Research On Constacyclic Codes In Management Information

Management informatics is an integration of management science and information technology, which is based on information theory. Reliability of Information transformission is an important aspect of information theory, and error-correcting coding theory is its theoretical basis. Through more than sixty years of development, classical error-correcting codes over finite fields have become mature in theory, and attained extensive applications in practice. With a more deeply study in error-correcting coding theory, the theoretical value and practical significance of error-correcting codes over finite rings have gradually been realized. Constacyclic codes (including cyclic codes and negacyclic codes) and self-dual codes over finite rings have become a focus of the study of error-correcting codes over finite rings.This dissertation investigates the structure properties of constacyclic codes over finite chain rings, in particular constacyclic self-dual codes. LetR be a finite chain ring and (?) be the residue field of R . Let a be a fixed generator of R and p be the characteristic of (?) . Firstly, we introduce (1 + au)-constacyclic codes over R , where u is a unit in R . The structure of (1 + au)-constacyclic codes overR of length p^s is obtained and the Hamming and homogenous distances of these codes are determined. By using a ring isomorphism, the structure of (1 + au)-constacyclic codes over R of length N = p^sn(n prime to p ) is established. On one hand, this structure is applied to analyze binary cyclic self-dual codes and negacyclic self-dual codes over Z₄, and bounds for distances of these two classes of codes are given. On the other hand, by combining this structure with the discrete Fourier transform, constacyclic self-dual codes over Z₂^a and cyclic self-dual codes over 2rF are explored. The structure ofη-constacyclic self-dual codes over Z₂^t is determined, whereη= -1 or -1 + 2^t-1, and some constacyclic self-dual codes over Z₂^a are found. It is shown that all cyclic self-dual codes over F₂^r are Type I. Secondly, we study negacyclic self-dual codes over R under the condition that the length N is relatively prime to p . Necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over R are given. As an application, negacyclic MDR self-dual codes over GR ( p^t ,m) of length p^m+1 are constructed. Finally, we discuss two classes of repeated-root constacyclic codes of length 2^e over Z₄ and GR (4,2). The Hamming distance of cyclic codes of length 2^e over Z₄ is determined, and the exact Lee distance of some cyclic codes of length 2^e over Z₄ are also obtained. It is shown that the Gray image of a negacyclic code of length 2^e over GR (4,2) is a distance-invariant linear quasicyclic code of index 2 and length 2^e+2 over .