Self-dual bases in \(\mathbb{F}_{q^ n}\)
Let \(F_{q^ n}\) denote the finite field of order \(q^ n\) so that \(F_{q^ n}\) may be viewed as a vector space of dimension \(n\) over \(F_ q\). Given a basis \(A=\{\alpha_ 0,\dots, \alpha_{n-1}\}\) of \(F_{q^ n}\) over \(F_ q\), the dual basis is given by \(\{\beta_ 0,\dots,\beta_{n-1}\}\) where \(\text{tr}(\alpha_ i\beta_ j)= \delta_{ij}\) for \(0\leq i,j<n\), where \(\text{tr}(\;)\) denotes the trace function. The basis \(A\) is self-dual if \(\beta_ i= \alpha_ i\), \(0\leq i<n\) and it is said to be weakly self-dual if there exists a \(\gamma\in F_{q^ n}\) and a permutation \(\pi\) of the indices \(\{0,1,\dots,n-1\}\) so that \(\beta_ i= \gamma\alpha_{\pi(i)}\) for all \(0\leq i<n\). A polynomial basis is a basis of the form \(\{1,\alpha,\dots, \alpha^{n- 1}\}\) where \(\alpha\) is a root of an irreducible of degree \(n\) over \(F_ q\). Weakly self-dual polynomial bases are studied and the following important and useful characterization is proved: The field \(F_{q^ n}\) has a weakly self-dual polynomial basis if and only if there exists an irreducible trinomial \(p(x)= x^ n- cx^ k -1\) or an irreducible binomial \(p(x)= x^ n -d\). The permutation \(\pi\) yielding the dual basis is given by \(\pi(i)= k-1-i\pmod n\) (if \(p(x)\) is a binomial, set \(k=0\)). Weakly self-dual normal bases are also considered and applications of weakly self-dual bases to dual basis multipliers are discussed.
scientific article; zbMATH DE number 474717
scientific article