The Algebraic Method Construction Of LDPC Code

In coding theory with applications in computer science and telecommunication and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels.Many communication channels are subject to channel noise,and thus errors may be introduced during transmission from the source to a receiver.Error detection techniques allow detecting such errors,while error correction enables reconstruction of the original data in many cases.The general idea for achieving error detection and correction is to add some redundancy(i.e.,some extra data)to a message,which receivers can use to check consistency of the delivered message,and to recover data determined to be corrupted.Error-correcting codes are usually distinguished between convolutional codes and black codes:Convolutional codes are processed on a bit-by-bit basis.They are particularly suitable for implementation in hardware.Block codes are processed on a block-by-block basis.Early examples of block codes are repetition codes,Hamming codes and multidimensional parity-check codes.They were followed by a number of efficient codes,Reed–Solomon codes being the most notable due to their current widespread use.Low-density parity-check codes(LDPC)are relatively new constructions that can provide almost optimal efficiency.In information theory,a low-density parity-check(LDPC)code is a linear error correcting code,a method of transmitting a message over a noisytransmission channel.[1][2] An LDPC is constructed using a sparse matrix.[3] LDPC codes are capacity-approaching codes,which means that practical constructions exist that allow the noise threshold to be set very close(or even arbitrarily close on the BEC)to the theoretical maximum(the Shannon limit)for a symmetric memoryless channel.LDPC codes are finding increasing use in applications requiring reliable and highly efficient information transfer over bandwidth or return channel-constrained links in the presence of corrupting noise.Error correcting code is an important tool to realize reliable data transmission in communication system.In the information transmission,we hope to achieve a lower error decoding in a relatively short time,In order to solve this problem,in 1962,Gallager Robert proposed a low density parity check code for the first time in the paper [1],LDPC code,This is a packet error correcting code with sparse parity check matrix.It presents a new method of constructing the encoding structure in a new way to solve the problem of long code decoding using the "sparsity" of the parity check matrix,namely,the number of the number of the 1 "" is much less than the number of "0".Its excellent error correction performance and high degree of decoding scheme has attracted many scholars to study [2]-[6],and they found that the performance ofLDPC codes is close to the [4] limit [5],Shannon.Since then,the design,construction and application of LDPC codes have become a focus of the research.In most of the practical applications,we need to design a good LDPC code with short length.Many scholars try to construct the [7]-[23]code with different length and ratio LDPC.In this article,we will be on the basis of the above LDPC code on the finite field and the algebraic structure of the ring.This paper consists of four chapters,each chapter is as follows:The first chapter introduces the related to the relevant algebra basic concepts and theorems;the second chapter LDPC codes introduced basic coding theory;the third chapter,introduce additive group of finite field and two algebraic methods for constructing q-ary LDPC codes.