Abstract
LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.
Highlights
Normal basis is one important type of basis over Galois fields, because it is computationally manageable
Some researcher are interested in finding a way to construct self-dual normal basis over Galois fields; see [3, 4]
Normal basis and its variants such as self-dual are important in Galois ring, especially for computations in codes over this ring [2, 4]
Summary
Normal basis is one important type of basis over Galois fields, because it is computationally manageable. One type of normal basis which has applications in cryptography and coding theory is self-dual normal basis; see [1, 2]. As a generalization of Galois fields, Galois rings have several connections with coding theory; see [5, 6]. Normal basis and its variants such as self-dual are important in Galois ring, especially for computations in codes over this ring [2, 4]. We will show some properties of self-dual normal basis of a Galois ring similar to the properties of self-dual normal basis over finite fields. We give an application of normal basis, especially self-dual normal basis, in encoding certain cyclic codes over Zpk
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