Bases of the Galois Ring GR(pr, m)over the Integer Ring Zpr

Issue Date

10-2014

Abstract

The Galois ring GR(pr, m)of characteristic prand cardinality prm,where pis a prime and r, m ≥1are integers, is a Galois extension of theresidue class ring Zprby a root ωof a monic basic irreducible polynomial ofdegree mover Zpr. Every element of GR(pr, m)can be expressed uniquelyas a polynomial in ωwith coefficients in Zprand degree less than or equalto m−1, thus GR(pr, m)is a free module of rank mover Zprwith basis{1, ω, ω2, . . . , ωm−1}. The ring Zprsatisfies the invariant dimension prop-erty, hence any other basis of GR(pr, m), if it exists, will have cardinalitym.This paper was motivated by the code-theoretic problem of finding thehomogeneous bound on the pr-image of a linear block code over GR(pr, m)with respect to any basis. It would be interesting to consider the dual andnormal bases of GR(pr, m).By using a Vandermonde matrix over GR(pr, m)in terms of the gen-eralized Frobenius automorphism, a constructive proof that every basis ofGR(pr, m)has a unique dual basis is given. The notion of normal bases wasalso generalized from the classic case for Galois fields.Keywords – Galois rings, trace function, Frobenius automorphism, Vander-monde matrix, dual basis, normal basis.

Source or Periodical Title

International Electronic Journal of Algebra

Volume

28

Page

206-219

Document Type

Article

Physical Description

figures; references

Language

English

Subject

Dual basis, Galois ring, Normal basis, Vandermonde matrix

Identifier

DOI: 10.24330/ieja.768265

Digital Copy

yes

Link to Full Text

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