The complexity of the dual basis of a type I optimal normal basis of Fqn over Fq was determined to be 3n − 3 or 3n − 2 according as q is even or odd, respectively, by Z.-X. Wan and K. Zhou in 2007. We give a new proof to this result by clearly deriving the dual of a type I optimal basis with the aid of a lemma on the dual of a polynomial basis.

Computing the Discrete Fourier Transform (DFT) of signals over some nite eld Fq often requires an extension to a large eld Fqn containing an appropriate primitive root of unity. The Algebraic Discrete Fourier Transforms (ADFTs) avoid the extension of the base eld Fq and can be used to compute the spectrum of the DFT. We derive a complete parametrization of ADFT matrices and show how this knowle...

A new construction is given of cyclic semifields of orders q2n, n odd, with kernel (left nucleus) Fqn and right and middle nuclei isomorphic to Fq2 , and the isotopism classes are determined. Furthermore, this construction is generalized to produce potentially new semifields of the same general type that are not isotopic to cyclic semifields. In particular, a new semifield plane of order 45 and...

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for Fqn over Fq, where r is a prime and nk = r− 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = φ(r), and find in many cases smaller values of k than i...

In this paper we give some experimental results on the multiplicative orders of optimal normal basis generators in F2n over F2 for n < 1200 whenever the complete factorization of 2" 1 is known. Our results show that a subclass of optimal normal basis generators always have high multiplicative orders, at least 0((2" \)/n), and are very often primitive. For a given optimal normal basis generator ...

The Chevalley–Warning Theorem is a result on the solution set of system polynomial equations f1,…,fr in n variables over finite field Fq low degree case d≔∑j=1rdeg(fj)<n. In this note we reformulate that terms fibers associated map and, following Heath-Brown, show something weaker continues to hold when d=n. This invites search for homogeneous polynomials which function Fqn→Fq not surjective, a...