Optimal normal bases

Let K L be a finite Galois extension of fields, of degree n. Let G be the Galois group, and let (<)_<G be a normal basis for L over K. An argument due to Mullin, Onyszchuk, Vanstone and Wilson (Discrete Appl. Math. 22 (1988/89), 149–161) shows that the matrix that describes the map x x on this basis has at least 2n - 1 nonzero entries. If it contains exactly 2n - 1 nonzero entries, then the normal basis is said to be optimal. In the present paper we determine all optimal normal bases. In the case that K is finite our result confirms a conjecture that was made by Mullin et al. on the basis of a computer search.