On Error-correcting Codes And Primitive Elements Over Finite Fields

In this thesis, we discuss the m-MDS codes which are a special classes oferror-correction codes and the primitive elements in finite fields. We prove the followingmain results:(1) Several sufcient and necessary conditions for that C=[n, k, d]qsatisfiesS(C)=S(C⊥)=m are obtained, where C⊥stands for the dual code of C, S(C)=n+1k d=m(0). Furthermore, all q-ary Hamming m-MDS codes are determined.(2) A sufcientcondition for the existence of the finite fieldFqin which there exists two primitive elementsα, β∈Fq, such that α+β is also a primitive element ofFq. Furthermore, for finite fieldssatisfying this condition, by using character sums, we get a lower bound of the correspondingq. Finally, by using mathematical software—Maple, we verify that the condition is true forall finite fields with characteristic2exceptF2andF4.