Abstract
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 <img src=image/13424212_13.gif>. Let <img src=image/13424212_02.gif> be the weight enumerator of the code <img src=image/13424212_13.gif>. Fujii and Oura showed that <img src=image/13424212_02.gif> is generated by <img src=image/13424212_03.gif> and <img src=image/13424212_04.gif>. Indeed, we show that <img src=image/13424212_02.gif> is an element of the polynomial ring <img src=image/13424212_05.gif>. We know that the weight enumerator of all self-dual double-even (Type II) code is generated by <img src=image/13424212_03.gif> and <img src=image/13424212_06.gif>. Recall <img src=image/13424212_13.gif> is a type II code. Thus, <img src=image/13424212_02.gif> is an element of the polynomial ring <img src=image/13424212_07.gif> and <img src=image/13424212_08.gif>. One of the motivations of this research is to investigate the connection between these two polynomial rings in representing <img src=image/13424212_02.gif>. Let <img src=image/13424212_09.gif> and <img src=image/13424212_10.gif> be the coefficients of polynomial that represent <img src=image/13424212_02.gif> as an element of <img src=image/13424212_07.gif> and <img src=image/13424212_08.gif>, respectively. We find that <img src=image/13424212_10.gif> is an element of the polynomial <img src=image/13424212_11.gif>. In addition, we also show that there are no weight enumerators of Type II code generated by <img src=image/13424212_03.gif> and <img src=image/13424212_12.gif> that can be written uniquely as isobaric polynomials in five homogeneous polynomial elements of degrees 8, 24, 24, 24, 24.
Highlights
Let C be a linear code of length n
We show that Wdn+ (x, y) is an element of the polynomial ring Z[214][Wd8+ (x, y), Wd2+4(x, y)]
It is known that the weight enumerator of all Type II codes can be written uniquely as isobaric polynomials in Wd8+ (x, y) = x8 + 14x4y4 + y8 and φ24(x, y) = x4y4(x4 − y4)4 with integer coefficients [1]-[4]
Summary
Let C be a linear code of length n. It is known that the weight enumerator of all Type II codes can be written uniquely as isobaric polynomials in Wd8+ (x, y) = x8 + 14x4y4 + y8 and φ24(x, y) = x4y4(x4 − y4) with integer coefficients [1]-[4]. 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 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
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