On the Representation of the Weight Enumerator of 𝒅 𝒏+

The weight enumerator of a code is a homogeneous polynomial that provides a lot of information about the code. In this case, for the development of a code, research on the weight enumerator is very important. In this study, we focus on the code d n+ . Let W d n+ (x, y) be the weight enumerator of the code d n+ . Fujii and Oura showed that W d n+ (x, y) is generated by W d 8+ (x, y) and W d 24+ (x, y) . Indeed, we show that W d n+ (x, y) is an element of the polynomial ring ℤ[ 1 24 ][W d 8+ (x, y), W d 24+ (x, y)] . We know that the weight enumerator of all self-dual double-even (Type II) code is generated by W d 8 + ( x, y ) and 𝜑 24 ( 𝑥, 𝑦 ) = x 4 y 4 ( x 4 − y 4 ) 4 . Recall d n+ is a type II code. Thus, W d n+ (x, y) is an element of the polynomial ring ℤ[W d 8+ (x, y), 𝜑 24 (𝑥, 𝑦)] and ℤ[ 1 24 ][W d 8+ (x, y), W d 24+ (x, y)] . One of the motivations of this research is to investigate the connection between these two polynomial rings in representing W d n+ (x, y) . Let 𝑎 𝑖 and 𝑏 𝑖 be the coefficients of polynomial that represent W d n+ (x, y) as an element of ℤ[W d 8+ (x, y), 𝜑 24 (𝑥, 𝑦)] and ℤ[ 1 24 ][W d 8+ (x, y), W d 24+ (x, y)] , respectively. We find that 𝑏 𝑖 is an element of the polynomial ℤ [ 1 24 ] [𝑎 𝑖 ] . In addition, we also show that there are no weight enumerators of Type II code generated by W d 8+ (x, y) and φ 24 (x, y) that can be written uniquely as isobaric polynomials in five homogeneous polynomial elements of degrees 8, 24, 24, 24, 24.

We know that the code + is a type II code [11]- [15]. Thus, it is clear that the weight enumerator of the code + ,  [19]. Previously, in genus 2 showed that there exist no weight enumerators of Type II code could be written uniquely as an isobaric polynomial in homogeneous polynomials of degrees 8, 24, 24, 32, 40. Theorem 2.1 in this paper shows that, in genus 1, there is also no weight enumerator of Type II code that can be written uniquely as isobaric polynomials in five homogeneous polynomial elements of degrees 8, 24, 24, 24, 24. where ∈ and = for all but a finite number of values of . If for some ≥ it is true that ≠ , the largest such value of is the degree of ( ). If all = , then the degree of ( ) is undefined.

Result and Discussion
Furthermore, for ≥ ≥ 0, we will investigate the coefficient of ( 8 + ) 8 −3 ( 24 + ) on the right-hand side of (2). We consider that the coefficient of is (−1) − ( ). As a result, we get the coefficient of ] since ∈ ℤ. This completes the proof. Generally, based on Theorem 2.1, in terms of finding the value of b i for the same n, we need to construct as many as k + 1 equations. Meanwhile, to find the value of a i , we only need as many as k equations. Thus, to find the value of b i will be more efficient when using the value of a i , since b i depends on a i . It is clear that we can only use a 1 and a 2 to find b 0 , b 1 , dan b 2 . We can simply construct and solve a system of equations with only two variables based on (4) ; 1 = 49 9 ; dan 2 = 28 9 We need to show that ∈ ℤ for all ∈ ℕ. We next use induction on . Based on theorem 1, for = , we have 1 = 24 = ∈ ℤ. So, the statement is true for = 1. Next, suppose it is true for = . Then, we have

Lemma 3.2
There exist no weight enumerators of Type II code generated by W d 8 + and 24 that can be written uniquely as isobaric polynomial in five homogeneous polynomial elements of degrees 8, 24, 24, 24, 24.
Proof. Suppose there exists such a weight enumerator of a Type II code (call it code * ). Then we have a representation of weight enumerator of the code * as a polynomial in W d 8 + and 24 and it can be written as * ( , ) = ∑ =0 W d 8 + 8 −3 24 , Substitute (6) and (8)  Which also means 1 = 2 = 3 = ⋯ = −1 = = 0. Now, we have a claim that the weight enumerator of code C is not generated by 24 . So, we get a contradiction. This completes the proof of the Theorem. Here we have claim that the weight enumerator of the code d 40 + has at least two different representations as isobaric polynomials in five elements of polynomials of degree 8, 24, 24, 24, and 24.

Conclusions
In this paper, we have shown that the weight enumerator of the code + is an element of ring ℤ[ representing W d n + were also presented in this paper.