Almost weakly self-dual bases for finite fields (Q1352414)

It is well-known that polynomial self-dual bases for a finite field \(\text{GF}(q^n)\) over the ground field \(\text{GF}(q)\) never exist (which is unfortunate, since such bases would be very useful in constructing hardware devices for finite field multiplication called ``bit serial dual basis multipliers''). This fact has prompted the study of more general notions, in particular, that of weakly self-dual bases; see Chapter 4 of [\textit{D. Jungnickel}, Finite fields: structure and arithmetics, B. I.-Wissenschaftsverlag (1993; Zbl 0779.11058)] for background. The authors study a further generalization, namely almost weakly self-dual polynomial bases; essentially, these correspond to the polynomial bases associated with irreducible trinomials. Their results finish the classification of the polynomial bases with excess \(0\); the paper also contains some results on bases with larger excess.