Constructions Of Several Classes Of Linear Codes Over Finite Chain Rings

With the sustained and rapid development of communication technology and information industry,all kinds of communication mode appear constantly and quickly get extensive applications and the digital information storage and exchange capacity increasing.Which make all aspects of people's life very convenient.As we all know that sometimes the digital signals will be affected by external interference in the process of wireless or wired launch so that the receiving end will get error.Error correction coding which is also called the channel coding is an important part of the modern coding theory and aims at let the communication system has the capability of automatic error detection and error correction.Whose main contents include how to overcome the interference of the outside world,increase the accuracy of information transmission,improve the quality of communication.Recently error correction coding technology has been widely used in mobile communications,satellite communications,computer(data storage and computing system),image recording,digital audio and broadcast,radar,remote control and telemetry,aerospace and other fields.Linear code is a very important kind of error-correcting code,which has a good algebraic structure and its encoding and decoding algorithms are easy to implement.So it has been widely used in cryptography and communication systems.Compared with over finite rings,it has been proved to have more codewords than the ones over finite fields with the same length.For example,the Preparata code have twice as many codewords of error-correcting BCH code.Recently,linear codes constructed from defining sets have been extensively studied.This construction was proposed in 2007 by Ding et al.It has been proved that many classes of known codes as well as linear codes with a few weights can be produced by selecting the suitable defining set.Linear codes with a few weights have widely applications in secret sharing schemes,strongly regular graphs,association schemes,authentication codes,and constant composition codes,evenmore in consumer electronics,communication and data storage systems.For computer processing of data is a form of 0 and 1,we mainly consider the binary linear codes from defining sets over the finite chain ring F2+uF2+…+ukF2.A class of binary three-weight linear codes from defining sets over the finite field F2 is constructed by Kelai Ding and Cunsheng Ding.We generalize its results to construct some linear codes over the finite chain ring F2+uF2+…+ukF2(1?k?Z+,uk+1=0).In which there are many linear codes with a few nonzero weights.Then we investigate the Lee-weight distribution of these linear codes and determine the explicit complete weight enumerator for the images of these linear codes under the Gray map.Furthermore,a class of constant weight codes that meets the Griesmer bound for constructing optimal constant composition codes achieving the LVFC bound is also presented.