A new characterization of dual bases in finite fields and its applications

We propose a new characterization of dual bases in finite fields. Let A=(α1,…,αn)A=(α1,…,αn) be a basis of FF over FqFq and its dual basis B=(β1,…,βn)B=(β1,…,βn) with the transition matrix C∈GLn(Fq)C∈GLn(Fq) such that (β1,…,βn)=(α1,…,αn)C(β1,…,βn)=(α1,…,αn)C. We show that TkT=C-1TkC holds for all 1⩽k⩽n1⩽k⩽n, where Tk∈Mn(Fq)Tk∈Mn(Fq) satisfies αk(α1,…,αn)=(α1,…,αn)Tkαk(α1,…,αn)=(α1,…,αn)Tk. Conversely, suppose F=Fq(αk′)F=Fq(αk′) and Tk′T=G-1Tk′G for some 1⩽k′⩽n1⩽k′⩽n and G∈GLn(Fq)G∈GLn(Fq), then B   is equivalent to (α1,…,αn)G(α1,…,αn)G. As applications, we can construct the dual basis of a given basis AA or determine whether the dual basis of AA satisfies the desired conditions from TkTk. This generalizes the results obtained by Liao and Sun for normal bases. Furthermore, we give a simple proof of the theorem of Gollmann, Wang and Blake for polynomial bases.