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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 theresidue class ring $Z_{p^r}$ by a root $\omega$ of a monic basic irreduciblepolynomial of degree $m$ over $Z_{p^r}$. Every element of $GR(p^r,m)$ can beexpressed 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 ofrank $m$ 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 otherbasis of $GR(p^r,m)$, if it exists, will have cardinality $m$.  This paper was motivated by the code-theoretic problem of finding thehomogeneous 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 andnormal bases of $GR(p^r,m)$.  By using a Vandermonde matrix over $GR(p^r,m)$ in terms of the generalizedFrobenius 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 alsogeneralized from the classic case for Galois fields.

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