Explicit iterative constructions of normal bases and completely free elements in finite fields (Q1908792)

This interesting and well-written paper is rather technical and builds on earlier work by several different authors, especially the author's [Des. Codes Cryptography 4, 129-143 (1994; Zbl 0823.11071)]. We do not see how to improve upon the author's summary. ``A characterization of normal bases and complete normal bases in \(\text{GF}(q^{r^n})\) over \(\text{GF}(q)\), where \(q> 1\) is any prime power, \(r\) is any prime number different from the characteristic of \(\text{GF}(q)\), and \(n\geq 1\) is any integer, leads to a general construction scheme of series \((v_n)_{n\geq 0}\) in \(\text{GF}(q^{r^\infty}):= \bigcup_{n\geq 0} \text{GF}(q^{r^n})\) having the property that the partial sums \(w_n:= \sum^n_{i:= 0} v_i\) are free or completely free in \(\text{GF}(q^{r^n})\) over \(\text{GF}(q)\), depending on the choice of \(v_n\). In the case where \(r\) is an odd prime divisor of \(q- 1\) or where \(r= 2\) and \(q\equiv 1\text{ mod } 4\), for any integer \(n\geq 1\), all free and completely free elements in \(\text{GF}(q^{r^n})\) over \(\text{GF}(q)\) are explicitly determined in terms of certain roots of unity. In the case where \(r= 2\) and \(q\equiv 3\text{ mod } 4\), for any \(n\geq 1\), in terms of certain roots of unity, an explicit recursive construction for free and completely free elements in \(\text{GF}(q^{2^n})\) over \(\text{GF}(q)\) is given. As an example, for a particular series of completely free elements the corresponding minimal polynomials are given explicitly''.