BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$

Authors : Virgilio P SISON

Pages : 206-219

Doi:10.24330/ieja.768265

View : 13 | Download : 12

Publication Date : 2020-07-14

Article Type : Research Paper

Abstract :Let $GRinsert ignore into journalissuearticles values(p^r,m);$ denote the Galois ring of characteristic $p^r$ and cardinality $p^{rm}$ seen as a free module of rank $m$ over the integer ring $\mathbb{Z}_{p^r}$. A general formula for the sum of the homogeneous weights of the $p^r$-ary images of elements of $GRinsert ignore into journalissuearticles values(p^r,m);$ under any basis is derived in terms of the parameters of $GRinsert ignore into journalissuearticles values(p^r,m);$. By using a Vandermonde matrix over $GRinsert ignore into journalissuearticles values(p^r,m);$ with respect to the generalized Frobenius automorphism, a constructive proof that every basis of $GRinsert ignore into journalissuearticles values(p^r,m);$ 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.
Keywords : Galois ring, Vandermonde matrix, dual basis, normal basis