On the Representation of the Weight Enumerator of <img src=image/13424212_01.gif>

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.