Weight Enumerators of codes over $\mathbb{F}_2$ and over $\mathbb{Z}_4$
A. K. M. Selim Reza, Manabu Oura, Nur Hamid
TL;DR
The paper investigates whether symmetrized weight enumerators of Type II codes over $\mathbb{F}_2$ and $\mathbb{Z}_4$ realize the invariant ring of a finite group. It builds an algebraic framework using a group action on polynomial rings and employs invariant-theory tools (Molien series, Hironaka decomposition) to analyze degree-8 and degree-16 invariants. The main results show that the degree-8 invariant space is generated by two weight enumerators from $(\mathscr{E}_8,\mathscr{Q}_8)$ and $(\mathscr{E}_8,\mathscr{K}_8)$, while six independent degree-16 invariants—constructed from direct sums and higher-order codes like $\mathscr{D}_{16}$ and $\mathscr{K}_{16}$—form a basis for the corresponding subspace. Computational verification with Sage and Magma confirms independence and identifies invariant bases, highlighting a deep link between weight enumerators and invariant rings.
Abstract
Weight enumerators are important tools for deciphering the algebraic structure of the related code spaces and for understanding group actions on these spaces. Our study focuses on symmetrized weight enumerators of pairs of Type II codes over the finite field $\mathbb{F}_{2}$ and the ring $\mathbb{Z}_{4}$. These pairs have been examined as invariants for a specified group. In particular, we concentrate on the scenarios where the space of the invariant ring is of degree 8 and 16. Our findings show that in certain situations, the ring produced by the symmetrized weight enumerators precisely matches with the invariant ring of the designated group. This coincidence points to a profound relationship between the invariant ring's structure and the algebraic characteristics of the weight enumerators.