EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6275 ISSN 1307-5543 – ejpam.com Published by New York Business Global Design and Analysis of Shortened Bose-Chaudhuri-Hocquenghem Codes for Efficient Data Transmission over the Eisenstein Fields Muhammad Sajjad1,∗, Rida Asghar2, Mushtaq K. Abdalrahem3, Adnan Burhan Rajab4, Mahesha Narayana5, Moin-ud-Din Junjua6,∗, Salman A. AlQahtani7 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 College of Pharmacy, University of Al-Ameed 4 Department of Computer Engineering, College of Engineering, Knowledge University, Erbil 44001, Iraq 5 Department of Mathematics, The University of the West Indies, Kingston 7, Jamaica 6 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China 7 New Emerging Technologies and 5G Network and Beyond Research Chair, Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia Abstract. This article focuses on the constructing and decoding of Shortened Bose-Chaudhuri- Hocquenghem (BCH) codes over the Eisenstein fields, based on the Berlekamp-Massey Algorithm (BMA) with improved decoding performance. Thus, Eisenstein fields, being a natural generaliza- tion of Gaussian fields, form a well-devised algebraic structure for error-correcting codes and have better parameters for transmission of information in noisy channels. The work starts with the defi- nition of shortened BCH codes over the Eisenstein fields with the mention of generator polynomials and the study of certain algebraic characteristics. These codes are designed to provide the best balance between the word’s length and the ability to cope with error detection and correction in a variety of high-efficiency applications. The modified BMA is presented as a decoding mechanism that is equipped for the translation of Eisenstein field arithmetic, which is different in nature. The modification concerns an improved approach to the residual terms defined by the classes of higher level, which leads to the efficiency of convergence and the reduction of numerical complexity in contrast to the BMA. The imitations show that the proposed codes offer a higher error-correcting capability and faster computation compared to the traditional BCH codes over finite fields when reliability and low latency are desired. 2020 Mathematics Subject Classifications: 94B75, 11T71, 94A24, 68P30, 14G50, 94A05 Key Words and Phrases: Eisenstein field, Shortened BCH codes, Modified BMA ∗Corresponding author. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6275 Email addresses: muhammad.sajjad@nutech.edu.pk (M. Sajjad), moinuddin@zjnu.edu.cn (M. Junjua) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 2 of 36 1. Introduction Nowadays transmission of data is necessary, and error-correcting hyphenate codes are critically important for realizing reliable data transmission via noisy channels. Of them, BCH codes are being widely studied because of their algebraic nature and superlative error correction capability. Originally built over finite fields, BCH codes are widely used in storage equipment, wireless communication, and other data-applying fields. However, as the requirement for higher reliability and system efficiency increases in the next generation of communication systems, researchers are considering different algebraic contexts that will help improve these codes. Such field-based approaches are also possible, and Eisenstein fields appear to be logical extensions of Gaussian fields. Several algebraic characteristics of Eisenstein fields, including unique factorization and extended residue classes, can be critically exploited for efficient codes as well as being resilient to errors. Nonetheless, the use of the Eisenstein fields in coding theory has not been fully researched until now, especially in the construction and decoding of shortened BCH codes. This is especially due to the fact that shortened codes are renowned for their capability to sustain relatively short codes with good error correction capabilities. Shortened codes are well suited for any application that requires concise and efficient communication code. Other components of error-correcting codes as a standard are decoding algorithms. In a number of situations, the Berlekamp-Massey Algorithm (BMA) has been a standard tool in the decoding of BCH codes. Nevertheless, its use in Eisenstein fields may be problematic due to the fact that the residue classes are extended and the computations are more cumbersome. Overcoming these challenges calls for propositions on the classical BMA for faster and more accurate decoding for codes over Eisenstein fields [1–6]. Among the coding techniques, the error-correcting code, especially the shortened BCH code of a given order, has been an important topic in coding theory because of its random error-correcting capability and use in high-reliability communication systems. The theo- retical characteristics of shortened cyclic codes and their uses, especially for burst-error correction, were investigated by Kasami and Hsu. Further, they pointed out that, while optimizing code length, it is necessary to balance the amount of correction capacity [7, 8]. Helgert and Stinaff extended the construction and decoding of shortened BCH codes and the fact that these codes are well suited to any narrow bandwidth systems [9]. Tradition- ally linear codes have been discussed in numerous algebraic contexts, for example, BCH codes. Cyclotomic linear codes of order three were studied by Ding and Niederreiter, and cyclic codes with certain specified weight distributions were investigated by Ding et al. [10, 11], because such codes have been deemed useful in application, particularly in efficient error correction. Shah et al. introduced codes and decoding using generalized polynomials and sequences in [12, 13]. Linear codes from 2-designs have also been designed by Ding, who offered a new way to define codes with performance characteristics that would be suitable for various practical uses [14]. An important part of the development of coding theory demands the consideration of shortened and punctured codes. Goldwasser studied the relation between code shortening and MacWilliams identities by giving significant in- formation about the structure of these codes [15]. Yardi and Pellikaan also investigated M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 3 of 36 the behaviour of shortened cyclic codes in more detail regarding that being useful for solv- ing error correction for a constraint [16]. Gaussian fields are extended to Eisenstein fields, which have attracted attention as a sound algebraic structure for implementing richer error-correcting codes. Huber extended codes over Eisenstein-Jacobi integers, which laid down the groundwork for approaching these fields in coding theory [17]. Later on, Sajjad et al. did more advanced work on Eisenstein and Gaussian fields using cryptology and error correction codes, showing more efficiency in noisy channel communication systems [18, 19]. The Berlekamp-Massey Algorithm (BMA) has played an important role in decoding schemes of linear and cyclic codes, BCH codes. Appropriate overviews of the overall algo- rithm and explanations of its uses were given by Lin and Huffman and Pless for decoding finite geometry and cyclic codes [20, 21]. Nonetheless, the computational complexity of the classical BMA remains fairly high, and innovative modified algorithms have been adapted to solve concrete algebraic structures. The adoption of such modifications, mainly for Eisenstein fields, is a useful step forward in terms of simplification of decoding and in- creasing the convergence rate [19]. Other important works have brought to the knowledge of issues on the algebraic structures for the coding theory. The combinatorial characteris- tics of linear codes were examined using t-designs by Assmus and Mattson, while t-covers of the finite projective spaces were studied by Beutelspacher, which also focused on the geometry [20, 22]. These concepts have been further generalized to mixed binary/ternary codes by Brower et al. [16] and to codes obtained from Boolean functions by Tang et al. [23], thus proving the versatility of algebraic methods in terms of reliable code construc- tion. This paper extends the literature presented above by developing abbreviated BCH codes over Eisenstein fields and presenting an adapted BMA appropriate to the arithmetic of the shortened BCH codes. This work fills the gap of existing Eisenstein field theoretical developments and applies them to error correction and amplification of several features, promoting the reliability and performance of communication systems. Thus, this thesis supplements prior research and helps further develop the theory of coding for the next generations by utilizing the algebraic characteristics of Eisenstein fields and enhancing the decoding approach [7, 9, 17–19, 24]. Shortened Bose-Chaudhuri-Hocquenghem (BCH) codes have been found to be very use- ful in correcting errors, thereby facilitating smooth data communication in noisy media. Their flexibility and speed are ideal for the current data transmission networks where the probability of error and computational time are critical concerns [9, 24]. While the classical BCH codes over finite fields are highly efficient, they seem to have drawbacks that cropped up in some high-reliability applications, such as the trade-off between the code length and error-correcting capability [7, 8]. The proposed concept of Eisenstein fields as an algebraic structure is prospective to open a new direction toward research on the construction of higher coding schemes. Eisenstein fields are a natural generalization of Gaussian fields and offer a higher level of structural complexity, which can contribute to the improvement of the error-correcting code. A number of previous works, including Huber and Sajjad et al., have shown that Eisenstein and Gaussian fields could be applied in cryptographic M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 4 of 36 applications and coding theory; hence, more investigation needs to be made on the use of Eisenstein and Gaussian fields in the construction of BCH code [17, 18]. Moreover, current improvements in decoding techniques like the Berlekamp-Massey Algorithm (BMA) offer potentialities to make moderate decoding procedures and lessen the computational intri- cacy required to defend contemporary communication systems [19, 20]. This research is motivated by developing error-correcting codes with enhanced performance measures and channel decoding techniques using Eisenstein field advantages. The proposed modification of BMA and integration with these three fields is designed to contribute to the further development of coding theory and real communication applications. Despite significant advancements in coding theory, several research gaps persist in the development of shortened BCH codes and their decoding techniques. Although the fields of Eisenstein have been discussed in the sphere of cryptography and the theory of codes, the application of such fields to construct shortened BCH codes has not been thoroughly investigated. In finite and Gaussian fields, most of the previous studies have been per- formed on them [17–19]. The classical BMA has been studied in detail in literature, but its complexity is high and thus becomes a problem when one needs to use it in the real- time application. Extensions of the above-described BMA, especially those that take into account arithmetic properties of Eisenstein fields, are few and far between in the literature [19, 21, 25]. Previous research works that analyze the shortened BCH codes mostly focus on either the length reduction aspect or the error-correcting capability dimension without providing a balance between the two factors. Such balance is envisaged for applications that demand both high reliability and low latency [7–9]. Closing these gaps, this research is aimed at developing shortened BCH codes over the Eisenstein fields, deriving the gen- erator polynomials for them, and presenting the modified BMA for decoding. Therefore, this study seeks to solve the problem of developing an understanding of the trade-offs and performance measures to close the gap that exists between theoretical enhancements and application in coding theory. This work presents new approaches toward the construction and decoding of error- correcting codes using shortened BCH codes over the Eisenstein fields. The study con- structs these codes from a fresh perspective with reference to the properties of Eisenstein fields as an augmentation of Gaussian fields to attain optimal generator polynomials and algebraic architecture. The obtained codes strike a perfect balance between possible code length and the code’s ability to correct errors, making it perfect for applications where reliability is paramount and latency is low. An original approach proposed in this work is the modified Berlekamp-Massey Algorithm (BMA) suitable for the Eisenstein fields. The approach outlined in this modification is capable of utilizing more complex methods to deal with wider residue classes in order to offer better convergence and lower computational demands than needed for the classical BMA. In the analysis of performance, the work reveals that the proposed codes exhibit higher error correction and better computational complexity compared to conventional BCH over finite fields. In addition, this research expands on the cyclic subgroups of fields of Eisenstein for coding theory, as it now in- corporates settings outside of cryptographic applications, thereby finding its usefulness in next-generation communication systems. These contributions are firm to fill the existing M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 5 of 36 gaps in coding theory and also plan a strong base for many more inventions in the field. 2. Eisenstein Field From [26], assume ω being the cubic root of unity correspondingly 1 + ω + ω2 = 0. Assume that Z[ω] = {a+bω : a, b ∈ Z} be the Euclidean domain of the Eisenstein integers. Consequently, it Zp[ω] = {a+bω : a, b ∈ Zp} is a commutative ring with identity and Zp[ω] is an Eisenstein field (EF) if p ≡ 2 (mod 3). Illustration 2.1: From [24, 26], let Z2[ω] = {0, 1, ω, 1+ω} be the EF, as every nonzero element of Z2[ω] is a unit element and the cardinality of Z2[ω] is 2 2 = 4. Remark 2.1 [26]: Zp[ω] has p 2 elements if p ≡ 2 (mod 3). 2.1. Eisenstein Field Extension From [26], assume Zp[ω] with p ≡ 2 (mod 3) is an EF, then Zp[ω][x] is a Euclidean domain. For the extension of EF, Zp[ω] m, we have: Zp[ω] m = Zp[ω][x] ⟨H(x)⟩ ∼= GF(p2m), where ⟨H(x)⟩ is the maximal ideal generated by an irreducible polynomial H(x) of degree k in Zp[ω][x]. Let β be the coset x+ ⟨H(x)⟩, so that H(β) = 0. Moreover, Zp[ω] m = { a0 + a1β + a2β 2 + · · ·+ am−1β m−1 : ∀a0, a1, . . . , am−1 ∈ Zp[ω] } . Zp[ω] m is an m-degree extension field of Zp[ω]. The multiplicative group Zp[ω] ∗m = Zp[ω] m \ {0} is a cyclic group (CG) of order p2m − 1. Remark 2.2 [26, Remark 3]: The cardinality of Zp[ω] m is p2m. Illustration 2.2: Assume that the quotient ring Z2[ω][x] ⟨x2 + ωx+ ω⟩ = {a0 + a1x : ∀a0, a1 ∈ Z2[ω]} = Z2[ω] 2, is generated by the primitive irreducible polynomial H(x) = x2 + ωx+ ω over Z2[ω]. Let α be a root of H(x) in the extension field Z2[ω][x], then H(α) = 0 as α2 + ωα + ω = 0, and Z2[ω] ∗2 = Z2[ω] 2 \ {0} is a cyclic group (CG) of order 22·2 − 1 = 15 as shown in Table 1. Now, for finding the total number of possible polynomials, the possible irreducible polynomials, and the number of primitive irreducible polynomials over the Eisenstein Field (EF) Z2[ω][x] ⟨f(x)⟩ for f(x) being a polynomial of degree 2, consider Table 2. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 6 of 36 Table 1: Cyclic Group over Eisenstein Integers of Order 15 S. No. Power of α’s S. No. Power of α’s 1 α 9 ω + 1 + αω 2 αω + ω 10 1 + ω 3 α+ ω + 1 11 α+ αω 4 α+ ω 12 1 + α 5 ω 13 ω + α(1 + ω) 6 αω 14 1 + α+ αω 7 α(1 + ω) + 1 + ω 15 1 8 αω + 1 Illustration 2.3: Let the quotient ring Z2[ω][x] ⟨x3 + x2 + x+ 1 + ω⟩ = { a0 + a1x+ a2x 2 | a0, a1, a2 ∈ Z2[ω] } = Z2[ω][x] 3, be generated by the irreducible primitive polynomial H(x) = x3+x2+x+1+ω over Z2[ω], and let α be the root of H(x) in the extension field Z2[ω][x]. Then H(α) = 0 implies α3 + α2 + α+ 1 + ω = 0 ⇒ α3 = α2 + α+ 1 + ω. Therefore, Z2[ω] ∗ 3 = Z2[ω] 3 \ {0} is a cyclic group of order 22·3 − 1 = 64 − 1 = 63, as illustrated in Table 3. Remark 2.4: The order of a subgroup of Eisenstein integers must divide the order of the corresponding group of Eisenstein integers. Illustration 2.4: If we require a subgroup of five elements over the extension field Z2[ω][x], then we know that we cannot directly obtain it from any polynomial f(x) of degree 2 over Z2[ω][x] ⟨f(x)⟩ . We obtain a cyclic group of order n = ( (2)2 )2 − 1 = 16− 1 = 15. The desired subgroup is obtained by dividing n by 5, i.e., 15 5 = 3. If α is the root of the extension field Z2[ω][x], then α3 is the generator of the required subgroup of Z2[ω][x] 2 for n = 5. Similarly, we can find a subgroup for n = 3. Illustration 2.5: If we want a subgroup of twenty-one elements over Z2[ω][x], we cannot directly obtain it from any polynomial f(x) of degree 3 over Z2[ω][x] ⟨f(x)⟩ . M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 7 of 36 Table 2: Details of Irreducible and Primitive Irreducible Polynomials Polynomials of degree two over Z2[ω] Irreducible Primitive Irreducible x2 × × x2 + 1 × × x2 + ω ✓ × x2 + 1 + ω ✓ × x2 + x × × x2 + ωx × × x2 + (1 + ω)x × × x2 + x+ 1 × × x2 + x+ ω ✓ ✓ x2 + x+ 1 + ω ✓ ✓ x2 + ωx+ 1 ✓ × x2 + ωx+ ω ✓ ✓ x2 + ωx+ 1 + ω × × x2 + (1 + ω)x+ 1 × × x2 + (1 + ω)x+ ω × × x2 + (1 + ω)x+ 1 + ω ✓ × Instead, we obtain a cyclic group of order n = ( (2)2 )3 − 1 = 64− 1 = 63. The desired subgroup is obtained by dividing n by 21, i.e., 63 21 = 3. If α is the root of the extension field Z2[ω][x], then α3 is the generator of the required subgroup of Z2[ω][x] for n = 21. Similarly, subgroups of orders 3, 7, and 9 can be found with generators α21, α9, and α7 respectively. 3. Shortened BCH Codes over the Eisenstein Field Shortened BCH codes over the Eisenstein fields involve taking a longer BCH code, perhaps over a bigger Eisenstein field, and shortening it to achieve specific design goals, such as minimizing complexity or meeting certain constraints. It involves nearly the same steps as shortened BCH codes over Eisenstein fields [13, 17, 26]. 3.1. Encoding of Shortened BCH Code over the Eisenstein Field First of all, obtain the shortened BCH code from the full length BCH code over the extension field Zp[ω], where p ≡ 2 (mod 3). Assume that the integers c, n, k, d > 0, such M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 8 of 36 Table 3: Cyclic Group over Eisenstein Integers of Order 63 S. No. Power of α’s S. No. Power of α’s 1 α 33 (α+ 1 + ωα)2 2 α2 34 (α(1 + ω) + 1 + (1 + ω)α)2 3 α2 + α+ 1 + ω 35 ω + ωα 4 1 + ω + ωα 36 (ωα+ ωα)2 5 α+ ωα+ α2ω 37 ωα+ 1 6 α2 + ωα+ 1 38 α+ ωα2 7 α2(1 + ω) + 1 + ω 39 (1 + ω)α2 + 1 + ωα 8 α2(1 + ω) + ω 40 α2 + ω + ωα 9 α+ α2(1 + ω) + ω 41 1 + ω + (1 + ω)α2 + (1 + ω)α 10 α+ ω + ωα2 42 ω 11 α2(1 + ω) + 1 43 ωα 12 (1 + ω)α2 + ω + ωα 44 ωα2 13 α+ ω + α2 45 ωα2 + 1 + ωα 14 1 + ω + (1 + ω)α 46 (1 + ω)α+ 1 15 (1 + ω)α2 + ωα 47 α+ (1 + ω)α2 16 ω + (1 + ω)α 48 ωα2 + ω + (1 + ω)α 17 (1 + ω)α2 + ωα 49 α2 + 1 18 α2 + ω + (1 + ω)α 50 α2 + 1 + ω 19 ωα2 + 1 + ω + (1 + ω)α 51 1 + ω + α2 + ωα 20 α+ α2 + 1 52 1 + ω + (1 + ω)α2 + ωα 21 1 + ω 53 α2 + ω 22 (1 + ω)α 54 α2 + 1 + ω + (1 + ω)α 23 (1 + ω)α2 55 1 + ω + ωα2 + ωα 24 (1 + ω)α2 + ω + (1 + ω)α 56 α+ 1 25 α+ ω 57 α+ α2 26 α2 + ωα 58 α+ 1 + ω 27 α+ 1 + ω + (1 + ω)α2 59 α2 + (1 + ω)α 28 ωα2 + ω 60 α+ ω + 1 + ωα2 29 ωα2 + 1 61 α+ (1 + ω)α2 + 1 30 ωα2 + (1 + ω)α+ 1 62 ω + ωα+ ωα2 31 α2 + (1 + ω)α+ 1 63 1 32 ωα2 + 1 + ω that k is a prime power, 2 ≤ d ≤ n−1, and gcd(n, k) = 1. Assume that there is a smallest positive integer b such that p2b ≡ 1 (mod n). Then, by Euler’s theorem, if p2φ(n) ≡ 1 (mod n), then b divides φ(n), where φ is the Euler phi function. Thus, n | (p2b − 1). Let β be an element of the extension field Zp[ω] b. Consider the minimal polynomials bi(x) ∈ Zp[ω][x] of β i. The least common multiple (lcm) of all distinct polynomials bi(x), for i = c, c+ 1, c+ 2, . . . , c+ d− 2, is known as the generator polynomial g(x), i.e., g(x) = lcm{bi(x) | i = c, c+ 1, c+ 2, . . . , c+ d− 2}. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 9 of 36 Since all minimal polynomials divide xn−1, the generator polynomial g(x) also divides xn − 1. Let C be the cyclic code generated by g(x) in the ring Zp[ω][x] m, then C is called a BCH code of length n over the extension field Zp[ω] with designed distance d. Now, if the code length is shorter than the full length BCH code, then it is known as a shortened BCH code. If the code is a narrow-sense shortened BCH code, then c = 1. Pseudocode of the encoding algorithm is given in Algorithm 3.1. Theorem 3.1 [26] : Assume α is an element of the extension field Zp[ω] m where p ≡ 2 (mod 3). Then, the elements αp0 , αp2 , αp4 , αp6 , . . . have minimal polynomials over the Eisenstein field Zp[ω]. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 10 of 36 Illustration 3.1: Construct a Shortened BCH code for a degree-two polynomial by taking n = 3 over the Eisenstein field Z2[ω] 2. Let us take a degree-two polynomial f(x) = x2 + ωx+ ω, which is primitive and irreducible over Z2[ω]. As proved in Section 2, the cardinality of the cyclic group generated by f(x) is 15, and since 15/3 = 5, our required cyclic subgroup is generated by ⟨β = α5⟩, where β is the root of f(x). The required cyclic subgroup is G∗ = {α5, α10, α15 = 1} = {ω, 1 + ω, 1}. Thus, there is only one case for the construction of Shortened BCH codes over Z2[ω] 2 with parameters n = 3 and d = 3. • For i = 1, let β ∈ Z2[ω] 2. Then, by Theorem 3.1, β has a minimal polynomial φ1(x) = (x− β) = x+ ω. • For i = 2, let β2 ∈ Z2[ω] 2. Then, by Theorem 3.1, β2 has a minimal polynomial by itself: φ2(x) = (x− β2) = x+ (1 + ω). Now, the generator polynomial is G(x) = φ1(x) · φ2(x) = (x+ ω)(x+ (1 + ω)) = x2 + x+ 1. Since k = 3−2 = 1, the parameters of the Shortened BCH code are (n, k, d) = (3, 1, 3). Illustration 3.2: Construct a Shortened BCH code for a degree-two polynomial by taking n = 5 over the Eisenstein field Z2[ω] 2. Let us take a degree-two polynomial f(x) = x2 + ωx+ ω, which is primitive and irreducible over Z2[ω]. As proved in Section 2, the cardinality of the cyclic group generated by f(x) is 15 and 15/5 = 3. So, our required cyclic subgroup is generated by ⟨β = α3⟩, where β is the root of f(x). The required cyclic subgroup is G∗ = {α3, α6, α9, α12, α15 = 1} = {1 + ω + α, αω, 1 + ω + αω, α+ 1, 1}. Thus, there are two cases for the construction of Shortened BCH codes over Z2[ω] 2. Case 1: For n = 5 and d = 3, i = 1, 2. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 11 of 36 • For i = 1, let β ∈ Z2[ω] 2. By Theorem 3.1, β and β4 have the same minimal polynomial: φ1(x) = (x− β)(x− β4) = x2 − (β + β4)x+ β5 = x2 + ωx+ 1. • For i = 2, let β2 ∈ Z2[ω] 2. By Theorem 3.1, β2 and β3 have the same minimal polynomial: φ2(x) = (x− β2)(x− β3) = x2 − (β2 + β3)x+ β5 = x2 + (1 + ω)x+ 1. The generator polynomial is G(x) = φ1(x) · φ2(x) = (x2 + ωx+ 1)(x2 + (1 + ω)x+ 1) = x4 + x3 + x2 + x+ 1. Since k = 5−4 = 1, the parameters of the Shortened BCH code are (n, k, d) = (5, 1, 3). Case 2: For n = 5 and d = 5, i = 1, 2, 3, 4. • For i = 1, as before, φ1(x) = x2 + ωx+ 1. • For i = 2, φ2(x) = x2 + (1 + ω)x+ 1. • For i = 3, by Theorem 3.1, β3 and β2 have the same minimal polynomial, so φ3(x) = φ2(x) = x2 + (1 + ω)x+ 1. • For i = 4, φ4(x) = φ1(x) = x2 + ωx+ 1. The generator polynomial is again G(x) = φ1(x) · φ2(x) = (x2 + ωx+ 1)(x2 + (1 + ω)x+ 1) = x4 + x3 + x2 + x+ 1. Since k = 5−4 = 1, the parameters of this Shortened BCH code are (n, k, d) = (5, 1, 5). Illustration 3.3: Construct a Shortened BCH code for a degree-three polynomial by taking n = 3 over the Eisenstein field Z2[ω] 3. Let us take a degree-three polynomial f(x) = x3 + x2 + x+ 1 + ω, which is primitive and irreducible over Z2[ω]. As proved in Section 2, the cardinality of the cyclic group generated by f(x) is 63, and 63 3 = 21. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 12 of 36 So, our required cyclic subgroup is generated by ⟨β = α21⟩, where β is the root of f(x). The required cyclic subgroup is G∗ = {α21, α42, α63 = 1} = {1 + ω, ω, 1}. Hence, there is only one case for the construction of Shortened BCH codes over the Eisenstein field Z2[ω] 3. For n = 3 and d = 3, i = 1, 2: • For i = 1, let β ∈ Z2[ω] 3. By Theorem 3.1, β has minimal polynomial itself: φ1(x) = (x− β) = x+ (1 + ω). • For i = 2, let β2 ∈ Z2[ω] 3. By Theorem 3.1, β2 has minimal polynomial itself: φ2(x) = (x− β2) = x+ ω. The generator polynomial is G(x) = φ1(x) · φ2(x) = (x+ ω)(x+ (1 + ω)) = x2 + x+ 1. Since k = 3−2 = 1, the parameters of this Shortened BCH code are (n, k, d) = (3, 1, 3). Illustration 3.4: Construct a Shortened BCH code for three-degree polynomial by taking n = 9 over the EF Z2[ω] 3. Let us take f(x) = x3 + x2 + x+ 1 + ω, which is primitive and irreducible over Z2[ω]. As we proved in Section 2, the cardinality of the cyclic group of f(x) is 63 and 63 9 = 7. So, our required cyclic subgroup is generated by ⟨β = α7⟩ by taking β to be the root of f(x). Then cyclic subgroup is G8 = {β, β2, β3, β4, β5, β6, β7, β8, β9 = 1} = {α7, α14, α21, α28, α35, α42, α49, α56, α63 = 1} = {α2(1 + ω) + 1 + ω, 1 + ω + (1 + ω)α, 1 + ω, ωα2 + ω, ω + ωα, ω, α2 + 1, α+ 1, 1}. So, there are four cases for the construction of Shortened BCH codes over the EF Z2[ω] 3. Case 1: For n = 9 and d = 3, i = 1, 2. • For i = 1, let β ∈ Z2[ω] 3. By Theorem 3.1, β, β4, β7 have the minimal polynomial φ1(x) = (x− β)(x− β4)(x− β7) = (x2 + (β + β4)x+ β5)(x− β7) = (x2 + (β + β4)x+ β5)(x+ β7) = x3 + (β + β4 + β7)x2 + (β2 + β5 + β8)x+ β3 = x3 + 0 · x2 + 0 · x+ β3 = x3 + 1 + ω. • For i = 2, let β2 ∈ Z2[ω] 3. By Theorem 3.1, β2, β5, β8 have the minimal polynomial φ2(x) = (x− β2)(x− β5)(x− β8) = (x2 + (β2 + β5)x+ β7)(x+ β8) = x3 + (β2 + β5 + β8)x2 + (β + β4 + β7)x+ β6 = x3 + 0 · x2 + 0 · x+ β6 = x3 + ω. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 13 of 36 Now the generator polynomial is G(x) = φ1(x) · φ2(x) = (x3 + ω)(x3 + 1 + ω) = x6 + ωx3 + (1 + ω)x3 + 1 = x6 + x3 + 1. Hence, k = 9− 6 = 3. So the shortened BCH code parameters are (n, k, d) = (9, 3, 3). Case 2: For n = 9 and d = 5, i = 1, 2, 3, 4. • For i = 1, φ1(x) = x3 + 1 + ω (as in Case 1). • For i = 2, φ2(x) = x3 + ω (as in Case 1). • For i = 3, let β3 ∈ Z2[ω] 3 then by Theorem 3.1, φ3(x) = (x− β3) = (x+ 1 + ω). • For i = 4, φ4(x) = φ1(x) = x3 + 1 + ω. The generator polynomial is G(x) = lcm(φ1(x), φ2(x), φ3(x), φ4(x)) = (x3 + ω)(x3 + 1 + ω)(x+ 1 + ω) = (x6 + ωx3 + (1 + ω)x3 + 1)(x+ 1 + ω) = x7 + x6(ω + 1) + x4 + x3(1 + ω) + x+ 1 + ω. Hence, k = 9− 7 = 2. So the shortened BCH code parameters are (9, 2, 5). Case 3: For n = 9 and d = 7, i = 1, 2, 3, 4, 5, 6. • For i = 1, φ1(x) = x3 + 1 + ω. • For i = 2, φ2(x) = x3 + ω. • For i = 3, φ3(x) = x+ 1 + ω. • For i = 4, φ4(x) = φ1(x). • For i = 5, φ5(x) = φ2(x). • For i = 6, let β6 ∈ Z2[ω] 3, then by Theorem 3.1, φ6(x) = (x− β6) = (x+ ω). The generator polynomial is G(x) = lcm(φ1(x), φ2(x), φ3(x), φ4(x), φ5(x), φ6(x)) = (x3+ω)(x3+1+ω)(x+1+ω)(x+ω) = (x6 + ωx3 + (1 + ω)x3 + 1)(x+ 1 + ω)(x+ ω) = (x6 + x3 + 1)(x+ 1 + ω)(x+ ω) = (x7 + x6(ω + 1) + x4 + x3(1 + ω) + x+ 1 + ω)(x+ ω) = x8 + x7 + x6 + x5 + x4 + x3 + x2 + x+ 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 14 of 36 Hence, k = 9− 8 = 1. So the shortened BCH code parameters are (9, 1, 7). Illustration 3.5: Construct a Shortened BCH code for a three-degree polynomial by taking n = 21 over the Eisenstein field Z2[ω] 3. Let us take f(x) = (x3 + x)2 + x+ 1 + ω which is primitive and irreducible over Z2[ω]. As shown in Section 2, the cardinality of the cyclic group generated by f(x) is 63. Since 63 21 = 3, the required cyclic subgroup is generated by ⟨β = α3⟩, where β is a root of f(x). Then the cyclic subgroup is G∗ = {β, β2, . . . , β21 = 1} = {α3, α6, α9, . . . , α63}. Thus, explicitly: G∗ = {α3, α6, α9, α12, α15, α18, α21, α24, α27, α30, α33, α36, α39, α42, α45, α48, α51, α54, α57, α60, α63 = 1}. So, there are ten cases for the construction of Shortened BCH codes over the EF Z2[ω] 3. Case 1: For n = 21, d = 3, i = 1, 2. For i = 1, let β ∈ Z2[ω] 3. By Theorem 3.1, the minimal polynomial of β, β4, β16 is φ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. For i = 2, the minimal polynomial of β2, β8, β11 is φ2(x) = (x−β2)(x−β8)(x−β11) = x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21 = x3+ωx2+1. Generator polynomial: G(x) = φ1(x)φ2(x) = (x3 + (1 + ω)x2 + 1)(x3 + ωx2 + 1) = x6 + x5 + x4 + x2 + x+ 1. Therefore, k = 21− 6 = 15, and the BCH code is (n, k, d) = (21, 15, 3). Case 2: For n = 21, d = 5, i = 1, 2, 3, 4. The minimal polynomials are: φ1(x) = x3 + (1 + ω)x2 + 1, φ2(x) = x3 + ωx2 + 1, φ3(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1, φ4(x) = φ2(x). Generator polynomial: G(x) = φ1(x)φ2(x)φ3(x) = (x6 + x5 + x4 + x2 + x+ 1)(x3 + x+ 1) = x9 + x8 + x5 + 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 15 of 36 Therefore, k = 21− 9 = 12, and the BCH code is (n, k, d) = (21, 12, 5). Case 3: For n = 21, d = 7, i = 1, 2, 3, 4, 5, 6. The minimal polynomials are: φ1(x) = x3 + (1 + ω)x2 + 1, φ2(x) = x3 + ωx2 + 1, φ3(x) = x3 + x+ 1, φ4(x) = φ2(x), φ5(x) = (x− β5)(x− β17)(x− β20) = x3 + (1 + ω)x+ 1, φ6(x) = φ3(x). Generator polynomial: G(x) = φ1(x)φ2(x)φ3(x)φ5(x). [Polynomial multiplication can be performed explicitly if required.] Therefore, k = 21− (degG(x)), and the BCH code is (n, k, d) = (21, 9, 7). Case 4: For n = 21 and d = 9, i = 1, 2, 3, 4, 5, 6, 7, 8. • For i = 1, Let β ∈ Z2[ω] 3, then by Theorem 3.1, β, β4, β16 has the minimal polynomial φ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. • For i = 2, Let β2 ∈ Z2[ω] 3, then φ2(x) = (x−β2)(x−β8)(x−β11) = x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21 = x3+ωx2+1. • For i = 3, Let β3 ∈ Z2[ω] 3, then φ3(x) = (x−β3)(x−β6)(x−β12) = x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21 = x3+x+1. • For i = 4, Let β4 ∈ Z2[ω] 3, then φ4(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = φ2(x). • For i = 5, Let β5 ∈ Z2[ω] 3, then φ5(x) = (x−β5)(x−β17)(x−β20) = x3+(β5+β17+β20)x2+(β+β4+β16)x+β21 = x3+(1+ω)x+1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 16 of 36 • For i = 6, Let β6 ∈ Z2[ω] 3, then φ6(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = φ3(x). • For i = 7, Let β7 ∈ Z2[ω] 3, then φ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3, then φ8(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = φ2(x). Now Generator polynomial is G(x) = φ1(x)φ2(x)φ3(x)φ5(x)φ7(x) = (x12 + x11 + (1 + ω)x10 + ωx9 + (1 + ω)x6 + x5 + x3 + (1 + ω)x+ 1)(x+ 1 + ω) = x13+ωx12+x9+(1+ω)x7+(1+ω)x6+(1+ω)x5+x4+(1+ω)x3+(1+ω)x2+(1+ω)x+1+ω. Now k = 21− 13 = 8. So (n, k, d) = (21, 8, 9) is a shortened BCH code. Case 5: For n = 21 and d = 11, i = 1, 2, 3, . . . , 10. • For i = 1, Let β ∈ Z2[ω] 3, then by Theorem 3.1, β, β4 and β16 have the minimal polynomial ϕ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. • For i = 2, Let β2 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial ϕ2(x) = (x−β2)(x−β8)(x−β11) = x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21 = x3+ωx2+1. • For i = 3, Let β3 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial ϕ3(x) = (x−β3)(x−β6)(x−β12) = x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21 = x3+x+1. • For i = 4, Let β4 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial ϕ4(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = ϕ2(x). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 17 of 36 • For i = 5, Let β5 ∈ Z2[ω] 3, then by Theorem 3.1 β5, β17, and β20 have the minimal polynomial ϕ5(x) = (x−β5)(x−β17)(x−β20) = x3+(β5+β17+β20)x2+(β+β4+β16)x+β21 = x3+(1+ω)x+1. • For i = 6, Let β6 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial ϕ6(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = ϕ3(x). • For i = 7, Let β7 ∈ Z2[ω] 3, then by Theorem 3.1, β7 has the minimal polynomial ϕ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial ϕ8(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = ϕ2(x). • For i = 9, Let β9 ∈ Z2[ω] 3, then by Theorem 3.1, β9, β18, and β15 have the minimal polynomial ϕ9(x) = (x−β9)(x−β18)(x−β15) = x3+(β9+β15+β18)x2+(β3+β6+β12)x+β21 = x3+x2+1. • For i = 10, Let β10 ∈ Z2[ω] 3, then by Theorem 3.1, β10, β13, and β19 have the minimal polyno- mial ϕ10(x) = (x−β10)(x−β13)(x−β19) = x3+(β10+β13+β19)x2+(β2+β8+β11)x+β21 = x3+ωx+1. Now the generator polynomial is G(x) = ϕ1(x)ϕ2(x)ϕ3(x)ϕ5(x)ϕ7(x)ϕ9(x)ϕ10(x) = x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω. Now k = 21− 19 = 2. So the parameters are: (n, k, d) = (21, 2, 11) Shortened BCH code. Case 6: For n = 21 and d = 13, i = 1, 2, . . . , 12. • For i = 1, Let β ∈ Z2[ω] 3, then by Theorem 3.1, β, β4, and β16 have the minimal polynomial φ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 18 of 36 • For i = 2, Let β2 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ2(x) = (x−β2)(x−β8)(x−β11) = x3+(β2+β8+β11)x2+(β10+β13+β19)x+β21 = x3+ωx2+1. • For i = 3, Let β3 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial φ3(x) = (x−β3)(x−β6)(x−β12) = x3+(β3+β6+β12)x2+(β9+β18+β15)x+β21 = x3+x+1. • For i = 4, Let β4 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ4(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = φ2(x). • For i = 5, Let β5 ∈ Z2[ω] 3, then by Theorem 3.1, β5, β17, and β20 have the minimal polynomial φ5(x) = (x−β5)(x−β17)(x−β20) = x3+(β5+β17+β20)x2+(β+β4+β16)x+β21 = x3+(1+ω)x+1. • For i = 6, Let β6 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial φ6(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = φ3(x). • For i = 7, Let β7 ∈ Z2[ω] 3, then by Theorem 3.1, β7 has the minimal polynomial φ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ8(x) = x3 + ωx2 + 1 = φ2(x). • For i = 9, Let β9 ∈ Z2[ω] 3, then by Theorem 3.1, β9, β18, and β15 have the minimal polynomial φ9(x) = x3 + x2 + 1. • For i = 10, Let β10 ∈ Z2[ω] 3, then by Theorem 3.1, β10, β13, and β19 have the minimal polyno- mial φ10(x) = x3 + ωx+ 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 19 of 36 • For i = 11, Let β11 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ11(x) = x3 + ωx2 + 1 = φ2(x). • For i = 12, Let β12 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial φ12(x) = x3 + x+ 1 = φ3(x). Now, the generator polynomial is G(x) = φ1(x)φ2(x)φ2(x)φ5(x)φ7(x)φ9(x)φ10(x) = x19+(1+ω)x18+x16+(1+ω)x15+(1+ω)x14+(1+ω)x13+x8+ωx7+(1+ω)x6+x5+(1+ω)x4+x2+ωx+1+ω. Hence, k = 21− 19 = 2. So the parameters of the shortened BCH code are: (n, k, d) = (21, 2, 13). Case 7: For n = 21 and d = 15, i = 1, 2, 3, . . . , 14. • For i = 1, Let β ∈ Z2[ω] 3. Then by Theorem 3.1, β, β4 and β16 have the minimal polynomial φ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. • For i = 2, Let β2 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ2(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1. • For i = 3, Let β3 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ3(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1. • For i = 4, Let β4 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ4(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = φ2(x). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 20 of 36 • For i = 5, Let β5 ∈ Z2[ω] 3. Then by Theorem 3.1, β5, β17 and β20 have the minimal polynomial φ5(x) = (x− β5)(x− β17)(x− β20) = x3 + (1 + ω)x+ 1. • For i = 6, Let β6 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ6(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = φ3(x). • For i = 7, Let β7 ∈ Z2[ω] 3. Then by Theorem 3.1, β7 has the minimal polynomial φ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ8(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1 = φ2(x). • For i = 9, Let β9 ∈ Z2[ω] 3. Then by Theorem 3.1, β9, β18 and β15 have the minimal polynomial φ9(x) = (x− β9)(x− β18)(x− β15) = x3 + x2 + 1. • For i = 10, Let β10 ∈ Z2[ω] 3. Then by Theorem 3.1, β10, β13 and β19 have the minimal polyno- mial φ10(x) = (x− β10)(x− β13)(x− β19) = x3 + ωx+ 1. • For i = 11, Let β11 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ11(x) = (x− β2)(x− β8)(x− β11) = φ2(x). • For i = 12, Let β12 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ12(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = φ3(x). • For i = 13, Let β13 ∈ Z2[ω] 3. Then by Theorem 3.1, β10, β13 and β19 have the minimal polyno- mial φ13(x) = (x− β10)(x− β13)(x− β19) = x3 + ωx+ 1 = φ10(x). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 21 of 36 • For i = 14, Let β14 ∈ Z2[ω] 3. Then by Theorem 3.1, β14 has the minimal polynomial φ14(x) = (x− β14) = x+ ω. Now the generator polynomial is G(x) = φ1(x) · φ2(x) · φ3(x) · φ5(x) · φ7(x) · φ9(x) · φ10(x) · φ14(x). Expanding, G(x) = x20 + x18 + x17 + x16 + ωx15 + ωx14 + x13 + x9 + x5 + x4 + x3 + 1. Since k = 21 − 20 = 1, the code parameters are (n, k, d) = (21, 1, 15), which is a shortened BCH code. Case 8: For n = 21 and d = 17, i = 1, 2, 3, . . . , 16. • For i = 1, Let β ∈ Z2[ω] 3. Then by Theorem 3.1, β, β4 and β16 have the minimal polynomial φ1(x) = (x− β)(x− β4)(x− β16) = x3 + (1 + ω)x2 + 1. • For i = 2, Let β2 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ2(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1. • For i = 3, Let β3 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ3(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1. • For i = 4, Let β4 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ4(x) = (x− β2)(x− β8)(x− β11) = φ2(x). • For i = 5, Let β5 ∈ Z2[ω] 3. Then by Theorem 3.1, β5, β17 and β20 have the minimal polynomial φ5(x) = (x− β5)(x− β17)(x− β20) = x3 + (1 + ω)x+ 1. • For i = 6, Let β6 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ6(x) = (x− β3)(x− β6)(x− β12) = φ3(x). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 22 of 36 • For i = 7, Let β7 ∈ Z2[ω] 3. Then by Theorem 3.1, β7 has the minimal polynomial φ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ8(x) = (x− β2)(x− β8)(x− β11) = φ2(x). • For i = 9, Let β9 ∈ Z2[ω] 3. Then by Theorem 3.1, β9, β18 and β15 have the minimal polynomial φ9(x) = (x− β9)(x− β18)(x− β15) = x3 + x2 + 1. • For i = 10, Let β10 ∈ Z2[ω] 3. Then by Theorem 3.1, β10, β13 and β19 have the minimal polyno- mial φ10(x) = (x− β10)(x− β13)(x− β19) = x3 + ωx+ 1. • For i = 11, Let β11 ∈ Z2[ω] 3. Then by Theorem 3.1, β2, β8 and β11 have the minimal polynomial φ11(x) = (x− β2)(x− β8)(x− β11) = φ2(x). • For i = 12, Let β12 ∈ Z2[ω] 3. Then by Theorem 3.1, β3, β6 and β12 have the minimal polynomial φ12(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1 = φ3(x). • For i = 13, Let β13 ∈ Z2[ω] 3. Then by Theorem 3.1, β10, β13 and β19 have the minimal polyno- mial φ13(x) = (x− β10)(x− β13)(x− β19) == x3 + ωx+ 1 = φ10(x). • For i = 14, Let β14 ∈ Z2[ω] 3. Then by Theorem 3.1, β14 has the minimal polynomial φ14(x) = (x− β14) = x+ ω. • For i = 15, Let β15 ∈ Z2[ω] 3. Then by Theorem 3.1, β9, β18 and β15 have the minimal polyno- mial φ15(x) = (x− β9)(x− β18)(x− β15) = φ9(x). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 23 of 36 • For i = 16, Let β16 ∈ Z2[ω] 3. Then by Theorem 3.1, β, β4 and β16 have the minimal polynomial φ16(x) = (x− β)(x− β4)(x− β16) = x3 + (1 + ω)x2 + 1 = φ1(x). Now the generator polynomial is G(x) = φ1(x) · φ2(x) · φ3(x) · φ5(x) · φ7(x) · φ9(x) · φ10(x) · φ14(x) = ( x19 + (1 + ω)x18 + x16 + (1 + ω)x15 + (1 + ω)x14 + (1 + ω)x13 + x8 + ωx7 + (1 + ω)x6 + x5 + (1 + ω)x4 + x2 + ωx+ 1 + ω ) (ω + x) = x20 + x18 + x17 + x16 + ωx15 + ωx14 + x13 + x9 + x5 + x4 + x3 + 1. Now, k = 21 − 20 = 1. So the parameters of the shortened BCH code are (n, k, d) = (21, 1, 17). Case 9: For n = 21 and d = 19, i = 1, 2, 3, . . . , 18. • For i = 1, Let β ∈ Z2[ω] 3, then by Theorem 3.1, β, β4 and β16 have the minimal polynomial ϕ1(x) = (x− β)(x− β4)(x− β16) = x3 + (β + β4 + β16)x2 + (β5 + β17 + β20)x+ β21 = x3 + (1 + ω)x2 + 1. • For i = 2, Let β2 ∈ Z2[ω] 3, then β2, β8, β11 have ϕ2(x) = (x− β2)(x− β8)(x− β11) = x3 + (β2 + β8 + β11)x2 + (β10 + β13 + β19)x+ β21 = x3 + ωx2 + 1. • For i = 3, Let β3 ∈ Z2[ω] 3, then β3, β6, β12 have ϕ3(x) = (x− β3)(x− β6)(x− β12) = x3 + (β3 + β6 + β12)x2 + (β9 + β18 + β15)x+ β21 = x3 + x+ 1. • For i = 4, ϕ4(x) = ϕ2(x) = x3 + ωx2 + 1. • For i = 5, ϕ5(x) = (x− β5)(x− β17)(x− β20) = x3 + (β5 + β17 + β20)x2 + (β + β4 + β16)x+ β21 = x3 + (1 + ω)x+ 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 24 of 36 • For i = 6, ϕ6(x) = ϕ3(x) = x3 + x+ 1. • For i = 7, ϕ7(x) = (x− β7) = x+ 1 + ω. • For i = 8, ϕ8(x) = ϕ2(x) = x3 + ωx2 + 1. • For i = 9, ϕ9(x) = (x− β9)(x− β18)(x− β15) = x3 + (β9 + β15 + β18)x2 + (β3 + β6 + β12)x+ β21 = x3 + x2 + 1. • For i = 10, ϕ10(x) = (x− β10)(x− β13)(x− β19) = x3 + (β10 + β13 + β19)x2 + (β2 + β8 + β11)x+ β21 = x3 + ωx+ 1. • For i = 11, ϕ11(x) = ϕ2(x) = x3 + ωx2 + 1. • For i = 12, ϕ12(x) = ϕ3(x) = x3 + x+ 1. • For i = 13, ϕ13(x) = ϕ10(x) = x3 + ωx+ 1. • For i = 14, ϕ14(x) = (x− β14) = x+ ω. • For i = 15, ϕ15(x) = ϕ9(x) = x3 + x2 + 1. • For i = 16, ϕ16(x) = ϕ1(x) = x3 + (1 + ω)x2 + 1. • For i = 17, ϕ17(x) = ϕ5(x) = x3 + (1 + ω)x+ 1. • For i = 18, ϕ18(x) = ϕ9(x) = x3 + x2 + 1. Now the generator polynomial is G(x) = ϕ1(x) · ϕ2(x) · ϕ3(x) · ϕ5(x) · ϕ7(x) · ϕ9(x) · ϕ10(x) · ϕ14(x) = (x19 + (1 + ω)x18 + x16 + (1 + ω)x15 + (1 + ω)x14 + (1 + ω)x13 + x8 + ωx7 + (1 + ω)x6 + x5 + (1 + ω)x4 + x2 + ωx+ 1 + ω)(ω + x) = x20 + x18 + x17 + x16 + ωx15 + ωx14 + x13 + x9 + x5 + x4 + x3 + 1. Thus, the dimension of the code is k = 21− 20 = 1. Case 10: For n = 21 and d = 21, i = 1, 2, 3, . . . , 20. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 25 of 36 • For i = 1, Let β ∈ Z2[ω] 3, then by Theorem 3.1, β, β4, and β16 have the minimal polynomial φ1(x) = (x−β)(x−β4)(x−β16) = x3+(β+β4+β16)x2+(β5+β17+β20)x+β21 = x3+(1+ω)x2+1. • For i = 2, Let β2 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ2(x) = (x− β2)(x− β8)(x− β11) = x3 + ωx2 + 1. • For i = 3, Let β3 ∈ Z2[ω] 3, then by Theorem 3.1, β3, β6, and β12 have the minimal polynomial φ3(x) = (x− β3)(x− β6)(x− β12) = x3 + x+ 1. • For i = 4, Let β4 ∈ Z2[ω] 3, then by Theorem 3.1, β2, β8, and β11 have the minimal polynomial φ4(x) = φ2(x) = x3 + ωx2 + 1. • For i = 5, Let β5 ∈ Z2[ω] 3, then by Theorem 3.1, β5, β17, and β20 have the minimal polynomial φ5(x) = (x− β5)(x− β17)(x− β20) = x3 + (1 + ω)x+ 1. • For i = 6, Let β6 ∈ Z2[ω] 3, then φ6(x) = φ3(x) = x3 + x+ 1. • For i = 7, Let β7 ∈ Z2[ω] 3, then φ7(x) = x+ 1 + ω. • For i = 8, Let β8 ∈ Z2[ω] 3, then φ8(x) = φ2(x) = x3 + ωx2 + 1. • For i = 9, Let β9 ∈ Z2[ω] 3, then φ9(x) = x3 + x2 + 1. • For i = 10, Let β10 ∈ Z2[ω] 3, then φ10(x) = x3 + ωx+ 1. • For i = 11, Let β11 ∈ Z2[ω] 3, then φ11(x) = φ2(x) = x3 + ωx2 + 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 26 of 36 • For i = 12, Let β12 ∈ Z2[ω] 3, then φ12(x) = φ3(x) = x3 + x+ 1. • For i = 13, Let β13 ∈ Z2[ω] 3, then φ13(x) = φ10(x) = x3 + ωx+ 1. • For i = 14, Let β14 ∈ Z2[ω] 3, then φ14(x) = x+ ω. • For i = 15, Let β15 ∈ Z2[ω] 3, then φ15(x) = φ9(x) = x3 + x2 + 1. • For i = 16, Let β16 ∈ Z2[ω] 3, then φ16(x) = φ1(x) = x3 + (1 + ω)x2 + 1. • For i = 17, Let β17 ∈ Z2[ω] 3, then φ17(x) = φ5(x) = x3 + (1 + ω)x+ 1. • For i = 18, Let β18 ∈ Z2[ω] 3, then φ18(x) = φ9(x) = x3 + x2 + 1. • For i = 19, Let β19 ∈ Z2[ω] 3, then φ19(x) = φ10(x) = x3 + ωx+ 1. • For i = 20, Let β20 ∈ Z2[ω] 3, then φ20(x) = φ5(x) = x3 + (1 + ω)x+ 1. Now the generator polynomial is G(x) = ϕ1(x) · ϕ2(x) · ϕ3(x) · ϕ5(x) · ϕ7(x) · ϕ9(x) · ϕ10(x) · ϕ14(x) = x20 + x18 + x17 + x16 + ωx15 + ωx14 + x13 + x9 + x5 + x4 + x3 + 1. Now, k = 21 − 20 = 1. So the parameters of the shortened BCH code are (n, k, d) = (21, 1, 21). 4. 4. Decoding Algorithm for Shortened BCH Codes over the Eisenstein Fields This section is based on the procedure of decoding Shortened BCH codes over Eisen- stein fields of length n, using a modified Berlekamp–Massey Algorithm (BMA). The pro- cedure follows the same principles as decoding conventional BCH codes. Theorem 4.1 [6, Theorem 4.2]: Let C be a BCH or Shortened BCH code of length n, defined over the Eisenstein field Zp[ω], with designed distance d. Then, the code C is the null space of the matrix N , where M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 27 of 36 N =  1 βc β2c · · · β(n−1)c 1 βc+1 β2(c+1) · · · β(n−1)(c+1) ... ... ... . . . ... 1 βc+d−2 β2(c+d−2) · · · β(n−1)(c+d−2)  . Here, β is a primitive element in the extension field containing Zp[ω], and the matrix N consists of d− 1 rows and n columns. The codewords of C are those vectors v ∈ Zp[ω] n such that NvT = 0. 4.1. Algorithm for Decoding of Shortened BCH Codes over the Eisenstein Fields Assume that c is a codeword from a narrow-sense Shortened BCH code (n, k, d) over an Eisenstein field, and the received word is r. The designed distance is d. The follow- ing steps outline the decoding process using the modified Berlekamp–Massey algorithm (BMA). Step 1: Syndrome Calculation Let Si for i = c, c+1, . . . , c+ d− 2 denote the syndromes computed from the received word r and the matrix N : Si = rNT mod p = (Sc, Sc+1, . . . , Sc+d−2). Alternatively, each syndrome can be computed using: Si = rβi = b0 + b1β i + · · ·+ bn−1β (n−1)i, for i = c, c+ 1, . . . , c+ d− 2. If all Si = 0, then c = r, indicating no error. Otherwise, proceed to the next step. Step 2: Compute the Error Locator Polynomial ∆n(y) Apply the modified BMA to find ∆n(y), the error locator polynomial. Let ϑn be the discrepancy, and un = deg(∆n(y)). The maximum number of correctable errors is t. Table 4 shows the initialization values. Table 4: Error Correcting Polynomials by Modified BMA Iterations n ∆n(y) ϑn un n− un -1 1 1 0 -1 0 1 First non-zero syndrome 0 0 1 ... ... ... ... ... 2t M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 28 of 36 Case 1: If ϑn = 0, then: ∆n+1(y) = ∆n(y), un+1 = un. Case 2: If ϑn ̸= 0, choose m ≤ n− 1 such that n− um is maximal or equal to n− un. Then, using ϑn − zϑm = 0, solve for z and compute: ∆n+1(y) = ∆n(y)− zyn−m∆m(y), and update the discrepancy as: ϑn+1 = Sn+2 +∆ (n+1) 1 (y)Sn+1 +∆ (n+1) 2 (y)Sn + · · ·+∆(n+1) un+1 (y)Sn+2−un+1 . Step 3: Find the Reciprocal Function From ∆n(y), compute its reciprocal g(y). Let yj be the roots of g(y), and suppose xj = pj are error positions if xj − yj = 0, for 1 ≤ j ≤ n− 1. Step 4: Symmetric Function Determine the elementary symmetric function of the error locations: (y − x1)(y − x2) · · · (y − xv) = ∆0y v +∆1y v−1 + · · ·+∆v, where v is the total number of error locations. Step 5: Error Magnitudes by Forney’s Formula [27] The magnitude zi of the error at location yi is given by: zi = ∑v−1 l=0 ∆(i,l)Sv−l∑v−1 l=0 ∆(i,l)y v−l i , with initialization ∆0 = ∆(i,0) = 1, and recurrence: ∆(i,j) = ∆j + xi∆(i,j−1), j = 1, 2, . . . , v − 1, i = 1, 2, . . . , v. Step 6: Recover the Codeword Finally, compute the corrected codeword: c = r− e, where e is the error vector. The pseudocode of this decoding process is given in Algorithm 4.1. Illustration 4.1: Assume that [3, 1, 3] narrow-sense shortened BCH code over the Eisenstein field Z2[ω] 2 and the received vector r = (1, 1, ω). Determine the corrected codeword (if possible). M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 29 of 36 As t = ⌊ d−1 2 ⌋ = ⌊ 3−1 2 ⌋ = 1, the number of iterations will be 2t = 2. Let S = rHT = (1, 1, ω)  1 1 β β2 β2 β4 T = ( 1 ω ) , where S1 = β3 and S2 = β are the syndromes. Next, we find ∆2(y) using the modified Berlekamp–Massey Algorithm (BMA), sum- marized in Table 5. Iteration 1: The initial non-zero syndrome is 1. Applying Case 2 of Step 2 from the modified BMA, since ϑ0 ̸= 0, −1 ≤ −1, and 0− u−1 = 0 is the maximum value of the last column, we get ϑ0 − zϑ−1 = 0 ⇒ z = ϑ0 ϑ−1 = 1 1 = 1. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 30 of 36 Table 5: Single Error Locating Polynomials n ∆n(y) ϑn un n− un −1 1 1 0 −1 0 1 1 0 0 1 1 + y 1 + ω 1 0 2 1 + ωy Thus, ∆1(y) = ∆0(y)− (1)y0+1∆−1(y) = 1 + y. Now compute ϑ1 = S2 +∆ (1) 1 S1 = ω + (1)(1) = 1 + ω. Iteration 2: Since ϑ1 ̸= 0, and 1 − u0 = 1 − 0 = 1 is the highest value of the last column, we have: ϑ1 − zϑ0 = 0 ⇒ z = ϑ1 ϑ0 = 1 + ω 1 = 1 + ω. Hence, ∆2(y) = ∆1(y)− (1 + ω)y1−0∆0(y) = 1 + ωy. The reciprocal polynomial is g(y) = y + ω. The root of g(y) is ω, i.e., y1 = ω = β2, indicating that the error occurred at the second position of r. The elementary symmetric function (ESF) is given by: ∆0y v +∆1 = y − β2. Error Magnitude: z1 = ∆1,0S1 ∆1,0y1 = β3 β2 = β = 1 + ω. where ∆0 = 1, ∆1 = β2, and v = 1. Corrected Codeword: c = r − e = (1, 1, ω)− (0, 1 + ω, 0) = (1, 1, 1). Thus, the corrected codeword is c = (1, 1, 1), which is a valid codeword of the (3, 1, 3) BCH code. Illustration 4.2: Assume that [9, 3, 3] narrow sense shortened BCH code over the Eisenstein field Z2[ω] 2 and the received vector r = (1, 0, ω, 1, 0, 0, 1, 0, 0), determine the corrected code word (if possible). As t = ⌊ d−1 2 ⌋ = ⌊ 3−1 2 ⌋ = 1. So, the number of iterations will also be 2t = 2. Let S = rHT = (1, 0, ω, 1, 0, 0, 1, 0, 0) ( 1 β β2 · · · β8 1 β2 β4 · · · β16 )T = ( β8 β ) , M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 31 of 36 Table 6: Error Locating Polynomials n ∆n(y) ϑn un n− un −1 1 1 0 −1 0 1 β8 0 0 1 1 + β8y β4 1 0 2 1 + ωy where S1 = β8 and S2 = β are the syndromes. Next, we will find ∆2(y) (see Table 2) by applying the modified BMA shown in Table 6. Iteration 1: The initial non-zero syndrome is β8. We apply Case 2 of Step 2 from the modified BMA as ϑ0 ̸= 0, −1 ≤ −1, and 0− u−1 = 0 is the extreme value of the end column. Then, ϑ0 − zϑ−1 = 0 ⇒ z = ϑ0 ϑ−1 = β8 1 = β8. So, the polynomial: ∆1(y) = ∆0(y)− β8y0+1∆−1(y) = 1 + β8y. ϑ1 = S2 +∆1 1(y)S1 = β + β8 · β8 = β + β7 = β4. Iteration 2: Let ϑ1 ̸= 0 in iteration one, 0 = 1 − 1, and 1 − u0 = 1 − 0 = 1 is the highest value of the last column. Thus, ϑ1 − zϑ0 = 0 ⇒ z = ϑ1 ϑ0 = β4 β8 = β5. So, the polynomial: ∆2(y) = ∆1(y)− β5y1−0∆0(y) = 1 + (β5 + β8)y = 1 + β2y. Now, the reciprocal function of ∆2(y) = 1 + β2y is g(y) = y + β2. So β2 is the only root of g(y). Hence y1 = β2, and the error takes place in the third position of r. ∆0y v +∆1 = y − β2 is an error-locator polynomial (ESF). The error magnitude is: z1 = ∆1,0S1 ∆1,0y1 = β8 β2 = β6 = ω, where ∆0 = 1, ∆1 = β2, and v = 1. The revised code word is: c = r − e = (1, 0, ω, 1, 0, 0, 1, 0, 0)− (0, 0, ω, 0, 0, 0, 0, 0, 0) = (1, 0, 0, 1, 0, 0, 1, 0, 0). Thus, c is the corrected narrow sense shortened (9, 3, 3) BCH code. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 32 of 36 5. The Importance of Shortened Codes over EF in Modern-Day Transmission Being able to code and decode shortened BCH codes within the Eisenstein field Zp[ω], where p ≡ 2 (mod 3), is highly beneficial in modern data transmission systems. Defin- ing codes over Zp[ω] provides the codes with symmetry and enables shorter distances due to the identity ω3 = 1. When p ≡ 2 (mod 3) is selected, the corresponding poly- nomial becomes irreducible in Zp[ω], thereby preserving the field extension and making the construction of codes using Eisenstein arithmetic more convenient. Depending on the required block length and level of error correction, shortened BCH codes can be designed in this setting. These codes are particularly suitable for applications such as Internet of Things (IoT) devices, satellite communication, and mobile networks where strict power and bandwidth limitations exist. The anti-noise efficiency of these codes is enhanced ow- ing to dense encoding schemes and improved distances measured via both Euclidean and Eisenstein weights. In this context, coding algorithms, especially those based on syndrome decoding and the Berlekamp–Massey algorithm, exploit the algebraic structure of the field to efficiently correct errors. Overall, utilizing BCH codes over Zp[ω] presents a promising approach to enhancing the robustness and reliability of modern digital communication infrastructures. 6. Comparative Analysis In this section, we compare the performance of narrow-sense shortened BCH codes over the finite field GF(pm) and the Eisenstein field Zp[ω] m/2, where p ≡ 2 (mod 3). The comparison is made by identifying key parameters, including the code length (n), minimum distance (d), code dimension (k), number of codewords (|C|), and code rate (R). The analysis draws on prior research findings, particularly [6] and Sections 3 and 4 of this article. We provide a comparison based on the finite field GF(26) and the Eisenstein field Z2[ω] 3, as represented in Tables 7 and 8. From Tables 7 and 8, both GF(26) and Z2[ω] 3 yield shortened BCH codes with com- parable lengths and distances. The comparative outcomes are summarized below: Dimension: The dimensions of codes over Z2[ω] 3 often match or surpass their GF(26) counterparts. For instance, for n = 21, both fields provide k = 15 at d = 3, indicating equivalent capacity. However, Z2[ω] 3 offers greater dimensions at higher distances, indi- cating better payload accommodation. Number of Codewords: Due to the quadratic growth in the symbol set, codes over Z2[ω] 3 provide significantly more codewords for the same dimension k. For example, at n = 21 and d = 3, we obtain |C| = 1073741824 for Z2[ω] 3, compared to only |C| = 32768 in GF(26). This suggests an advantage in applications requiring high codeword dispersion. Code Rate: The code rate is equivalent in some cases. For n = 21 and d = 3, both fields yield R = 0.7143. Yet, for other configurations, Z2[ω] 3 offers better performance, M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 33 of 36 Table 7: Analysis of Shortened BCH Codes over the Galois Field GF(26) n d k |C| = 2k R = k/n 3 3 1 2 0.3333 7 3 4 16 0.5714 5 1 2 0.1429 7 1 2 0.1429 9 3 3 8 0.3333 5 1 2 0.1111 7 1 2 0.1111 9 1 2 0.1111 21 3 15 32768 0.7143 5 12 4096 0.5714 7 6 64 0.2857 9 4 16 0.1905 11 1 2 0.0476 13 NA NA NA 15 NA NA NA 17 NA NA NA 19 NA NA NA 21 NA NA NA such as R = 0.3900 at n = 21, d = 9, compared to R = 0.2857 in GF(26). In summary, the higher codeword diversity and greater dimensions offered by Zp[ω] m/2, where p ≡ 2 (mod 3), make it particularly suitable for applications demanding robust error correction. These benefits come at the cost of potentially lower data rates. Furthermore, the demonstrated higher-order minimum distances in the Eisenstein field indicate enhanced error resilience, positioning such codes well for next-generation communication systems and storage solutions requiring high reliability and minimal redundancy. 7. Conclusion and Future Directions This article proposed a new approach in shortened BCH codes over the Eisenstein fields and established its improved error correction performance through modified BMA. Due to the natural extension of Gaussian fields, the algebraic characteristics of Eisenstein fields turn out to be suitable for the development of efficient codes where high reliability and low latency are desired in noisy channels. Many of the changes that have been proposed for the BMA strongly contribute to the growth of the decoding efficiency by taking into account the computational concerns of using Eisenstein field arithmetic. Therefore, this study adds value to the development of the coding theory by bringing conceptuality in theory and idea in application in order to accommodate future issues with respect to reliable data transmission. Efficient implementations of the proposed codes and decoding algorithms in communication structures are possible if the suggested structures are used. The extension M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 34 of 36 Table 8: Analysis of Shortened BCH Codes over the Eisenstein Field Z2[ω] 3 n d k |C| = 22k R = k/n 3 3 1 4 0.3333 7 3 1 4 0.1429 5 2 16 0.2857 7 1 4 0.1429 9 3 1 4 0.1111 5 2 16 0.2222 7 1 4 0.1111 9 1 4 0.1111 21 3 15 1073741824 0.7143 5 12 16777216 0.5714 7 9 262144 0.4286 9 8 65536 0.3900 11 2 16 0.0952 13 1 2 0.0476 15 1 2 0.0476 17 1 2 0.0476 19 1 2 0.0476 21 1 2 0.0476 of these methods into multidimensional and spatially coupled codes holds great potential to be useful in new fields such as storage systems and sensor networks. Likely expanding the areas of application in QEC and in the further advancement of next-generation wireless networks, such as 6G, the techniques based on the Eisenstein field-based codes can bring added value to the transformative technologies. Acknowledgements This research is supported by the Ongoing Research Funding program - Research Chairs (ORF-RC-2025-5300), King Saud University, Riyadh, Saudi Arabia. Tribute We wish to express our heartfelt gratitude to our beloved supervisor, Professor Dr. Tariq Shah (late), whose exceptional guidance, profound knowledge, and unwavering sup- port profoundly shaped our journey as researchers in algebra, number theory, coding the- ory, and cryptography. His mentorship was a cornerstone of our academic and personal growth, and his presence continues to inspire our scholarly endeavours. May his soul rest in eternal peace. M. Sajjad et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6275 35 of 36 Data availability All the data given in this article. 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