A Sufficient and Necessary Condition for the Finite Field Which Has a Weakly Self-dual Normal Basis
Sun Qi · Chinese Annals of Mathematics · 2007
This paper expands self-dual bases to general weakly self-dual bases and gets a sufficient and necessary condition for the finite field which has a weakly self-dual normal basis as the following: Let q be the power of a prime, E = Fqn be the n-dimensional extension of the finite field F = Fq, and N = {αi=αqi|i= 0,1,…, n -1} be a normal basis of E over F. Then there exist some c∈F* and some r, 0≤r≤n-1 such thatβ= cαr generates the dual of N if and only if either q is even and n≠0 (mod 4); or n and q are odd; or q is odd, n is even, (-1) is a nonsquare in F and r is odd.