Cyclic Codes Over Galois Rings And Gray Map

Cyclic codes are a very important class of linear codes. Besides its goodalgebraic structure and cyclic character, encoding and decoding are very easy torealize by using linear shift registers.Therefore, cyclic codes have attracted manyscholar's attention. In 1957, Prang first began to study cyclic codes over fieldGF(q), and then the study of cyclic codes over field GF(q) had made greatprogress both on theory and practice. With the development of the theory of errorcorrectingcodes, the coding theory over finite field had completely developed. Inrecent years, the sdudy of cyclic codes has extended from field to finite ring. Inparticular, many coding experts have widely studied much of the Z₄ cyclic code(which is not necessarily linear) and its Gray map , and have obtained someproperties of Gray images of cyclic codes and negacyclic codes over 4 Z . Similarly,the definition of Gray map has extended to Z₂^K+1 and Z_P^k+1, and some properties of theimages of cyclic codes have obtained.Enlightened by these results, we perfect the relationship between the GrayimageΦ(C) of a linear code C on 4 Z and the Gray imageΦ(C⊥)of C's duallinear code C⊥. And then we present a necessary and sufficient condition that C isa self-dual linear code over 4 Z . Furthermore, we introduce a new isometricmap- k on pk 1 Z + and study a few properties of (1+ tpk )- cyclic codes over Z_P^k+1 byusing - k . We obtain a necessary and sufficient conditions that C is a(1+tpk )-cyclic code over Z_P^k+1 is that- k(C) is a (1+ tp)- quasi-cyclic code withorder p^k-1 and length p^k-1n .In this article, we mainly deal with cyclic codes overGR(4m ) . Firstly, we studythe relationship between the generators of a cyclic code onGR(4m ) and its dualcode. Secondly, we introduce the definition of Gray map onGR(4m ) , study theproperties of Gray images of cyclic codes and quasi-cyclic codes, and obtain anecessary and sufficient conditions under which C is a self-dual code on GR(4m ) .In summary, we expand codes over 4 Z and Z_P^k+1 toGR(4m ) , i.e., we extend theresearch range of this family.