Generalizations of the Normal Basis Theorem of Finite Fields

We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with OSri~"_2, " such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range~..0 ~ ~ 2 l. .-1 zero. Then, there exists an element ae GF(q") such that a ,a 9,a 9 , ... ,a .-19 fonn a basis of GF (q") over GF (q). This result, which includes the nonnal basis theorem as a particular case when ~Al= .•. =A.,.-1=1, is proved for all choices of Ao,)..lt .•. ,)..11-1 satisfying the above conditions when n is odd, and for more restricted sets of values ,(A.;) when n is even.