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

The Galois ring GR(p,m) of characteristic p and cardinality p, where p is a prime and r,m ≥ 1 are integers, is a Galois extension of the residue class ring Zpr by a root ω of a monic basic irreducible polynomial of degree m over Zpr . Every element of GR(p ,m) can be expressed uniquely as a polynomial in ω with coefficients in Zpr and degree less than or equal to m − 1, thus GR(p,m) is a free module of rank m over Zpr with basis {1, ω, ω, . . . , ω}. The ring Zpr satisfies the invariant dimension property, hence any other basis of GR(p,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-image of a linear block code over GR(p,m) with respect to any basis. It would be interesting to consider the dual and normal bases of GR(p,m). By using a Vandermonde matrix over GR(p,m) in terms of the generalized Frobenius automorphism, a constructive proof that every basis of GR(p,m) has a unique dual basis is given. The notion of normal bases was also generalized from the classic case for Galois fields.