Research Of Structures And Ranks Of Linear Codes Over Finite Ring

In the dissertation, the author mainly studies the structure,proposition, rank and the minimal generating set of codes, which are two hot topics of the study of error-correcting codes. The details are given as follows:(1) We firstly give the structure of generator matrixes of the linear codes over ring F₂ + vF₂. Secondly, we not only prove that the Gray images of the linear codesover ring F₂ + vF₂ are also linear codes, but also prove that the Gray images of themutual dual linear codes are also mutual dual linear codes. Thirdly, we present various formulas of the codes weight between codes and their Gray images, and then give their Macwilliams identities.(2) By discussing the structure of repeated-root cyclic codes and (1 + u) - cycliccodes over ring F₂ + uF₂, we gives their ranks and minimal generating sets, and prove that the Gray image of one kind of cyclic code over F₂ + uF₂ of length 2^e is cyclic codes, and establish the structure of linear cyclic codes over F₂ + uF₂ oflength 2^e when the Gray images are cyclic codes.(3) We present the generators of cyclic codes and their dual cyclic codes overring Z_p²[x]/(x^pe -1), and study the ranks of these codes and their minimal generatingsets.(4) We discuss the structures of cyclic codes,Negacyclic codes and their dual cycliccodes of odd length over the finite chain ring R and F_q + uF_q+…+ u^s-1 F_q, and we also explore the rank of these codes and their minimal generating sets.