On the number of self-dual bases of 𝐺𝐹(𝑞^{𝑚}) over 𝐺𝐹(𝑞)

Abstract

Let E = G F ( q m ) E = GF({q^m}) be the m m -dimensional extension of F = G F ( q ) F = GF(q) . We are concerned with the numbers s d ( m , q ) sd(m,q) and s d n ( m , q ) sdn(m,q) of self-dual bases and self-dual normal bases of E E over F F , respectively. We completely determine s d ( m , q ) sd(m,q) , en route giving a very simple proof for the Sempel-Seroussi theorem which states that s d ( m , q ) = 0 sd(m,q) = 0 iff q q is odd and m m is even. Using results of Lempel and Weinberger and MacWilliams, we can also determine s d n ( m , p ) sdn(m,p) for primes p p .