the natural projection map, π : A(ps, n)[x] → K[x], where A(ps, n)[x] denotes the ring of polynomials in the variable x with coefficients from A(ps, n), and is defined by: π(a(x)) = a(x). If f(x) is a monic polynomial of degree m such that π(f(x)) is irreducible over the residue field K, then f(x) is irreducible over A(ps, n). The ring R = A(ps, n)[x] ⟨f(x)⟩ is called the Galois extension of the Q-GR A(ps, n), and consists of the collection of residue class polynomials in the variable x over A(ps, n), modulo the polynomial f(x). The elements of R are of the form: R = { m−1∑ i=0 ciα i : ci ∈ A(ps, n) } , where α is a root of f(x), i.e., f(α) = 0. Let R∗ be the multiplicative Abelian group of the units of R, which can be expressed as the direct product of subgroups. A cyclic subgroup of R∗ is denoted by Gs. The unit elements of R can be described as: R∗ = { x = c0 + c1α+ · · ·+ cm−1α m−1 ∈ R : ∃ci ∈ U(A(ps, n)) for i = 0, 1, . . . ,m− 1 } , where U(A(ps, n)) indicates the group of units of the Q-GR A(ps, n). On the other hand, the nilradical of R, denoted by Nil(R), is defined as: Nil(R) = { c0 + c1α+ · · ·+ cm−1α m−1 : ci ∈ Nil(A(ps, n)) } . Similarly, the extension of the residue field K of A(ps, n) is: K ′ = (A(ps, n)/m(ps, n))[x] ⟨π(f(x))⟩ = K[x] ⟨π(f(x))⟩ , which has cardinality pms. The multiplicative group of units in K ′ is denoted by K ′∗. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 7 of 22 3.2. Generator Polynomial of BCH Codes using Maximal Cyclic Sub- group of Group Units of Q-GR The generator polynomial of a BCH code of length n is defined as g(x) = lcm{mi(x) : i = c, c+ 1, . . . , c+ d− 2}, where mi(x) are the minimal polynomials corresponding to each ψi, for i = 1, 2, . . . , d− 1. The parity-check matrix H of the BCH code with the generator polynomial g(x) is given by: H = 1 ψc ψ2c · · · ψ(n−1)c ... ... ... . . . ... 1 ψc+d−2 ψ2(c+d−2) · · · ψ(n−1)(c+d−2)  . Equivalently, the code C is the null space of the matrix H. The following steps are used to construct the generator polynomial of an n-length BCH code over a Galois ring: (i) Create the maximal cyclic subgroup of the group of units of order n. (ii) Compute the minimal polynomials for each design distance. (iii) Find the least common multiple of all the minimal polynomials. The pseudocode for construction of BCH codes over the Quasi-Galois ring is given in Algorithm 1. Example 3.1: BCH Codes over A(2, 2) Let A(2, 2) be a Quasi-Galois ring (QGRs), defined as: A(2, 2) = F2[y] ⟨y2⟩ = { 1∑ i=0 aiθ i : ai ∈ F2, θ 2 = 0 mod 2 } = {a0 + a1θ : a0, a1 ∈ F2 = {0, 1}} = {0, 1, θ, 1+θ}. The group of units in A(2, 2) is: U(A(2, 2)) = {1, 1 + θ}. Basic Irreducible Polynomial in A(2, 2)[x]: The residue field of A(2, 2) is F2, and the natural projection is: π : A(2, 2)[x] → F2[x]. Case 1: Let f(x) ∈ A(2, 2)[x] be a monic polynomial of degree 2 of the form: f(x) = a0 + a1x+ x2, where a0, a1 ∈ A(2, 2). Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 8 of 22 Choose a0 = 1, a1 = 1 + θ, then: f(x) = x2 + (1 + θ)x+ 1. Now we check if f(x) is irreducible over A(2, 2). Its image under projection is: π(f(x)) = x2 + x+ 1, which is a monic irreducible polynomial over F2. We now evaluate f(x) at all possible elements of A(2, 2): f(0) = 1 ̸= 0, f(1) = 1 + (1 + θ) + 1 = 1 + θ ̸= 0, Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 9 of 22 f(θ) = θ2 + (1 + θ)θ + 1 = 0 + θ + 1 + 1 = θ ̸= 0, f(1 + θ) = (1 + θ)2 + (1 + θ)2 + 1 = 1 + θ + 1 + θ + 1 = 1 ̸= 0. Thus, f(x) is irreducible over A(2, 2) and hence is a basic irreducible polynomial. Extension of A(2, 2) with Respect to f(x): The Galois extension of A(2, 2) by the basic irreducible polynomial f(x) is defined as: R = A(2, 2)[x] ⟨f(x)⟩ = { 1∑ i=0 ciα i : ci ∈ A(2, 2) } , where f(x) = x2 + (1 + θ)x+ 1 and f(α) = 0. Maximal Cyclic Subgroup of R∗: Given that α satisfies: α2 = (1 + θ)α+ 1, we compute the successive powers: α3 = (1 + θ)α2 + α = (1 + θ)((1 + θ)α+ 1) + α = (1 + θ)2α+ (1 + θ) + α = α+ 1 + θ + α = 1 + θ, α4 = α · α3 = α(1 + θ) = (1 + θ)α, α5 = α · α4 = α · (1 + θ)α = (1 + θ)α2 = (1 + θ)((1 + θ)α+ 1) = α+ 1, α6 = α · α5 = α(α+ 1) = α2 + α = (1 + θ)α+ 1 + α = 1. So, the order of α is 6. The maximal cyclic subgroup G3 ⊂ R∗ generated by γ = α2 is: G3 = {γ, γ2, γ3 = 1} = {1 + (1 + θ)α, (1 + θ)α, 1}. Generator Polynomial of BCH Code over A(2, 2): Since γ is a primitive cube root of unity in G3, and taking design distance d = 3, we need the minimal polynomials of γi for i = 1, 2. Both γ and γ2 share the same minimal polynomial: m1(x) = (x− γ)(x− γ2) = x2 + x+ 1. Hence, the generator polynomial is: g(x) = lcm(m1(x),m2(x)) = m1(x) = x2 + x+ 1. For r = 2, q = 2, and m = 2, the length of the narrow-sense primitive BCH code is: n = qm − 1 = 22 − 1 = 3. The cyclic code C generated by g(x) over G3 ⊂ R∗ ⊂ A(2, 2) has dimension 1. Thus, the BCH code C is a: [3, 1, 3] primitive narrow-sense BCH code over the maximal cyclic subgroup of the group of units in A(2, 2). Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 10 of 22 Case 2: BCH Codes over A(2, 2) with Degree 3 Extension Let f(x) ∈ A(2, 2)[x] be a monic polynomial of degree 3, of the form: f(x) = a0 + a1x+ x3, where a0, a1 ∈ A(2, 2). Choose a0 = 1, a1 = 1 + θ, then: f(x) = x3 + (1 + θ)x+ 1. Under the natural projection π : A(2, 2)[x] → F2[x], we get: π(f(x)) = x3 + x+ 1, which is a monic irreducible polynomial over F2. We now verify the irreducibility of f(x) over A(2, 2) by evaluating: f(0) = 1 ̸= 0, f(1) = 1 + (1 + θ) + 1 = 1 + θ ̸= 0, f(θ) = θ3 + (1 + θ)θ + 1 = 0 + θ + 1 + 1 = θ ̸= 0, f(1 + θ) = (1 + θ)3 + (1 + θ)2 + 1 = 1 + θ ̸= 0. Hence, f(x) is a basic irreducible polynomial in A(2, 2)[x]. Extension of A(2, 2) with Respect to f(x): Define the extension ring as: R = A(2, 2)[x] ⟨f(x)⟩ = { 2∑ i=0 ciα i : ci ∈ A(2, 2) } , where α is a root of f(x), i.e., f(α) = 0, implying: α3 = (1 + θ)α+ 1. Maximal Cyclic Subgroup of R∗: Since f(x) = x3 + (1 + θ)x+ 1 is basic irreducible and α is its root, the multiplicative group R∗ has a cyclic subgroup G7 of order 7 generated by γ = α2. Then: G7 = {γ, γ2, . . . , γ7 = 1} = {α2, α+ (1 + θ)α2, . . . , 1}. This is isomorphic to the residue field: K = F2[x] ⟨π(f(x))⟩ = F2[x] ⟨x3 + x+ 1⟩ . Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 11 of 22 Table 1: Multiplicative Group of Order 14 Exp. Polynomial Exp. Polynomial 1 α 8 α+ uα2 2 α2 9 α2 + uα+ u 3 (1 + u)α+ 1 10 α2u+ α+ 1 4 α+ (1 + u)α2 11 α2 + (1 + u)α+ u 5 α2 + α+ (1 + u) 12 1 + α+ (1 + u)α2 6 α2 + 1 13 α2 + u+ 1 7 αu+ 1 14 1 Generator Polynomial of BCH Code over A(2, 2): Since γ = α2 is a primitive cube root of unity in G7, and for design distance d = 3, we need the minimal polynomials of γi for i = 1, 2. Let m1(x) be the minimal polynomial of γ. Then γ, γ2, γ4 have the same minimal polynomial: m1(x) = (x− γ)(x− γ2)(x− γ4) = x3 + x+ 1. Hence, the generator polynomial is: g(x) = lcm{m1(x),m2(x)} = m1(x) = x3 + x+ 1. Given r = 3, q = 2, m = 3, and c = 1 (narrow-sense BCH code), the code length is: n = qm − 1 = 23 − 1 = 7. The cyclic code C generated by g(x) over G7 ⊂ R∗ ⊂ A(2, 2) has dimension 4. Thus, C is a: [7, 4, 3] primitive narrow-sense BCH code over the maximal cyclic subgroup of units in A(2, 2). Case 3: BCH Codes over A(2, 2) with Degree 4 Extension Let f(x) ∈ A(2, 2)[x] be a monic polynomial of degree 4: f(x) = a0 + a1x+ x4, where a0, a1 ∈ A(2, 2). Choose a0 = 1, a1 = 1 + θ, then: f(x) = x4 + (1 + θ)x+ 1. The projection map π : A(2, 2)[x] → F2[x] gives: π(f(x)) = x4 + x+ 1, which is a monic irreducible polynomial over F2. We now verify irreducibility over A(2, 2): Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 12 of 22 f(0) = 1 ̸= 0, f(1) = 1 + (1 + θ) + 1 = 1 + θ ̸= 0, f(θ) = θ4 + (1 + θ)θ + 1 = θ + θ + 1 = 1 ̸= 0, f(1 + θ) = (1 + θ)4 + (1 + θ)2 + 1 = 1 + θ ̸= 0. Thus, f(x) is a basic irreducible polynomial in A(2, 2)[x]. Extension of A(2, 2) with Respect to f(x): We define the Galois extension ring: R = A(2, 2)[x] ⟨f(x)⟩ = { 3∑ i=0 ciα i : ci ∈ A(2, 2) } , where f(x) = x4 + (1 + θ)x+ 1 and f(α) = 0, hence: α4 = (1 + θ)α+ 1. Maximal Cyclic Subgroup of R∗: The multiplicative group R∗ has a cyclic subgroup G15 of order 15 generated by γ = α2. This subgroup is isomorphic to the residue field: K = F2[x] ⟨π(f(x))⟩ = F2[x] ⟨x4 + x+ 1⟩ . Table 2: Cyclic Group of Order 30 Exp. Polynomial Exp. Polynomial 1 α 16 (1 + u)α 2 α2 17 (1 + u)α2 3 α3 18 (1 + u)α3 4 1 + (1 + uα) 19 (1 + u) + α 5 α+ (1 + uα2) 20 (1 + u)α+ α2 6 α2 + (1 + u)α3 21 (1 + u)α2 + α3 7 α3 + α+ 1 + u 22 1 + (1 + u)α+ (1 + u)α3 8 α2 + 1 23 (1 + u) + (1 + u)α2 9 α3 + α 24 (1 + u)α+ (1 + u)α3 10 α2 + (1 + u)α+ 1 25 (1 + u) + α+ (1 + u)α2 11 α3 + (1 + u)α2 + α 26 (1 + u)α+ α2 + (1 + u)α3 12 (1 + u)α3 + α2 + (1 + u)α+ 1 27 (1 + u) + α+ (1 + u)α2 + α3 13 α3 + (1 + u)α2 + (1 + u) 28 1 + α2 + (1 + u)α3 14 (1 + u)α3 + 1 29 (1 + u) + α3 15 1 + u 30 1 Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 13 of 22 Generator Polynomial of BCH Code over A(2, 2): Since γ = α2 is a primitive cube root of unity in G15, for design distance d = 3, we need the minimal polynomials of γi, i = 1, 2. Let m1(x) be the minimal polynomial of γ. Then γ, γ2, γ4, γ8 all share the same minimal polynomial: m1(x) = (x− γ)(x− γ2)(x− γ4)(x− γ8) = x4 + x+ 1. Therefore, the generator polynomial is: g(x) = lcm{m1(x),m2(x)} = m1(x) = x4 + x+ 1. Given r = 4, q = 2, m = 4, and c = 1, the code length is: n = qm − 1 = 24 − 1 = 15. The cyclic code C generated by g(x) over G15 ⊂ R∗ ⊂ A(2, 2) has dimension 11. Hence, C is a: [15, 11, 3] primitive narrow-sense BCH code over the maximal cyclic subgroup of units in A(2, 2). 4. Decoding of BCH Codes over QGRs and Their Significance [13, 14, 16] 4.1. Decoding of BCH Codes using Advanced Modified Berlekamp–Massey Algorithm Let C be a BCH code with length n, designed distance d, and received vector r. Let S denote the syndrome vector, computed as the multiplication of the parity-check matrix H and the transpose of the received vector: S = Hrt. Apply the Advanced Modified Berlekamp–Massey Algorithm (AMBMA) to find the error-locator polynomial δn(x) under the following initial conditions: d−1 = 1, δ−1(x) = 1, l−1 = 0, l0 = 0, δ0(x) = 1, d0 = first non-zero syndrome. Let dn represent the discrepancy at iteration n. If dn = 0 or is a zero divisor, then: δn(x) = δn+1(x), ln = ln+1. If dn is a nonzero unit element, choose m ≤ n− 1 such that n− lm is the largest or equal to the last column index satisfying the update condition. From dn − ydm = 0, compute y, and update the error-locator polynomial as: δn+1(x) = δn(x)− yxn−mδm(x). Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 14 of 22 The next discrepancy is given by: dn+1 = Sn+2 + δ (n+1) 1 Sn+1 + δ (n+1) 2 Sn + · · ·+ δ (n+1) ln+1 Sn+2−ln+1 . Next, compute the reciprocal polynomial g(x) of δn(x), and find its roots in the form xu. These roots correspond to the error locations. Choose elements zu such that (zu−xu) are either zero or zero divisors, where 1 ≤ u ≤ n− 1 and xu = αu. Now construct the error-locator polynomial using the elementary symmetric function: (x− z1)(x− z2) · · · (x− zv) = δ0x v + δ1x v−1 + · · ·+ δv, where z1, z2, . . . , zv represent the v error locations. Using Forney’s algorithm [13, 14, 16], compute the error magnitudes yj as: yj = v−1∑ l=0 δj,uSv−l v−1∑ l=0 δj,uz v−l j , where the recurrence relation for δj,u is: δ0 = δj,0 = 1, δj,u = δu + xj · δj,u−1, u = 1, 2, . . . , v − 1, j = 1, 2, . . . , v. The final error vector e is obtained, and the corrected codeword is: c = r − e. The pseudocode for the decoding of BCH codes over the Quasi-Galois ring using AMBMA is given in Algorithm 2. Example 4.1 Let (7, 4, 3) be a BCH code over the QGRs A(2, 2) as described in Section 4, and let the received vector be r = (0, 0, 0, . . . , α2u+ α+ 1)1×7. Compute the syndrome vector S as: S = HrT = ( 1 γ γ2 · · · γ6 1 γ2 γ4 · · · γ12 )  0 0 0 ... α2u+ α+ 1  = ( γ11 γ12 ) = ( γ4 γ3 ) = ( α+ uα2 α2 + 1 ) = ( S1 S2 ) . Apply the AMBMA to determine δn(x) using the iteration steps shown in Table 3. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 15 of 22 Table 3: AMBMA for finding the linear polynomial Iterations δn(x) dn ln n − ln −1 1 1 0 −1 0 1 α+ uα2 0 0 1 1 + γ4x 1 1 0 2 1 + γ6x Thus, it follows that: δ2(x) = 1 + γ6x, and the reciprocal function is g(x) = γ6 + x. The root of g(x) is γ6, which indicates the error is located at position 7 in the received Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 16 of 22 vector r. The error-locator polynomial δ0x v + δ1 = x− γ6 is a symmetric function for v = 1. Compute the error magnitude using Forney’s formula, y1 = δ1,0S1 δ1,0x1 = S1 z1 = γ4 γ6 = γ5 = α2u+ α+ 1, where δ0 = 1, δ1 = −α2u− α− 1, and v = 1. Therefore, the error vector is: e = (0, 0, 0, . . . , α2u+ α+ 1), and the corrected codeword is: c = r − e = (0, 0, 0, . . . , 0). Example 4.2 Let (15, 11, 3) be a BCH code over the QGRs A(2, 2) as described in Section 4, and let the received vector be r = (0, (1 + u)α, 0, 0, . . . , 0)1×15. Compute the syndrome vector S as: S = HrT = ( 1 γ γ2 · · · γ14 1 γ2 γ4 · · · γ28 )  0 (1 + u)α 0 ... 0  = ( (1 + u)α3 (1 + u)α+ α2 ) = ( S1 S2 ) = ( γ9 γ10 ) . Apply the AMBMA to determine δn(x) using the iteration steps shown in Table 4. Table 4: AMBMA for finding the polynomial Iterations δn(x) dn ln n − ln −1 1 1 0 −1 0 1 (1 + u)α3 0 0 1 1 + γ9x (1 + u)α+ (1 + u)α3 1 0 2 1 + γx Hence, we find: δ2(x) = 1 + γx, and the reciprocal function is g(x) = γ + x. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 17 of 22 The root of g(x) is γ, indicating that the error occurred at a position 2 in the received vector r. The symmetric form of the error-locator polynomial is: δ0x v + δ1 = x− γ, where v = 1. Using Forney’s formula, compute the error magnitude y1 = δ1,0S1 δ1,0x1 = S1 z1 = γ9 γ = γ8 = α16 = (1 + u)α, where δ0 = 1, δ1 = −(1 + u)α, and v = 1. Therefore, the error vector is e = (0, (1 + u)α, 0, 0, . . . , 0), and the corrected codeword is: c = r − e = (0, 0, 0, . . . , 0). 4.2. Significance of Quasi-Galois Ring-Based BCH Code Construction and Decoding in Modern Data Transmission The use of Quasi-Galois Rings (QGRs) in the construction and decoding of BCH codes marks a significant advancement in the field of error-correcting codes, particularly for modern data transmission applications. QGRs extend the algebraic framework of classical Galois rings, enabling richer structural properties that lead to greater design flexibility. This allows for BCH codes with improved minimum distances, larger codeword sets, and higher code rates—key features for ensuring data integrity in bandwidth-intensive and noise-prone communication environments such as satellite systems, mobile networks, and high-density data storage devices. Moreover, the decoding strategies developed for BCH codes over QGRs enhance the reliability and speed of error correction. These methods exploit the algebraic depth of QGRs to support efficient syndrome computation and error location, making them suitable for real-time and low-latency systems. As data commu- nication continues to evolve, especially with emerging technologies requiring robust and scalable coding schemes, QGR-based BCH codes offer a future-ready solution by balancing strong theoretical foundations with practical performance benefits. 5. Comparison and Discussion This section provides a comparative analysis of BCH codes over Galois rings [3] and Quasi-Galois Rings (QGRs). Let the parameters be as follows: code length n, designed distance d, dimensions k1 and k2, code rates R1 and R2, and the number of codewords qk for BCH codes defined over Galois rings GR(pn, s) and Quasi-Galois rings A(ps, n). Comparative results using the examples GR(2, 2) and A(2, 2) are summarized in Tables 5 Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 18 of 22 Table 5: Code analysis of BCH codes over Galois rings GR(2, 2) Characteristic n d k 2k R1 2 3 3 1 21 0.3333 Table 6: Code analysis of BCH codes over Quasi-Galois rings A(2, 2) Characteristic n d k 2k R2 2 15 3 11 244 0.7333 2 15 5 7 228 0.4647 2 15 7 5 220 0.3333 2 15 9 1 24 0.0667 2 15 11 1 24 0.0667 2 15 13 1 24 0.0667 Figure 1: Designed distances vs. Code rates of BCH codes over GR(2, 2) and A(2, 2) and 6. Furthermore, graphical comparisons of the designed distances, code rates, and dimensions are shown in Figures 1 and 2. From the analysis above, we observe the following: • In the case of the Galois ring GR(2, 2), only one possibility exists for constructing BCH codes, whereas six distinct possibilities are available for A(2, 2). • The maximum code length achievable over GR(2, 2) is 3, while for A(2, 2) it is 15. • Designed distance values for BCH codes over A(2, 2) range from 3 to 13, whereas for GR(2, 2) it is only 3. This indicates that BCH codes over Quasi-Galois rings offer superior error-correction capabilities. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 19 of 22 Figure 2: Designed distances vs. Dimensions of BCH codes over GR(2, 2) and A(2, 2) • Furthermore, the number of codewords over A(2, 2) is significantly larger than that of BCH codes over GR(2, 2), enhancing their usefulness in applications requiring higher data capacity and reliability. 6. Conclusions and Future Directions The study of BCH codes over Quasi-Galois Rings (QGRs), as presented in this ar- ticle, demonstrates their strong potential in enhancing modern communication systems. Through a comprehensive analysis of the underlying theoretical foundations, construction methodologies, and decoding algorithms, it is evident that the extended algebraic structure and intrinsic properties of QGRs contribute significantly to improving both code rates and the total number of codewords. These enhanced attributes translate into superior error correction capabilities, making QGR-based BCH codes promising candidates for contem- porary and next-generation communication technologies. The ability to construct longer codes with greater flexibility and efficiency over QGRs, as compared to classical Galois rings, further substantiates their relevance. Future research can be directed toward refining the construction and decoding pro- cesses of BCH codes over QGRs, with a focus on optimizing algorithmic complexity and implementation feasibility. Additionally, exploring the practical deployment of QGR-based codes in real-world communication systems could provide valuable insights into their per- formance under practical constraints. Another promising direction involves the integration of BCH codes over QGRs with higher layers of coding and modern cryptographic protocols. Such hybrid approaches may lead to more robust and secure communication architectures. Therefore, building upon the findings of this article, the continued development of BCH codes over Quasi-Galois Rings can significantly contribute to the advancement of reliable and efficient communication networks. Muhammad Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6414 20 of 22 Acknowledgements This research is supported by Universidad Pedagógica y Tecnológica de Colombia (SGI 3725) and Minciencias (Conv. 934). Data availability All the data is given in this study. Tribute We would like to express our heartfelt gratitude to our beloved supervisor, Professor Dr. Tariq Shah (late), whose exceptional guidance, profound expertise, and steadfast support were instrumental in shaping our academic journey. His mentorship not only nurtured our growth as researchers in algebra, number theory, coding theory, and cryptography but also profoundly influenced our personal and professional development. His legacy of wisdom, integrity, and inspiration continues to guide us. May his soul rest in eternal peace. Figure 3: Prof. Dr. Tariq Shah References [1] E. F. Jr. Assmus and H. F. Mattson. Error-correcting codes: An axiomatic approach. Information and Control, 6(4):315–330, 1963. Muhammad Sajjad et al. / Eur. J. Pure Appl. 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