Optimal normal bases (Q1203948)

Let \(K\subset L\) be a finite Galois extension of fields, \(n\) the degree of the extension, and \(G\) the Galois group. A basis of \(L\) over \(K\) is called normal if it is of the form \((\sigma(\alpha))_{\sigma\in G}\) for some \(\alpha\in L\). The matrix that describes the map \(x\mapsto\alpha x\) on this basis has at least \(2n-1\) non zero-entries [\textit{R. C. Mullin}, \textit{I. M. Onyszchuk}, \textit{S. A. Vanstone} and \textit{R. M. Wilson}, Discrete Appl. Math. 22, 149-161 (1989; Zbl 0661.12007)]; in the case of equality, the normal basis is called optimal. In the paper, all optimal normal bases are determined. It is shown that the constructions from (loc. cit.) exhaust all optimal normal bases. The dual basis of \((\sigma(\alpha))_{\sigma\in G}\) plays an important role.