| Cyclic codes are very important linear codes. They are built on the basis of strict algebraic theory. Cyclic codes have a strong error correction and error detection capabilities and play an important role in practice. So far, there are a lot of references about cyclic codes. Residue codes are cyclic codes which have good characters. There are a large number of references to study residue codes. Generating polynomials of high-order residue codes are factors of x n?1. When n is very large, it is very difficult to decompose of x n-1over the finite field F3 . MacWilliams and Sloane define residue codes in their book with generating idempotents. If we determine generating idempotents of the high-order residue codes, high-order residue codes can be determined without decomposition x n?1 over the finite field. Thus it is important to determine generating idempotents when we study high-order residue codes. This thesis first defines cubic and quartic residue codes over the finite field F3 , and then give expressions of generating idempotents of cubic and quartic residue codes over the finite field F3 .We have four parts in this thesis.In the first part we review some research results about residue codes over the finite field F3 and introduce the main contributions in this thesis.In the second part we give some preliminaries about the residue codes and generating idempotent.In the third part we study the character of the cubic residue codes over the finite field F3 , the cubic residue and their generating idempotents over the finite field F3 , and give expressions of generating idempotents of cubic residue codes over the finite field F3 . An example has been given for illustrating the theory.Finally in the fourth part we study the character of the quartic residue codes over the finite field F3 , quartic residue codes and their generating idempotents over the finite field F3 , and give expressions of generating idempotents of quartic residue codes over the finite field F3 . |
