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

Year 2020, Volume: 28 Issue: 28, 206 - 219, 14.07.2020

Virgilio P. Sıson

https://doi.org/10.24330/ieja.768265

Abstract

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 $\mathbb{Z}_{p^r}$. 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.

Keywords

Galois ring, Vandermonde matrix, dual basis, normal basis

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