Note on E-polynomials associated to $\mathbb{Z}_4$-codes

TL;DR

This paper explores the algebraic structure of E-polynomials linked to $bZ_4$-codes, identifying minimal generators and invariant rings, thereby connecting coding theory with invariant theory and number theory.

Contribution

It determines the minimal generators of E-polynomial rings related to $bZ_4$-codes and describes the invariant rings associated with Type II $bZ_4$-codes.

Findings

Identified minimal generators of E-polynomial rings for $bZ_4$-codes.

Described generators of invariant rings from E-Polynomials and complete weight enumerators.

Connected coding theory with invariant theory through algebraic structures.

Abstract

The invariant theory of finite groups can connect the coding theory to the number theory. In this paper, under this conformity, we obtain the minimal generators of the rings of E-polynomials constructed from the groups related to $\mathbb{Z}_4$-codes. In addition, we determine the generators of the invariant rings appearing by E-Polynomials and complete weight enumerators of Type II $\mathbb{Z}_4$-codes.