On Self-Dual Bases of the Extensions of the Binary Field

There are at least two points of view when representing elements of $\mathbb F_{2^n}$, the field of 2n elements. We could represent the (nonzero) elements as powers of a generating element, the exponent ranging from 0 to 2n–2. On the other hand, we could represent the elements as strings of n bits. In the former representation, multiplication becomes a very easy task, whereas in the latter one, addition is obvious. In this note, we focus on representing $\mathbb F_{2^n}$ as strings of n bits in such a way that the natural basis (1,0,...,0), (0,1,...,0), ..., (0,0,...,1) becomes self-dual. We also outline an idea which leads to a very simple algorithm for finding a self-dual basis. Finally we study multiplication tables for the natural basis and present necessary and sufficient conditions for a multiplication table to give $\mathbb F_2^n$ a field structure in such a way that the natural basis is self-dual.