https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2695 On Additive Complementary Dual Codes over ℤ2RS and their MacWilliams identities R. Gokul1* , G. Karthick2 , M. Cruz3 , C. Durairajan4 1,4Department of Mathematics, Bharathidasan University, Tiruchirappalli – 620 024, Tamil Nadu, India. 1gokulradha0598@gmail.com,4cdurai66@bdu.ac.in 2Department of Mathematics, SASTRA Deemed University, Thanjavur-613401, Tamil Nadu, India. karthygowtham@gmail.com 3Department of Mathematics, Bishop Heber College, Tiruchirapalli - 620 017, Tamil Nadu, India. cruzmohan@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, we study Additive Complementary Dual (ACD) codes over a mixed alphabet ℤ2RS, where R = ℤ2 + uℤ2 and S = ℤ2 + uℤ2 + vℤ2 + uvℤ2, under the conditions u2 = 0, v2 = 0, and uv = vu. An additive code will prove to be an ACD code under certain conditions. In addition, it is a necessary and sufficient condition for a separable additive code to be an ACD code. Certain additive codes improve into binary linear complementary dual codes under a gray map that we investigate. Furthermore, a few types of weight enumerators are also calculated, and the associated MacWilliams identities are discussed with supportive examples. Keywords: Linear code, Additive code, Gray map, Weight Enumerator, Mac Williams Identities. 1. Introduction A linear code with a complementary dual property was defined in [11]. If a linear code C satisfies C∩C⊥ = {0}, it is called an LCD code. In the same paper, Massey showed that asymptotically good LCD codes exist and highlighted applications such as providing an optimal linear coding solution for the two-user binary adder channel. In [4], LCD codes were utilized in the context of digital communication security. Recently, many authors have studied codes over rings due to their emerging role in algebraic coding theory and successful application in combined coding and modulation. Li et al. established families of reversible codes and demonstrated that some are optimal [7]. In [8], LCD codes were explored over finite chain rings. The concept of LCD codes was generalized to additive complementary dual (ACD) codes, and these were studied over ℤ2 × ℤ4 in [2]. MacWilliams established a relationship between a code’s weight distribution and that of its dual in [10], leading to the development of MacWilliams identities and consideration of weight enumerators for codes over finite Frobenius rings. Various MacWilliams identities over ℤ4 were https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2696 studied in [6]. Weight enumerators over ℤl and conditions ensuring the presence of such identities were discussed in [12] and [14]. In 2014, the investigation of linear codes over ℤ4 +uℤ4 and their weight enumerators was presented in [16]. ℤ2 ℤ2 [u]-additive codes and their standard generator matrices, along with the link between their weight enumerators and their duals, were introduced in [1]. Further research into MacWilliams identities for various weight enumerators was conducted in [13], examining linear codes over the new ring S = 𝔽2 +u𝔽2 +v𝔽2 +uv𝔽2, with identities for the Lee weight enumerator derived using a Gray map from Sn to (𝔽2 + u𝔽2) n . Additionally, self-dual and cyclic codes over S were studied. Motivated by the aforementioned research, we explore ACD codes over the finite commutative Frobenius non-chain ring ℤ2 RS, where R = ℤ2 + uℤ2 and S = ℤ2 + uℤ2 + vℤ2 + uvℤ2 . Throughout this discussion, we let Q = RS. We discuss a necessary condition for a ℤ2Q-additive code to be a ℤ2Q-ACD code. We examine a Gray map wherein specific additive codes over ℤ2Q correspond to binary LCD codes. We explore different weight enumerators of additive codes over ℤ2Q and the corresponding MacWilliams identities, providing examples to demonstrate our findings. All computations in this paper are performed using the Magma Computational Algebra System [3]. This paper is structured as follows: In Section 3, we examine ℤ2Q-ACD codes and determine a condition for an additive code to be an ACD code. In Section 4, we explore a Gray map and demonstrate how some additive codes are mapped to binary LCD codes. Weight enumerators of ℤ2Q- additive codes are studied in Section 5, and the corresponding MacWilliams identities are discussed. 2. Preliminary Let ℤ2 be the binary finite field. Then Q = (ℤ2 + uℤ2 ) × (ℤ2 + uℤ2 + vℤ2 +uvℤ2 ), where u2 = 0, v2 = 0 and uv = vu, is a commutative ring with characteristic 2. Observe that Q is a local ring with its group of units U (Q) = {(1 + ub, 1 + ub + vc + uvd) | b, c, d ∈ ℤ2 } and maximal ideal M = {(ub, ub + vc + uvd) | b, c, d ∈ ℤ2 } such that M = Q \ U (Q). Multiplication in ℤ2Q is defined as follows: 𝑥 ⋆ (𝑧, 𝑟, 𝑠) = (𝜂(𝑥)𝑧, 𝛿(𝑥)𝑟, 𝑥𝑠), (2.1) where η : S→ ℤ2 is defined by η (a + ub + vc + uvd) = a and δ : S→ R is defined by δ (a + ub + vc + uvd) = a + ub. Throughout the manuscript, we will use (x, y) to represent (z, r, s), where x = z and y = (r, s). Multiplication will be represented by 𝑏 ⋆ (𝑥, 𝑦) = (𝜂(𝑏)𝑥, 𝑏𝑦). (2.2) Let us take ℤ2Q as a cross product of the rings ℤ2 and Q. Observe that the ring ℤ2Q is an S-module with respect to usual addition and ⋆-multiplication. Definition 2.1. A subset C of ℤ2 𝑝𝑄𝑞 is said to be an additive code with block length (p, q) if it is a S- submodule of ℤ2 𝑝𝑄𝑞 . Note that if p = q, then ℤ2 𝑝𝑄𝑝= (ℤ2Q)p . It can be treated as a ℤ2Q-additive code with block length (p, p), equivalent to a ℤ2Q-additive code of length p. Definition 2.2. The inner product in ℤ2 𝑝𝑄𝑞 is defined as follows: https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2697 • Let w = (α, β) and w' = (α', β') be elements of Qp = (Rp × Sp ). Then ⟨·|·⟩ : Qp × Qp→ S is defined as follows: ⟨𝑤,𝑤′⟩ = ⟨(𝛼, 𝛽), (𝛼′, 𝛽′)⟩ = 𝛼𝛼′ + 𝛽𝛽′. (2.3) • By using Equation 2.3, we define an inner product over ℤ2 𝑝 𝑄𝑝 as ⟨·|·⟩ : ℤ2 𝑝 𝑄𝑝× ℤ2 𝑝 𝑄𝑝→ S. Let x = (v, w) and x' = (v', w') be elements of ℤ2 𝑝𝑄𝑝. Then ⟨𝑥, 𝑥′⟩ = ⟨(𝑣,𝑤), (𝑣′, 𝑤′)⟩ = 𝑢𝑣⟨𝑣, 𝑣′⟩ + ⟨𝑤,𝑤′⟩. (2.4) Definition 2.3. Let C ⊆ ℤ2 𝑝𝑄𝑞 be an additive code. The dual code C⊥ is defined as: C⊥ = {x ∈ ℤ2 𝑝𝑄𝑞 | ⟨x, y⟩ = 0 for all y ∈ C} where ⟨·, ·⟩ denotes an appropriate inner product over the module ℤ2 𝑝𝑄𝑞. Definition 2.4. The subset A = {v1 , v2 , . . . , vt } of Qq is called a Q-linearly independent set if the equation λ1v1 + λ2v2 + · · · + λtvt = 0, where λ1 , . . . , λt ∈ Q, has only the trivial solution λi = 0 for all i ∈ {1, 2, . . . , t}. Definition 2.5. An additive code C over ℤ2Q is said to be an Additive Complementary Dual (ACD) code if 𝐶 ∩ 𝐶⊥= {0}. If p = 0, then C is an ACD code of length q, and if q = 0, then C is a binary LCD code of length p. An additive code C over a finite ring ℤ2Q is called self-orthogonal (self-dual) if C ⊆ C⊥ (C=C⊥). If C has a basis over ℤ2Q, then it is called a free additive code and the cardinality of the basis is called the rank of C. 3. Additive Complementary Dual Code over ℤ2Q Let Cp be a code over ℤ2 generated by an m × p matrix Gp , and let Cq be a code over Q generated by an m × q matrix Gq . Then the m × (p + q) matrix G = [Gp | Gq ] over ℤ2Q generates an additive code C. Now, we have using the inner product equation 2.4. We define the matrix product as follows: 𝐺 ◦ 𝐺𝑡 = 𝑢𝑣𝐺𝑝𝐺𝑝 𝑡 + 𝐺𝑞𝐺𝑞 𝑡, (3.1) where Gt is the transpose matrix of G. Theorem 3.1. Let C be a ℤ2Q- additive code with the generator matrix G of size m × (p + q). Let nonzero vector vi be the ith row of G such that ⟨vi , vj ⟩ ∈ {0, uv} and ⟨vi , vi ⟩ ∈ U (Q) for all i, j ∈ {1, 2, . . . , m} such that i ≠ j, then C is a ℤ2Q – ACD code. Proof. Let u be any nonzero codeword of C. If u ≠ C⊥ , then C is a ℤ2Q-ACD codes. Since u ∈ C, then u = Σi∈J λi vi , where J = {1, 2, . . . , m} and λi ∈ Q. Firstly, we assume that there exist j ∈ J such that λj ∈ U (Q). Then, https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2698 ⟨𝑢, 𝑣𝑗⟩ = ∑ 𝜆𝑖𝑖∈𝐽 ⟨𝑣𝑖 , 𝑣𝑗⟩ =∑ 𝜆𝑖𝑖∈𝐽{𝑗 ⟨𝑣𝑖 , 𝑣𝑗⟩ + 𝜆𝑗⟨𝑣𝑗, 𝑣𝑗⟩. For i ≠ j, since ⟨vi , vj ⟩ ∈ {0, uv}, then λi ⟨vi , vj ⟩ ∈ {0, uv}. Since ⟨vj , vj ⟩ ∈ U (Q), then λj ⟨vj , vj ⟩ ∈ U (Q). Thus ⟨u, vj ⟩ ≠ 0 and u ≠ C⊥. Further, if λi ∈ m, the maximal ideal. Let j ∈ J such that λj ∈ m \ {0}. Since ⟨vi , vj ⟩ ∈ {0, uv}, then λi ⟨vi , vj ⟩ = 0. Thus, ⟨𝑢, 𝑣𝑗⟩ = ∑ 𝜆𝑖𝑖∈𝐽{𝑗 ⟨𝑣𝑖, 𝑣𝑗⟩ + 𝜆𝑗⟨𝑣𝑗, 𝑣𝑗⟩. = 𝜆𝑗⟨𝑣𝑗, 𝑣𝑗⟩. Since ⟨vj , vj ⟩ ∈ U (Q), then λj ⟨vj , vj ⟩ ≠ 0. Thus ⟨u, vj ⟩ ≠ 0 and u ≠ C⊥ . Thus C is a ℤ2Q- ACD codes. Hence the proved. Theorem 3.1 directly leads to the deduction of the following corollaries. Corollary 3.2. Let C be a ℤ2Q- additive code with the generator matrix G of size m × (p + q) and G ◦ Gt = [vij ]m×m . If vii ∈ U (Q) and vij ∈ {0, uv} for i ≠ j, 1 ≤ i, j ≤ m, then C is a ℤ2Q-ACD code. It is important to note that the conditions presented in Theorem 3.1 and Corollaries 3.2 are solely sufficient to establish that C is a ℤ2Q-ACD code. In general, the converse statements do not hold true. Following example justifies this. Example 3.3. Consider an additive code C of block length (2, 4) over ℤ2Q generated by 𝐺 = ( 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 + 𝑢 0 0 0 0 0 0 1 + 𝑢 0 0 0 0 0 0 1 + 𝑣 0 0 0 0 0 0 1 + 𝑢 + 𝑣) = [Gp | (Gq = R | S)]. From this C is an ACD code and it is C∩C⊥ = {0}. But if u1 is a first row of G, then the (1, 1) entry of G ◦ Gt is v11 = ⟨v1 , v1 ⟩ = uv ∉ U (Q). Through the use of corollary 3.2, the following result is obtained. Proposition 3.4. Let G = (Gp | λIm ) where 𝜆 ∈ 𝑈(𝑄), Gp is an m × p matrix over ℤ2 and Im is the identity matrix of size m × m. Then G generates an ACD code over ℤ2Q. Proof. Consider G ◦ Gt = uvGp G t p + λIm It m = uvGp G t p + λIm G ◦ Gt = [vij ]m×m . So, vij ∈ {0, uv} for i ≠ j and vii ∈ {λ, λ + uv} ⊂ U (Q) for 1 ≤ i ≤ m. Therefore, using Corollary 3.2 G generates an ACD code over ℤ2Q. https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2699 Example 3.5. Let us take 𝐺2 = ( 0 1 1 0 0 0 0 0 )over ℤ2 and λ = 1 + u. Then the ℤ2Q-additive code generated by G 𝐺 = (𝐺2|𝜆𝐼4) = ( 0 1 1 + 𝑢 0 0 0 1 0 0 1 + 𝑢 0 0 0 0 0 0 1 + 𝑢 0 0 0 0 0 0 1 + 𝑢 ) is an ACD code. In [2], the authors defined separable codes over a ring. Now, we generalize it. An additive code C over ℤ2Q of block length (p, q) is said to be a separable code if C is the direct product of Cp and Cq . In this case, C⊥ is the direct product of 𝐶𝑝 ⊥and 𝐶𝑞 ⊥. Thus, C⊥ is separable. The following theorem gives a necessary and sufficient condition for an additive separable code to be an ACD code. Theorem 3.6. Let C be an additive separable code over ℤ2Q of block length (p, q). Then Cp and Cq are LCD codes over ℤ2 and Q, respectively, if and only if C is an ACD code. Proof. Given that C = Cp × Cq is an additive separable code over ℤ2Q of block length (p, q). Then Cp ⊆ ℤ2 𝑝 and Cq ⊆ Qq , and hence 𝐶⊥ = 𝐶𝑝 ⊥ ∩ 𝐶𝑞 ⊥. Suppose Cp and Cq are LCD codes over ℤ2 and Q, respectively. Let (u, u′)∈ 𝐶 ∩ 𝐶⊥,then we have u∈ 𝐶𝑝 ∩ 𝐶𝑝 ⊥ = {0}and u′∈ 𝐶𝑞 ∩ 𝐶𝑞 ⊥ = {0}, since C is separable. Therefore, the intersection of C and C⊥ is trivial, and hence C is an ACD code. Conversely, assume that C is an ACD code. Let u ∈ 𝐶𝑝 ∩ 𝐶𝑝 ⊥ and u′ ∈ 𝐶𝑞 ∩ 𝐶𝑞 ⊥, then (u, u′) ∈ 𝐶 ∩ 𝐶⊥ = {0}, and hence u = 0 and u′ = 0. This implies that 𝐶𝑝 ∩ 𝐶𝑝 ⊥ = {0}and 𝐶𝑞 ∩ 𝐶𝑞 ⊥ = {0}. Hence, Cp and Cq are LCD codes over ℤ2 and Q. Example 3.7. Consider the additive code C over ℤ2Q generated by ( 1 0 0 1 0 0 0 0 0 1 + 𝑢 0 0 1 + 𝑢𝑣 0 0 0 0 0 0 1 0 1 0 0 0 0 1 + 𝑢 0 0 0 1 + 𝑢𝑣 0 0 0 0 0 0 1 0 1 0 1 + 𝑢 1 + 𝑢 0 1 + 𝑢𝑣 0 1 + 𝑢𝑣 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 ) Clearly, C is a separable code. Then 𝐶 ∩ 𝐶⊥ = {0}. Thus C is ACD code over ℤ2Q. Moreover, we have 𝐶𝑝 ∩ 𝐶𝑝 ⊥ = {0}and 𝐶𝑞 ∩ 𝐶𝑞 ⊥ = {0}. Thus Cp and Cq is an LCD code over ℤ2 and Q. https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2700 4. Gray map In this section, we have introduced a new Gray map from ℤ2Q to ℤ2 7 and have discussed the properties in it. Define the gray map 𝜙: Q →ℤ2 6 by 𝜙(b + ud, e + uf + vg + uvh) = (d + b, d, h + g + f + e, h + g, h + f, h) for all element in Q. Let (u, u′), (u1, u′1)∈ℤ2 𝑝𝑄𝑞, then the Hamming weight of (u, u′) is the number of non-zero coordinates of (u, u′) and is denoted by WH (u, u′). We define the Lee weight of (u, u′) ∈ℤ2 𝑝𝑄𝑞 by WL (u, u′) = WH (𝜙(u, u′)) = WH (u) + WH (𝜙(u′)) and the Lee distance between (u, u′) and (u1 , u′1) by dL ((u, u′), (u1 , u′1)) = WL (u − u1 , u′ − u′1) = WH (u − u1) + WH (𝜙(u′ − u′1)). The Hamming distance between (u, u′) and (u1 , u′1) is defined by dH ((u, u′), (u1 , u′1 )) = WH (u − u1 , u′ − u′1 ). Then the Lee weight of elements of Q : Element in the ring Q its Lee weight (0,0) 0 (0,1), (0,1+u), (0,1+v), (0,1+u+v+uv), (1,0), (1+u,0) 1 (0,u), (0,v), (0,u+v), (0,u+uv), (0,v+uv), (0,u+v+uv), (1,1), (1,1+u),(1,1+v), (1,1+u+v+uv), (u,0), (1+u,1), (1+u,1+u), (1+u,1+v),(1+u,1+u+v+uv) 2 (0,1+uv), (0,1+u+v), (0,1+u+uv), (0,1+v+uv), (1,u), (1,v), (1,u+v),(1,u+uv), (1,v+uv), (1,u+v+uv), (u,1), (u,1+u), (u,1+v), (u,1+u+v+uv),(1+u,u), (1+u,v), (1+u,u+v), (1+u,u+uv), (1+u,v+uv), (1+u,u+v+uv) 3 (0,uv), (1,1+uv), (1,1+u+v), (1,1+u+uv), (1,1+v+uv), (u,u), (u,v), (u,u+v),(u,u+uv), (u,v+uv), (u,u+v+uv), (1+u,1+uv), (1+u,1+u+v), (1+u,1+u+uv),(1+u,1+v+uv) 4 (1,uv), (u,1+uv), (u,1+u+v), (u,1+u+uv), (u,1+v+uv), (1+u,uv) 5 (u,uv) 6 Lemma 4.1. The Gray map 𝜙 is ℤ2 -linear. Proof. The Lee weight of x ∈ Q is the Hamming weight of 𝜙(x). Then the Lee distance between two elements is given as dL (x, y) is a Lee weight of x − y for all x, y ∈ Q. See that table 1 is the Lee weight of element in Q. https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2701 For p, q ∈ ℤ, p, q ≥ 0,the gray mapping 𝜙 extended to 𝜙 : ℤ2 𝑝 × Qq →ℤ2 𝑝+6𝑞 where 𝜙(𝑢0, 𝑢1, . . . , 𝑢𝑝−1, 𝑢′0, 𝑢′1, . . . , 𝑢′𝑞−1) = ((𝑢0, 𝑢1, . . . , 𝑢𝑝−1), 𝜙(𝑢′0), 𝜙(𝑢′1). . . , 𝜙(𝑢′𝑞−1)). The Lee weight of (𝑢0, 𝑢1, . . . , 𝑢𝑝−1, 𝜙(𝑢′0), 𝜙(𝑢′1), . . , 𝜙(𝑢′𝑞−1)) ∈ ℤ2 𝑝𝑄𝑞 is 𝛴𝑖=0 𝑝−1𝑤𝑡𝐻(𝑢𝑖) + 𝛴𝑗=0 𝑞−1𝑤𝑡𝐿(𝑢′𝑗) and the Lee distance between x, y ∈ℤ2 𝑝𝑄𝑞 is d L (x, y) = wtL (x − y). In this section, we study certain conditions that make the gray image of an additive code over ℤ2Q the binary LCD code. Theorem 4.2. Let C be an additive code over ℤ2Q generated by the matrix G = (Gp |Gq)m×(p+q) where Gp generates a self-orthogonal code Cp over ℤ2 and Gq generates a additive code Cq over Q. If the rows of the matrics Gq is Q-linealy independent and 𝜙 (Cq) is an binary LCD codes, then the binary Gray image 𝜙 (C) is an LCD code. Proof. Let 𝜙(𝑥) = (𝑢, 𝜙(𝑢′)) ∈ 𝜙(𝐶) ∩ 𝜙(𝐶)⊥ where x = (u, u′) ∈ C. Since for all y = (v, v′) ∈ C, 𝜙(𝑦) = (𝑣, 𝜙(𝑣′)) ∈ 𝜙(𝐶)and 𝜙(𝑥) ∈ 𝜙(𝐶)⊥, (𝑢, 𝜙(𝑢′)). (𝑣, 𝜙(𝑣′)) = 0 for all y = (v, v′) ∈ C (𝑢. 𝑣) + (𝜙(𝑢′). 𝜙(𝑣′)) = 0 Since u, v ∈ Cp and CP is a self-orthogonal code, (u, v)=0. Therefore,(𝜙(𝑢′), 𝜙(𝑣′)) = 0. Since 𝜙(𝑢′), 𝜙(𝑣′) ∈ 𝜙(𝐶𝑞)and Cq is a binary LCD code, implies (𝜙(𝑢′), 𝜙(𝑣′)) = 0 and hence 𝜙(𝑢′) = 0, thus u′ = 0 because 𝜙 is linear. We will prove that for any c = (u, u′) ∈ C, if u′ = 0, then the c = 0. For i ∈ {0, 1, · · · , m − 1}, let gi = (vi , vi′) be the ith row of G where vi and vi′ denotes the ith row of Gp and Gq , respectively. Since c ∈ C, there exist scalars λ0 , λ1 , · · · , λ m−1 ∈ Q such that 𝑐 = (𝑢, 𝑢′) = ∑𝑖=0 𝑚−1𝜆𝑖𝑔𝑖 = (∑𝑖=0 𝑚−1𝜂(𝜆𝑖)𝑣𝑖 , ∑𝑖=0 𝑚−1𝜆𝑖𝑣𝑖′) Thus 𝑢 = ∑𝑖=0 𝑚−1𝜂(𝜆𝑖)𝑣𝑖 and 𝑢′ = ∑𝑖=0 𝑚−1𝜆𝑖𝑣𝑖 = 0. Since the set of all rows of Gq is Q-linealy independent, λi = 0 for all i ∈ {0, 1, · · · , m − 1}. Therefore (u, u′) = 0 and hence 𝜙(𝑢, 𝑢′) = 0. Here is an example to illustrate Theorem 4.2 Example 4.3. Let C be an additive code over ℤ2Q generated by the matrix https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2702 𝐺 = ( 1 1 0 0 𝑢 + 1 𝑢 𝑢𝑣 + 𝑣 + 𝑢 + 1 𝑣 + 1 0 0 1 1 0 𝑢 + 1 𝑢𝑣 + 𝑣 𝑣 + 𝑢 0 0 0 0 0 0 𝑣 + 1 𝑣 )= [Gp | (Gq = R | S)]. Then clearly, Gp generates a self-orthogonal code Cp and the set of all row vectors of Gq are Q-linearly independent. The Gray image𝜙(𝐶𝑞)of Cq is a binary [64, 12, 25] linear LCD code. 5. Weight Enumerators and MacWilliams Identities In this section, we present the weight enumerators of some ℤ2Q-additive codes and discuss related results. 5. 1. Complete Weight Enumerators and MacWilliams Identity We arrange the elements of ℤ2Q in a fixed order as follows and denote them as {f1, f2, f3, . . . , f128 }: ℤ2Q = {(0, 0, 0), (0, 0, 1), (0, 0, u), (0, 0, v), (0, 0, uv), (0, 0, 1 + u), (0, 0, 1 + v), (0, 0, 1 + uv), (0, 0, u + v), (0, 0, u + uv), (0, 0, v + uv), (0, 0, u + v + uv), (0, 0, 1 + u + v), (0, 0, 1 + u + uv), (0, 0, 1 + v + uv), (0, 0, 1 + u + v + uv), (0, 1, 0), (0, 1, 1), (0, 1, u), (0, 1, v), (0, 1, uv), (0, 1, 1 + u), (0, 1, 1 + v), (0, 1, 1 + uv), (0, 1, u + v), (0, 1, u + uv), (0, 1, v + uv), (0, 1, u + v + uv), (0, 1, 1 + u + v), (0, 1, 1 + u + uv), (0, 1, 1 + v + uv), (0, 1, 1 + u + v + uv), (0, u, 0), (0, u, 1), (0, u, u), (0, u, v), (0, u, uv), (0, u, 1 + u), (0, u, 1 + v), (0, u, 1 + uv), (0, u, u + v), (0, u, u + uv), (0, u, v + uv), (0, u, u + v + uv), (0, u, 1 + u + v), (0, u, 1 + u + uv), (0, u, 1 + v + uv), (0, u, 1 + u + v + uv), (0, 1 + u, 0), (0, 1 + u, 1), (0, 1 + u, u), (0, 1 + u, v), (0, 1 + u, uv), (0, 1 + u, 1 + u), (0, 1 + u, 1 + v), (0, 1 + u, 1 + uv), (0, 1 + u, u + v), (0, 1 + u, u + uv), (0, 1 + u, v + uv), (0, 1 + u, u + v + uv), (0, 1 + u, 1 + u + v), (0, 1 + u, 1 + u + uv), (0, 1 + u, 1 + v + uv), (0, 1 + u, 1 + u + v + uv), (1, 0, 0), (1, 0, 1), (1, 0, u), (1, 0, v), (1, 0, uv), (1, 0, 1 + u), (1, 0, 1 + v), (1, 0, 1 + uv), (1, 0, u + v), (1, 0, u + uv), (1, 0, v + uv), (1, 0, u + v + uv), (1, 0, 1 + u + v), (1, 0, 1 + u + uv), (1, 0, 1 + v + uv), (1, 0, 1 + u + v + uv), (1, 1, 0), (1, 1, 1), (1, 1, u), (1, 1, v), (1, 1, uv), (1, 1, 1 + u), (1, 1, 1 + v), (1, 1, 1 + uv), (1, 1, u + v), (1, 1, u + uv), (1, 1, v + uv), (1, 1, u + v + uv), (1, 1, 1 + u + v), (1, 1, 1 + u + uv), (1, 1, 1 + v + uv), (1, 1, 1 + u + v + uv), (1, u, 0), ( 1, u, 1), (1, u, u), (1, u, v), (1, u, uv), (1, u, 1 + u), (1, u, 1 + v), (1, u, 1 + uv), (1, u, u + v), (1, u, u + uv), (1, u, v + uv), (1, u, u + v + uv), (1, u, 1 + u + v), (1, u, 1 + u + uv), (1, u, 1 + v + uv), (1, u, 1 + u + v + uv), (1, 1 + u, 0), (1, 1 + u, 1), (1, 1 + u, u), (1, 1 + u, v), (1, 1 + u, uv), (1, 1 + u, 1 + u), (1, 1 + u, 1 + v), (1, 1 + u, 1 + uv), (1, 1 + u, u + v), (1, 1 + u, u + uv), (1, 1 + u, v + uv), (1, 1 + u, u + v + uv), (1, 1 + u, 1 + u + v), (1, 1 + u, 1 + u + uv), (1, 1 + u, 1 + v + uv), (1, 1 + u, 1 + u + v + uv)}. The complete weight enumerator of an additive code C over ℤ2Q is defined by CWEC (x1 , x2 , . . . , x128 ) =∑ 𝑥1 𝑤𝑓1 (𝑐) 𝑐∈𝐶 𝑥2 𝑤𝑓2 (𝑐) 𝑥3 𝑤𝑓3(𝑐) . . . 𝑥128 𝑤𝑓128 (𝑐) (5.1) where 𝑤𝑓𝑖 (𝑐)=|{j | cj = f i , 1 ≤ j ≤ n}| for 1 ≤ i ≤ 128 and c = (c1 , c2 , c3 , . . . , cn ) ∈C. https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2703 Remark 5.1. Note that CWEC (x1 , x2 , . . . , x128 ) is a homogeneous polynomial. The total degree of each monomial in CWEC (x1 , x2 , . . . , x128 ) is n. The term xn 1 always appears in CWEC (x1 , x2 , . . . , x128 ) as (0, 0, 0, . . . , 0) is a codeword in C. Furthermore, note that CWEC (1, 1, 1, . . . , 1) = |C| and CWEC (a, 0, 0, . . . , 0) = an. Let R be a finite commutative ring with unity. A character 𝜒 of R is a group homomorphism from R to C*. In [5], the authors defined that if R is a Frobenius ring and 𝑅 is the character module of R such that 𝑅 is R-module isomorphic to R, and if γ : R→ 𝑅 is an R-module isomorphism, then 𝜒:= γ(1) is said to be a generating character of R. Lemma 5.2. [5] Let 𝜒 be a character of a finite commutative ring R. Then 𝜒 is a generating character if and only if ker (𝜒) contains no nonzero ideals of R. Definition 5.3. The generating character : ℤ2Q → C* is defined as 𝜒(𝑎, 𝑏 + 𝑑, 𝑒 + 𝑓 + 𝑔 + ℎ) = (−1)𝑎+𝑏+𝑑+𝑒+𝑓+𝑔+ℎ. (5.2) Observe that the restriction of 𝜒 to every nonzero ideal of ℤ2Q is nontrivial. Therefore, by Lemma 5.2, it is a generating character of the ring ℤ2Q. Lemma 5.4. For a nonzero ideal I of ℤ2Q, we have∑ 𝜒𝑥∈𝐼 (𝑥) = 0 where 𝜒 is defined in Equation 5.2. Proof. Suppose that I is an ideal generated by (0, u, uv). Then I = {(0, 0, 0), (0, 0, u), (0, 0, v), (0, 0, uv), (0, 0, u + v), (0, 0, u + uv), (0, 0, v + uv), (0, 0, u + v + uv), (0, u, 0), (0, u, u), (0, u, v), (0, u, uv), (0, u, u + v), (0, u, u + uv), (0, u, v + uv), (0, u, u + v + uv)} is a maximal ideal of ℤ2Q. Observe that, ∑𝜒 𝑥∈𝐼 (𝑥) = 1 + (−1) + (−1) + (−1) + 1 + 1 + 1 + (−1) + (−1) + 1 + 1 + 1 + (−1) + (−1) + (−1) + 1 = 0. Similarly, we can verify the equation if I is replaced by any other nonzero ideal of ℤ2Q. Suppose that 𝑇 = [𝑡𝑖,𝑗]128×128 is a matrix such that 𝑡𝑖,𝑗 = 𝜒(𝑓𝑖,𝑗), where fi , fj ∈ ℤ2Q for 1 ≤ i, j ≤ 128, and 𝜒 is the generating character defined in Equation 5.2. Then https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2704 𝑇 = [ 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴 𝐴] (5.3) 𝑤ℎ𝑒𝑟𝑒, 𝐴 = ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 −1 1 1 −1 1 1 −1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 1 −1 −1 1 1 1 −1 1 1 −1 1 1 −1 1 1 −1 −1 1 −1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 −1 1 1 −1 1 1 1 −1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 −1 1 1 1 1 1 −1 1 1 1 −1 1 1 −1 1 1 1 1 ) For further discussion, we need the following result from [15]. Theorem 5.5. If C is an additive code of length n over ℤ2Q, then CWEC⊥ (x1 , x2 , x3 , . . . , x128 ) = 1 |𝑐| CWEC (T · (x1 , x2 , x3 , . . . , x128 ) t ), where t is denoted by the transpose and T is an in Equation 5.3. Example 5.6. Let C be the R-submodule of (ℤ2Q)2 spanned by {(1, 0; 0, 0; 0, 0), (0, 0; 0, u; 0, 0), (0, 0; 0, 0; 0, uv), (0, 0; 0, u; 0, uv)}. Clearly, C is a vector space over ℤ2 of dimension 7. Then the dual C⊥ of the additive code C is the R-submodule of (ℤ2Q)2 spanned by {(0, 1; 0, 0; 0, 0), (0, 0; 0, u; 0, 0), (0, 0; 0, 0; 0, uv), (0, 0; 0, u; 0, uv)}, which is also a 7-dimensional vector space over ℤ2 . The complete weight enumerator of C and C⊥ are: CWEC (x1 , x2 , x3 , . . . , x128 ) = x2 1 + x2 5 + x2 33 + x2 37 + 2x1 x5 + 2x1 x33 + 2x1 x37 + x1 x3 + x1 x4+ x1 x9 + x1 x10 + x1 x11 + x1 x12 + x1 x35 + x1 x36 + x1 x41 + x1 x42 + x1 x43 + x1 x44 + x1 x65 + x1 x67 + x1 x68 + x1 x69 + x1 x73 + x1 x74 + x1 x75 + x1 x76 + x1 x97 + x1 x99 + x1 x100 + x1 x101 + x1 x105 + x1 x106 + x1 x107 + x1 x108 + 2x5 x33 + 2x5 x37 + x5 x3 + x5 x4 + x5 x9 + x5 x10 + x5 x11 + x5 x12 + x5 x35 + x5 x36 + x5 x41 + x5 x42 + x5 x43 + x5 x44 + x5 x65 + x5 x67 + x5 x68 + x5 x69 + x5 x73 + x5 x74 + x5 x75 + x5 x76 + x5 x97 + x5 x99 + x5 x100 + x5 x101 https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2705 + x5 x105 + x5 x106 + x5 x107 + x5 x108 + 2x33 x37 + x33 x3 + x33 x4 + x33 x9 + x33 x10 + x33 x11 + x33 x12 + x33 x35 + x33 x36 + x33 x41 + x33 x42 + x33 x43 + x33 x44 + x33 x65 + x33 x67 + x33 x68 + x33 x69 + x33 x73 + x33 x74 + x33 x75 + x33 x76 + x33 x97 + x33 x99 + x33 x100 + x33 x101 + x33 x105 + x33 x106 + x33 x107 + x33 x108 + x37 x3 + x37 x4 + x37 x9 + x37 x10 + x37 x11 + x37 x12 + x37 x35 + x37 x36 + x37 x41 + x37 x42 + x37 x43 + x37 x44 + x37 x65 + x37 x67 + x37 x68 + x37 x69 + x37 x73 + x37 x74 + x37 x75 + x37 x76 + x37 x97 + x37 x99 + x37 x100 + x37 x101 + x37 x105 + x37 x106 + x37 x107 + x37 x108 , CWEC ⊥ (x1 , x2 , x3 , . . . , x128) = x2 1 + x2 5 + x2 33 + x2 37 + 2x1 x5 + 2x1 x33 + 2x1 x37 + x1 x3 + x1 x4+ x1 x9 + x1 x10 + x1 x11 + x1 x12 + x1 x35 + x1 x36 + x1 x41 + x1 x42 + x1 x43 + x1 x44 + x1 x65 + x1 x67 + x1 x68 + x1 x69 + x1 x73 + x1 x74 + x1 x75 + x1 x76 + x1 x97 + x1 x99 + x1 x100 + x1 x101 + x1 x105 + x1 x106 + x1 x107 + x1 x108 + 2x5 x33 + 2x5 x37 + x5 x3 + x5 x4 + x5 x9 + x5 x10 + x5 x11 + x5 x12 + x5 x35 + x5 x36 + x5 x41 + x5 x42 + x5 x43 + x5 x44 + x5 x65 + x5 x67 + x5 x68 + x5 x69 + x5 x73 + x5 x74 + x5 x75 + x5 x76 + x5 x97 + x5 x99 + x5 x100 + x5 x101 + x5 x105 + x5 x106 + x5 x107 + x5 x108 + 2x33 x37 + x33 x3 + x33 x4 + x33 x9 + x33 x10 + x33 x11 + x33 x12 + x33 x35 + x33 x36 + x33 x41 + x33 x42 + x33 x43 + x33 x44 + x33 x65 + x33 x67 + x33 x68 + x33 x69 + x33 x73 + x33 x74 + x33 x75 + x33 x76 + x33 x97 + x33 x99 + x33 x100 + x33 x101 + x33 x105 + x33 x106 + x33 x107 + x33 x108 + x37 x3 + x37 x4 + x37 x9 + x37 x10 + x37 x11 + x37 x12 + x37 x35 + x37 x36 + x37 x41 + x37 x42 + x37 x43 + x37 x44 + x37 x65 + x37 x67 + x37 x68 + x37 x69 + x37 x73 + x37 x74 + x37 x75 + x37 x76 + x37 x97 + x37 x99 + x37 x100 + x37 x101 + x37 x105 + x37 x106 + x37 x107 + x37 x108. Definition 5.7. Let C be an additive code of length n over ℤ2Q. Then the Hamming weight enumerator of C over ℤ2Q is defined as 𝑊𝐶(𝑥, 𝑦) = ∑ 𝑥𝑛−𝑤𝑡𝐻(𝑐) 𝑐∈𝐶 𝑦𝑤𝑡𝐻(𝑐) = 𝐶𝑊𝐸𝐶(𝑥, 𝑦, 𝑦, . . . , 𝑦). (5.4) Note that the polynomial WC (x, y) is homogeneous of degree n. Theorem 5.8. Let C be an additive code of length n over ℤ2Q. Then WC ⊥ (x, y) = 1 |𝑐| WC (x + 127y, x − y) Proof. Consider WC ⊥ (x, y) = CWEC ⊥ (x, y, y, y, . . . , y) = 1 |𝑐| CWEC (T · (x, y, y, y, . . . , y)t ) = 1 |𝑐| CWEC (x + 127y, x − y, x − y, . . . , x − y) WC ⊥ (x, y) = 1 |𝑐| WC (x + 127y, x − y). https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2706 Example 5.9. Let C be the code in Example 5.6. Then WC (x, y) = x2 + 34xy + 93y 2 and WC ⊥ (x, y) = 1 128 {(x + 127y)2 + 93(x − y)2 + 34(x + 127y)(x − y)} = 1 128 {128x2 + 4, 352xy + 11, 904y 2 } WC ⊥ (x, y) =x2 + 34xy + 93y2 . 5. 2. Symmetrized Weight Enumerator and Lee Weight Enumerator The Lee weight of each element of ℤ2Q is listed below: Element in ℤ2Q Lee weight (0,0,0) 0 (0,0,1), (0,0,1+u), (0,0,1+v), (0,0,1+u+v+uv), (0,1,0), (0,1+u,0), (1,0,0) 1 (0,0,u), (0,0,v), (0,0,u+v), (0,0,u+uv), (0,0,v+uv), (0,0,u+v+uv), (0,1,1), (0,1,1+u), (0,1,1+v), (0,1,1+u+v+uv), (0,u,0), (0,1+u,1), (0,1+u,1+u), (0,1+u,1+v), (0,1+u,1+u+v+uv), (1,0,1), (1,0,1+u), (1,0,1+v), (1,0,1+u+v+uv), (1,1,0), (1,1+u,0) 2 (0,0,1+uv), (0,0,1+u+v), (0,0,1+u+uv), (0,0,1+v+uv), (0,1u), (0,1,v), (0,1,u+v), (0,1,u+uv), (0,1,v+uv), (0,1,u+v+uv), (0,u,1), (0,u,1+u), (0,u,1+v), (0,u,1+u+v+uv), (0,1+u,u), (0,1+u,v), (0,1+u,u+v), (0,1+u,u+uv), (0,1+u,v+uv), (0,1+u,u+v+uv), (1,0,u), (1,0,v), (1,0,u+v), (1,0,u+uv), (1,0,v+uv), (1,0,u+v+uv), (1,1,1), (1,1,1+u), (1,1,1+v), (1,1,1+u+v+uv), (1,u,0), (1,1+u,1), (1,1+u,1+u), (1,1+u,1+v), (1,1+u,1+u+v+uv) 3 (0,0,uv), (0,1,1+uv), (0,1,1+u+v), (0,1,1+u+uv), (0,1,1+v+uv), (0,u,u), (0,u,v), (0,u,u+v), (0,u,u+uv), (0,u,v+uv), (0,u,u+v+uv), (0,1+u,1+uv), (0,1+u,1+u+v), (0,1+u,1+u+uv), (0,1+u,1+v+uv), (1,0,1+uv), (1,0,1+u+v), (1,0,1+u+uv), (1,0,1+v+uv), (1,1,u), (1,1,v), (1,1,u+v), (1,1,u+uv), (1,1,v+uv), (1,1,u+v+uv), (1,u,1), (1,u,1+u), (1,u,1+v), (1,u,1+u+v+uv), (1,1+u,u), (1,1+u,v), (1,1+u,u+v), (1,1+u,u+uv), (1,1+u,v+uv), (1,1+u,u+v+uv) 4 (0,1,uv), (0,u,1+uv), (0,u,1+u+v), (0,u,1+u+uv), (0,u,1+v+uv), (0,1+u,uv), (1,0,uv), (1,1,1+uv), (1,1,1+u+v), (1,1,1+u+uv), (1,1,1+v+uv), (1,u,u), (1,u,v), (1,u,u+v), (1,u,u+uv), (1,u,v+uv), (1,u,u+v+uv), (1,1+u,1+uv), (1,1+u,1+u+v), (1,1+u,1+u+uv), (1,1+u,1+v+uv) 5 (0,u,uv), (1,1,uv), (1,u,1+uv), (1,u,1+u+v), (1,u,1+u+uv), (1,u,1+v+uv), (1,1+u,uv) 6 https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2707 (1,u,uv) 7 Definition 5.10. Let C be an additive code of length n over ℤ2Q. Its symmetrized weight enumerator is defined by SWEC (X, Y, Z, W, T, P, Q, N ) =CWEC (X, Y, Z, Z, T, Y, Y, W, Z, Z, Z, Z, W, W, W, Y, Y, Z, W, W,P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, W, T, P, P, N, T, T, Q, P, P, P, P, Q, Q, Q, T, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W ). where the Lee weight of the elements 0, 1, 2, 3, 4, 5, 6, and 7 is represented by the variables X, Y, Z, W, T, P, Q, and N , respectively. By definition, we have 𝑆𝑊𝐸𝐶(𝑋, 𝑌, 𝑍, 𝑊, 𝑇, 𝑃, 𝑄,𝑁) = ∑ 𝑋𝑛0(𝑐) 𝑐∈𝐶 𝑌𝑛1(𝑐)𝑍𝑛2(𝑐)𝑊𝑛3(𝑐)𝑇𝑛4(𝑐)𝑃𝑛5(𝑐)𝑄𝑛6(𝑐)𝑁𝑛7(𝑐) (5.5) where 𝑛0(𝑐) = 𝑤𝑓1 (𝑐), 𝑛1(𝑐) = 𝑤𝑓2 (𝑐) + 𝑤𝑓6 (𝑐) + 𝑤𝑓7 (𝑐) + 𝑤𝑓16 (𝑐) + 𝑤𝑓17 (𝑐) + 𝑤𝑓49 (𝑐) + 𝑤𝑓65 (𝑐), 𝑛2(𝑐) = 𝑤𝑓3 (𝑐) + 𝑤𝑓4 (𝑐) + 𝑤𝑓9 (𝑐) + 𝑤𝑓10 (𝑐) + 𝑤𝑓11 (𝑐) + 𝑤𝑓12 (𝑐) + 𝑤𝑓18 (𝑐) + 𝑤𝑓22 (𝑐) + 𝑤𝑓23 (𝑐) + 𝑤𝑓32 (𝑐) + 𝑤𝑓33 (𝑐) +𝑤𝑓50 (𝑐) + 𝑤𝑓54 (𝑐) + 𝑤𝑓55 (𝑐) + 𝑤𝑓64 (𝑐) + 𝑤𝑓66 (𝑐) + 𝑤𝑓70 (𝑐) + 𝑤𝑓71 (𝑐) + 𝑤𝑓80 (𝑐) + 𝑤𝑓81 (𝑐) + 𝑤𝑓113 (𝑐), 𝑛3(𝑐) = 𝑤𝑓8 (𝑐) + 𝑤𝑓13 (𝑐) + 𝑤𝑓14 (𝑐) + 𝑤𝑓15 (𝑐) + 𝑤𝑓19 (𝑐) + 𝑤𝑓20 (𝑐) + 𝑤𝑓25 (𝑐) + 𝑤𝑓26 (𝑐) + 𝑤𝑓27 (𝑐) + 𝑤𝑓28 (𝑐) + 𝑤𝑓34 (𝑐) +𝑤𝑓38 (𝑐) + 𝑤𝑓39 (𝑐) + 𝑤𝑓48 (𝑐) + 𝑤𝑓51 (𝑐) + 𝑤𝑓52 (𝑐) + 𝑤𝑓57 (𝑐) + 𝑤𝑓58 (𝑐) + 𝑤𝑓59 (𝑐) + 𝑤𝑓60 (𝑐) + 𝑤𝑓67 (𝑐) + 𝑤𝑓68 (𝑐) + 𝑤𝑓73 (𝑐) +𝑤𝑓74 (𝑐) + 𝑤𝑓75 (𝑐) + 𝑤𝑓76 (𝑐) + 𝑤𝑓82 (𝑐) + 𝑤𝑓86 (𝑐) + 𝑤𝑓87 (𝑐) + 𝑤𝑓90 (𝑐) + 𝑤𝑓97 (𝑐) + 𝑤𝑓114 (𝑐) + 𝑤𝑓118 (𝑐) + 𝑤𝑓119 (𝑐) + 𝑤𝑓128 (𝑐), 𝑛4(𝑐) = 𝑤𝑓5 (𝑐) + 𝑤𝑓24 (𝑐) + 𝑤𝑓29 (𝑐) + 𝑤𝑓30 (𝑐) + 𝑤𝑓31 (𝑐) + 𝑤𝑓35 (𝑐) + 𝑤𝑓36 (𝑐) + 𝑤𝑓41 (𝑐) + 𝑤𝑓42 (𝑐) + 𝑤𝑓43 (𝑐) + 𝑤𝑓44 (𝑐) +𝑤𝑓56 (𝑐) + 𝑤𝑓61 (𝑐) + 𝑤𝑓62 (𝑐) + 𝑤𝑓63 (𝑐) + 𝑤𝑓72 (𝑐) + 𝑤𝑓77 (𝑐) + 𝑤𝑓78 (𝑐) + 𝑤𝑓79 (𝑐) + 𝑤𝑓83 (𝑐) + 𝑤𝑓84 (𝑐) + 𝑤𝑓89 (𝑐) +𝑤𝑓90 (𝑐) + 𝑤𝑓91 (𝑐) + 𝑤𝑓92 (𝑐) + 𝑤𝑓98 (𝑐) + 𝑤𝑓102 (𝑐) + 𝑤𝑓103 (𝑐) + 𝑤𝑓112 (𝑐) + 𝑤𝑓115 (𝑐) + 𝑤𝑓116 (𝑐) + 𝑤𝑓121 (𝑐) + 𝑤𝑓122 (𝑐) +𝑤𝑓123 (𝑐) + 𝑤𝑓124 (𝑐), 𝑛5(𝑐) = 𝑤𝑓21 (𝑐) + 𝑤𝑓40 (𝑐) + 𝑤𝑓45 (𝑐) + 𝑤𝑓46 (𝑐) + 𝑤𝑓47 (𝑐) + 𝑤𝑓53 (𝑐) + 𝑤𝑓69 (𝑐) + 𝑤𝑓88 (𝑐) + 𝑤𝑓93 (𝑐) + 𝑤𝑓94 (𝑐) + 𝑤𝑓95 (𝑐) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2708 +𝑤𝑓99 (𝑐) + 𝑤𝑓100 (𝑐) + 𝑤𝑓105 (𝑐) + 𝑤𝑓106 (𝑐) + 𝑤𝑓107 (𝑐) + 𝑤𝑓108 (𝑐) + 𝑤𝑓120 (𝑐) + 𝑤𝑓125 (𝑐) + 𝑤𝑓126 (𝑐) + 𝑤𝑓127 (𝑐), 𝑛6(𝑐) = 𝑤𝑓37 (𝑐) + 𝑤𝑓85 (𝑐) + 𝑤𝑓104 (𝑐) + 𝑤𝑓109 (𝑐) + 𝑤𝑓110 (𝑐) + 𝑤𝑓111 (𝑐) + 𝑤𝑓117 (𝑐), 𝑛7(𝑐) = 𝑤𝑓101 (𝑐). Combining Theorem 5.5 and Definition 5.10, we have the following: Theorem 5.11. Suppose C is an additive code over ℤ2Q of length n. Then SWEC ⊥ (X, Y, Z, W, T, P, Q, N ) = 1 |𝑐| SWEC (S · (X, Y, Z, W, T, P, Q, N )t ), where 𝑆 = ( 1 7 21 35 35 21 7 1 1 5 9 5 −5 −9 −5 −1 1 3 1 −5 −5 1 3 1 1 1 −3 −3 3 3 −1 −1 1 −1 −3 3 3 −3 −1 1 1 −3 1 5 −5 −1 3 −1 1 −5 9 −5 −5 9 −5 1 1 −7 21 −35 35 −21 7 −1) Proof. Since C is an additive code over ℤ2Q, by Definition 5.10 and Theorem 5.5, we obtain SWEC ⊥ (X, Y, Z, W, T, P, Q, N ) = CWEC ⊥ (X, Y, Z, Z, T, Y, Y, W, Z, Z, Z, Z, W, W, W, Y, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, W, T, P, P, N, T, T, Q, P, P, P, P, Q, Q, Q, T, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W) = 1 |𝑐| CWEC{T · (X, Y, Z, Z, T, Y, Y, W, Z, Z, Z, Z, W, W, W, Y, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Y, Z, W, W, P, Z, Z, T, W, W, W, W, T, T, T, Z, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P, W, W, T, P, P, N, T, T, Q, P, P, P, P, Q, Q, Q, T, Z, W, T, T, Q, W, W, P, T, T, T, T, P, P, P,W )t } = 1 |𝑐| SWEC {X + 7Y + 21Z + 35W + 35T + 21P + 7Q + N, X + 4Y + 3Z + 2W − 2T − 3P − 4Q − N, X + 2Y − Z − 2W − 2T − P + 2Q + N, X − Y − Z − W + T + P + Q − N, X − Y − Z + W − T + P − Q + N, X − 2Y − Z − 2W+ 2T + P + 2Q − N, X − 4Y + 3Z https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2709 + 2W − 2T + 3P − 4Q + N, X − 7Y + 21Z − 35W + 35T − 21P + 7Q − N }. Hence, SWEC ⊥ (X, Y, Z, W, T, P, Q, N ) = 1 |𝑐| SWEC (S · (X, Y, Z, W, T, P, Q, N )t ). Example 5.12. Let C be the code in Example 5.6. Then SWEC (X, Y, Z, W, T, P, Q, N ) = X2 + 7Z2 + 7T2 + Q2 + 8XZ + 8XT + 2XQ + XN + XY+ 7XP + 7XW + 14ZT + 8ZQ + ZY + 7ZP + 7ZW + ZN+ 8TQ + T Y + 7T W + 7T P + T N + QY + 7QW + 7QP + QW. And SWEC ⊥ (X, Y, Z, W, T, P, Q, N ) = X2 + 7Z2 + 7T2 + Q2 + 8XT + 7ZP + XN + 8XZ + 14ZT+ 7XW + 7XP + 8ZQ + ZY + 2XQ + 7ZW + ZN + 8T Q+ 7T W + T Y + 7T P + T N + QY + 7QW + 7QP + QW + XY. Let c ∈ (Z2Q)n . The Lee weight of c is wtL (c) = n1(c) + 2n2(c) + 3n3(c) + 4n4(c) + 5n5(c) + 6n6(c) + 7n7(c), where ni (c) is defined in Equation 5.5 for 1 ≤ i ≤ 7. Definition 5.13. The Lee weight enumerator of an additive code C over ℤ2Q is given by 𝐿𝐶(𝑥, 𝑦) = ∑ 𝑥7𝑛−𝑤𝑡𝐿(𝑐) 𝑐∈𝐶 𝑦𝑤𝑡𝐿(𝑐). Theorem 5.14. Let C be a additive code of length n over ℤ2Q . Then LC (x, y) = SWEC (x7 , x6 y, x5 y2 , x4 y3 , x3 y4 , x2 y5 , xy6 , y7 ). Proof. Let c ∈ C. Then wt L (c) =∑ 𝑖7 𝑖=0 𝑛𝑖(𝑐) and n = ∑ 𝑛𝑖(𝑐) 7 𝑖=0 . By Definition 5.13, we have LC (x, y) =∑ 𝑥𝑐∈𝐶 7n−wtL (c) y wtL (c) = ∑ 𝑥7𝑛0(𝑐)+6𝑛1(𝑐)+5𝑛2(𝑐)+4𝑛3(𝑐)+3𝑛4(𝑐)+2𝑛5(𝑐)+6𝑛6(𝑐) 𝑐∈𝐶 𝑦𝑛1(𝑐)+2𝑛2(𝑐)+3𝑛3(𝑐)+4𝑛4(𝑐)+5𝑛5(𝑐)+6𝑛6(𝑐)+7𝑛7(𝑐) =∑ (𝑥7)𝑛0(𝑐) 𝑐∈𝐶 (𝑥6𝑦)𝑛1(𝑐)(𝑥5𝑦2)𝑛2(𝑐)(𝑥4𝑦3)𝑛3(𝑐)(𝑥3𝑦4)𝑛4(𝑐)(𝑥2𝑦5)𝑛5(𝑐)(𝑥𝑦6)𝑛6(𝑐). By Equation 5.5, LC (x, y) = SWEC (x7 , x6 y, x5 y2 , x4 y3 , x3 y4 , x2 y5 , xy6 , y7 ). Theorem 5.15. Let C be an additive code of length n over ℤ2Q. Then LC ⊥ (x, y) = 1 |𝑐| LC (x + y, x − y). https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2710 Proof. By Definition 5.13, LC ⊥ (x, y) =SWEC ⊥ (x7 , x6 y, x5 y2 , x4 y3 , x3 y4 , x2 y5 , xy6 , y7 ) = 1 |𝑐| SWEC (S ·(x7, x6 y, x5 y2 , x4 y3 , x3 y4 , x2 y5 , xy6 , y7 )t ) (By Theorem 5.11) = 1 |𝑐| SWEC ((x + y)7 , (x + y)6 (x − y), (x + y)5 (x − y)2 ,(x + y)4(x − y)3 ,(x +y)3 (x − y)4 , (x + y)2 (x − y)5 , (x + y)(x − y)6 , (x − y)7 ) = 1 |𝑐| LC (x + y, x − y) (By Theorem 5.14). Example 5.16. Suppose C is as in Example 5.6. Then by using Theorem 5.14, LC ⊥ (x, y) = SWEC ⊥ (x 7 , x6 y, x5 y2 , x4 y3 , x3 y4 , x2 y5 , xy6 , y7 ) = x14 + x13 y + 8x12 y2 + 8x11 y3 + 15x10y4 + 15x9 y5 + 16x8 y6 + 16x7 y7 + 15x6 y8 + 15x5 y9 + 8x4 y10 + 8x3 y11 + x2y12 + xy13 and LC (x + y, x − y) = SWEC ((x + y)7 , (x + y)6(x − y),(x + y)5(x − y)2 , (x + y)4 (x − y)3 , (x +y)3 (x − y)4 , (x + y)2 (x − y)5 , (x + y)(x − y)6 , (x − y)7 ) =128[x14 + x13 y + 8x12 y 2 + 8x11 y 3 + 15x10 y 4 + 15x9 y 5 + 16x8 y 6 +16x7y 7 + 15x6 y 8 + 15x5 y 9 + 8x4 y 10 + 8x3 y 11 + x2 y 12 + xy 13 ]. 6. 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