Weight Enumerators of codes over $\mathbb{F}_2$ and over $\mathbb{Z}_4$

TL;DR

This paper investigates symmetrized weight enumerators of pairs of Type II codes over finite fields and rings, revealing their connection to invariant rings and algebraic structures in specific cases.

Contribution

It establishes conditions under which the symmetrized weight enumerator rings coincide with the invariant rings of certain groups for Type II codes.

Findings

Invariant rings match symmetrized weight enumerator rings in specific degrees

Deep relationship between invariant ring structure and code algebraic properties

Results enhance understanding of code invariants over finite fields and 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.