Several Problems For Error-correcting Codes And Primitive Elements Over Finite Fields

In this thesis, we discuss the distribution of linear codes over some specialrings, g-ary codes correcting asymmetric errors and the existence for some special primitiveelements in finite fields. We prove the following main results: A formula for the depth dis-tribution of linear codes over some special rings is determined; For any positive integer g,a construction for g-ary codes correcting asymmetric errors is given and the correspondingdecoding algorithm is obtained also; By using the index sum method and Weil estimation,some sufcient conditions for that there exists some α∈Fqnsuch that both α and α+α1are primitive or α+α1is primitive and α is a primitive normal element ofFqnoverFq.