Bases of the Galois Ring $GR(p^r,m)$ over the Integer Ring $Z_{p^r}$

TL;DR

This paper explores the algebraic structure of Galois rings over integer residue rings, providing a basis and dual basis construction, with implications for coding theory and generalizations of normal bases.

Contribution

It offers a constructive proof of the uniqueness of dual bases in Galois rings and generalizes the concept of normal bases from Galois fields.

Findings

Every basis of $GR(p^r,m)$ has a unique dual basis.

A Vandermonde matrix over $GR(p^r,m)$ is used for basis construction.

Normal bases are generalized from Galois fields.

Abstract

The Galois ring $GR(p^r,m)$ of characteristic $p^r$ and cardinality $p^{rm}$, where $p$ is a prime and $r,m \ge 1$ are integers, is a Galois extension of the residue class ring $Z_{p^r}$ by a root $\omega$ of a monic basic irreducible polynomial of degree $m$ over $Z_{p^r}$. Every element of $GR(p^r,m)$ can be expressed uniquely as a polynomial in $\omega$ with coefficients in $Z_{p^r}$ and degree less than or equal to $m-1$, thus $GR(p^r,m)$ is a free module of rank $m$ over $Z_{p^r}$ with basis $\{1,\omega, \omega^2,..., \omega^{m-1} \}$. The ring $Z_{p^r}$ satisfies the invariant dimension property, hence any other basis of $GR(p^r,m)$, if it exists, will have cardinality $m$. This paper was motivated by the code-theoretic problem of finding the homogeneous bound on the $p^r$-image of a linear block code over $GR(p^r,m)$ with respect to any basis. It would be interesting to consider the dual and normal bases of $GR(p^r,m)$. By using a Vandermonde matrix over $GR(p^r,m)$ in terms of the generalized Frobenius automorphism, a constructive proof that every basis of $GR(p^r,m)$ has a unique dual basis is given. The notion of normal bases was also generalized from the classic case for Galois fields.