Research The Systematic Binary Quasi-cyclic Codes

In recent years,the rapid development of digital communication systems and applications to businesses,the urgent need to reduce the error rate during transmission,So that data can reproduce accurate,how to improve the coding error correction capability is an important issue.The systematic binary quasi-cyclic codes have widely used in existing digital communications systems because of circulation and good error correction capability,ensure the accuracy of the information can be transmitted to the receiving end.Construct the good systematic binary quasi-cyclic code,pursuing greater minimum distance,improving error detection and error correction capability,is the direction of scientific research scholars.This paper proposed a variable matroid search algorithm,because the matroid search algorithm is very complexity and the local matroid search algorithm cannot search all optimal codes.Construct the systematic binary quasi-cyclic codes based on this algorithm,this code have rate variable and is optimal codes.The codes feature is to meet the optimal code.This paper proposed the concept of shifting dual codes,because it is difficult to find the codeword for the dual code,combining with the nature of the shift dual codes,proposing algorithm design.Based on the algorithm construct shifting dual codes.The codes can be easily obtained its dual code by cyclically shifting,which rich rate 1/2 binary system for constructing quasi-cyclic codes.Specific content includes the following two parts: 1)Construct the systematic binary quasi-cyclic codes with rate 1/p based on variable matroid search algorithm.2)Construct of binary shift dual codes.The main work and research results are as follows:(1)In recent years,matroid theory has been used in coding field.And the relation between the minimum distance and the generator matrix can be obtained with matroid theory,the minimum distance theorem of quasi-cyclic codes has found,proposed matroid search algorithm and local matroid search algorithm to find the optimal codes.Construct rate 1/p systematic binary quasi-cyclic code based on the two algorithm has gotten remarkable achievements,which is better than the existing database of codes,it is difficult to ensure optimal code because of high complexity and non-global search.This paper proposed variable matroid search algorithm based on the previous algorithm.The algorithm reduce the repeated search by using the results of the previous step,which reduces computational complexity and can get the optimal code.A special kind of rate 1/p systematic binary quasi-cyclic codes have be constructed based on variable matroid search algorithm.The rate of the codes is variable with the change of integer p.And the generator matrix G of the rate 1/p optimal codes can be obtained with reduce or add a loop matrix.Through the experiments,two new codes of which the minimum distance larger than the existing best results have been worked out,verified the feasibility and superiority of the algorithm.(2)Researching dual code mainly focuses on the self-dual code,how to solve dual code of binary linear code,existing studies is less,methods of use can reduce the computational complexity,improve operational efficiency,but still have problems of high complexity.On the base of study dual code,combined with the characteristics of quasi-cyclic codes,we define shift-dual code with rate 1/2,designing method for constructing such code word,using the computer to search the optimal code.The algorithm uses the optimal code Theorem of shift dual code,reducing many of the operations,reduce complexity,and reduce time-consuming experiments.Through the experiment,the minimum distance of optimal code is equal mostly that of self-dual code,four is better.This approach not only reduces the complexity of researching optimal code,but also code word generated can get dual code easily through cyclic shift,and have the same minimum distance and weight distribution.