BASES an‎d AUTOMORPHISM MATRIX OF THE GALOIS RING G R ( p r , m ) GR(p ​r ​​ ,m) OVER Z p r Z ​p ​r ​​ ​​

Let GR(p r , m) denote the Galois ring of characteristic p r and cardinality p rm seen as a free module of rank m over the integer ring Zpr . A general formula for the sum of the homogeneous weights of the p r -ary images of elements of GR(p r , m) under any basis is derived in terms of the parameters of GR(p r , m). By using a Vandermonde matrix over GR(p r , m) with respect to the generalized Frobenius automorphism, a constructive proof that every basis of GR(p r , m) has a unique dual basis is given. It is shown that a basis is self-dual if and only if its automorphism matrix is orthogonal, and that a basis is normal if and only if its automorphism matrix is symmetric