Cyclically Covering Subspaces In Finite Fields

Let Fq be a finite field with q elements,Fqn be a n dimensional extension of Fq,Fqn be a row vector space with dimension n over Fq,where q is a power of the prime.For n ∈ N,let {e0,e1,…,en-1} be the standard basis for Fqn.In particular,we set en=e0.Define the cyclic shift operator τ:Fqn→Fqn by(?) We say that a subspace U is cyclically covering if(?) For any n ∈ N,let hq(n)denote the largest possible codimension of a cyclically covering subspace of Fn.In this paper,by the structure of vector spaces in finite fields,we get some sufficient and necessary conditions for the hyperplane to be a cyclically covering subspace from different standpoints,and give a sufficient condition for the cases hq(n)=0.Furthermore,for a general m(1 ≤m<n)-dimensional subspace W,two sufficient and necessary conditions for W to be a cyclically covering subspace are given by the relationship between the subspace and the linearized polynomial.