For q = p m and m ≥ 1, we construct systematic authentication codes over finite field F q using Galois rings. We give corrections of the construction of [2]. We generalize corresponding systematic authentication codes of [6] in various ways.

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Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider [Formula: see text]-Fibonacci cubes, which we obtain as subgraphs of Fibonacci cubes by eliminating certain edges during the fundamental recursion phase of their construction. These graphs have the same number of vertices as Fibonacci cubes, but their edge sets are determined by a parameter [Formula: see text]. We obtain properties of [Formula: see text]-Fibonacci cubes including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain.

In this work we present explicit classes of maximal and minimal Artin-Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1] at all.

In this article we give an explicit formula for the number of k-normal elements, hence answering Problem 6.1. of Huczynska et al. (Existence and properties of k-normal elements over finite fields, Finite Fields Appl 2013; 24: 170-183). Furthermore, for some cases we provide formulas that require the solutions of some linear Diophantine equations. Our results depend on the explicit factorization of cyclotomic polynomials.