Let q be an odd prime power and n integer. ??Fqn[x] a q-linearized t-scattered polynomial of linearized degree r. d=max?{t,r} number. In this paper we show that under these assumptions it follows ?=x. Our technique involves Galois theoretical characterization polynomials combined with the classification transitive subgroups general linear group over finite field Fq.

Based on the twisted Gabidulin codes obtained recently by Sheekey, we construct a new family of maximal rank distance codes as a set of qpolynomials over Fqn , which includes the generalized Gabidulin codes and the twisted Gabidulin codes. Their Delsarte duals and adjoint codes are investigated. We also obtain necessary and sufficient conditions for the equivalence between two members of our ne...

We interpret the finite Zariski tangent spaces Ta(X,Fqm) to an affine variety X ⊂ Fq n as linear codes. If the degree ofX is relatively prime to the characteristic p = charFq, we show that the minimum distance of Ta(X,Fqm) stabilizes while varying a ∈ X(Fqm) := X ∩ Fnqm and m ∈ N. The article provides a pseudocode for obtaining the generic minimum distance of a tangent code and points, at which...

Let X be an algebraic variety over a finite field Fq, homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N , the number of rational points of X over Fqn is a polynomial function of q with integer coefficients. Moreover, the shifted polynomials, where q is formally replaced with q +1, have non-negat...

The present paper is interested in a family of normal bases, considered by V. M. Sidel’nikov, with the property that all the elements in a basis can be obtained from one element by repeatedly applying to it a linear fractional function of the form φ(x) = (ax + b)/(cx + d), a, b, c, d ∈ Fq. Sidel’nikov proved that the cross products for such a basis {αi} are of the form αiαj = ei−jαi+ ej−iαj +γ,...

Let N and n be integers such that n|N and 2 < n < N . Let q be a prime power. The difference sets introduced by Gordon, Mills and Welch in [1] produce symmetric designs with the same parameters v = (q N −1)/(q−1), k = (q N−1−1)/(q−1), λ = (q N−2−1)/(q−1) as the point-hyperplane design of PG(N − 1, q). The purpose of this note is to prove that inequivalent difference sets of this sort produce no...

We study graphs coming from quadratic spaces over finite fields which generalize a recent result given by Bishnoi, Ihringer, and Pepe (2020). More precisely, we the graph Γ□(n,k,q) defined as follows: vertex set is of k-dimensional subspaces fixed space (Fqn,x12+⋯+xn−12+λxn2) isometrically isomorphic to x12+⋯+xk2. Here λ non-square in Fq, two vertices x,y are adjacent if x⊆y⊥.