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

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 $ω$ 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 $ω$ 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 $\{1,ω, ω^2,..., ω^{m-1} \}$. 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.