Self-Dual Normal Basis of a Galois Ring

Let and 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 for over , where is considered as a free module over . 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 for over and the set of all invertible, circulant, and orthogonal matrices over . 1. Introduction 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]. Therefore, some researcher are interested in finding a way to construct self-dual normal basis over Galois fields; see [3, 4]. On the other hand, as a generalization of Galois fields, Galois rings have also several connections with coding theory; see [5, 6]. Normal basis and its variants such as self-dual are also important in Galois ring, especially for computations in codes over this ring [2, 4]. Given two Galois rings and , where is an extension of , a normal basis for over is a basis which consists of all orbit of some from the action of into , where is Galois group of over . As a consequence of the work of Kanzaki [7] and the fact that Galois ring is a local ring [8, Lemma 2], there always exists normal basis for over . In this paper, 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 will also explain how to get normal basis generator and self-dual normal basis generator and show a connection between the set of generators of self-dual normal basis and the set of invertible circulant and orthogonal matrices over . Moreover, we give an application of normal basis, especially self-dual normal basis, in encoding certain cyclic codes over . 2. Normal Basis over Galois Ring 2.1. Some Properties of Normal Basis over Galois Ring Let be a Galois ring with characteristic and cardinality . If and , then is a subring of . Moreover, if is a basic primitive irreducible polynomial with degree , then is a Galois ring with characteristic and cardinality and contains as a subring. In other words, is an extension of of order . If is a root of and has order , then . Define a map where for every . We can prove that is an automorphism which left fixed. Furthermore, generates the Galois group of order [8]. Some detailed explanations about Galois ring