Research On The Period Distribution Of Cyclic Codes

The period distribution of error-correcting codes, in a classical coding theory, is a new investigation field. Research on the period distribution of cyclic codes must be propitious to know the structure of cyclic codes, to construct nonlinear cyclic constant weight codes and cyclic transpositional codes and to design signal with good speciality. So the study of it is significant from both a theoretical and a practical viewpoint. There has been a recent growth of interest in codes with respect to this period distribution of cyclic codes.In this paper, the period distribution of cyclic codes over finite fields and finite rings is explored at the base of previous work. The first part, we give the summarize for the condition and the significance; The second part, we give the preparation knowledge of the whole paper; The 3-rd part, the period distribution of cyclic codes and constant-cyclic codes over F_q were studied respectively, and exact calculation formulas are established too. The 4-th part, the definition of the period distribution of cyclic codes over Z_p^m is obtained, the period distribution of cyclic codes over Z₄ and Z_p^m were studied respectively, and exact calculation formulas were also established; The 5-th part, a kind of special cyclic codes over Z₄ and their images under the mapφμwere introduced, these images are all binary cyclic codes over the field F₂, the relation of period distributions between these two kinds of cyclic codes was presented, by this part a bridge was establish from part 3 to part 4; The 6-th part, we summarize the total paper, and put forword some problems which in perhaps development direction in the future.The period distribution and depth distribution of error-correcting codes are both important investigation fields in a classical coding theory. So the study of period distribution will have not only important theoretical significance to expand the coding theory but also very important practical value.