Research On Some Problems Of Linear Codes

At present,with the rapid development and perfection of error-correcting coding theory over the finite field,many good codes have emerged in large numbers. At the same time,many researches on error-correcting coding theory have drawn intensive attention on the methods of constructing new codes and good codes, especially the linear codes.For example,Xing Chaoping,Ling San,Aydin Nuh,etc., have put forward the methods of constructing good codes and given many good codes.On the other hand,with the publication of paper[1],researches find some good parameter nonlinear codes over the finite field be able to be obtained by the Gray map and linear codes on the finite ring.So,in recent decade,the reseach on codes on the finite ring have been popular.Hence,this dissertation are arranged as follows:1.We introduce the notion of new trace functions over,F_q,and using the new trace function and elements of some extensions of F_q,we develop Ling and Xing's idea of[2,3]to construct new q-ary linear codes.Our results show that some codes from this construction have good parameters based on Brouwer's table[4].2.On the base of quadratic residue codes on F₂,we introduce the notion of six cubic-residue codes over F₂.The relations between codes are discussed,and the generating idempotents and the dual codes are also studied.3.We define the generalized quasi cyclic codes over the finite field F_q, construct a kind of linear codes over F_q,and give the Hamming distance and dimension of them.4.The definition of constacyclic codes over finite fields F_p is extended to the finite ring Z_p^efirstly.The structure of constacyclic codes over ring Z_p^eis introduced in short.The equivalent relation between constacyclic codes and cyclic codes on some Galois ring of Z_p^eis proven.At last,some special constacyclic codes are given,which have similar properties to BCH codes of designed distance d over finite field: the Hamming distance≥d.