A Research On Constacyclic Codes Over Finite Fields

This dissertation is devoted to the classification of constacyclic codes over finite fields, and some classes constacyclic codes for their algebraic struc-tures and distance structures.This dissertation consists of five parts. In chapter1, we introduce the background and main ideas of the present research, and then we recall the necessary notations and known results.In chapter2, we introduce a concept "isometry" for the nonzero ele-ments of Fq to classify constacyclic codes over Fq such that the constacyclic codes belonging to the same isometry class have the same distance structures and the same algebraic structures. Some necessary and sufficient condition-s for any two elements of Fq*isometric to each other are established; as a consequence, the constacyclic codes isometric to cyclic codes are described.In chapter3, we classify the constacyclic codes of length (?)tps over FPm. into isometry classes, characterize explicitly the polynomial generators of the constacyclic codes of each isometry class, where (?) is a prime different from the characteristic of Fq, and s, t are positive integers.In chapter4, assuming that s is a positive integer such that X2s+1fac-tors completely into degree-one factors in Fq2[X], we obtain the polynomial generators of all self-dual and self-orthogonal negacyclic codes of length2s(?)t over Fq, where (?) is an odd prime coprime to the characteristic of Fq and t is a positive integer.In chapter5, we study minimal cyclic codes of length (?)m over a finite field Fq, where (?) is a prime divisor of q—1and m is a positive integer. Explicit expressions for the primitive idempotents, check polynomials, mini-mum Hamming distances and the dimensions of these codes are obtained.