Optimal Normal Bases

Let K⊂L be a finite Galois extension of fields, of degree n. Let G be the Galois group, and let (σα)σ∈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 x7→αx on this basis has at least 2n−1 non-zero entries. If it contains exactly 2n−1 non-zero 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.