On the self-dual normal bases and their distribution

In this paper, we discuss the correlation between the self-dual normal bases of FqvFqv and FqtFqt over FqFq with those of FqnFqn over FqFq, where n=vtn=vt and (v,t)=1(v,t)=1. In particular, we prove that if α and β   generate self-dual normal bases of FqvFqv and FqtFqt respectively over FqFq, then γ=αβγ=αβ generates a self-dual normal basis of FqnFqn over FqFq. We also explore the other direction and prove that if γ=αβγ=αβ (where α∈Fqvα∈Fqv, β∈Fqtβ∈Fqt and γ∈Fqnγ∈Fqn) generates a self-dual normal basis of FqnFqn over FqFq, then both α and β   generate trace-orthogonal normal bases. We also provide the possibility of a relation between the number of self-dual normal bases of FqnFqn over FqFq and those of FqmFqm over FqFq, where n=ptmn=ptm with (m,q)=1(m,q)=1.