Repairing Multiple Failures For Algebra Geometry Code

In recent years,with the rapid development of the global Internet and informa-tion technology,communication services have gradually penetrated into all aspects of our daily life,and the business of big data is growing day by day,so the storage requirements of big data are also increasing.However,most of the traditional stor-age schemes use network storage to solve the problem,unable to store massive data.Hence came the distributed storage.In a distributed storage system,a large file is encoded and distributed over many nodes.When a few nodes fail,one should be able to rebuild the replacement nodes efficiently by using the information from the remaining active nodes.In the process of communication,exact repair has the most practical significance.Exact repair refers to the repair of failure nodes exactly.One traditional solution to the exact repair problem has been to use Maximum Distance Separable(MDS)code,and in particular Reed-Solomon codes.However,it is not ideal to use the Reed-Solomon code to solve the problem of exact repair in traditional solution.Recently,wotters and Guruswami constructed a linear exact repair scheme for Reed-Solomon codes,which uses less bandwidth than traditional Reed-Solomon codes to repair failed nodes.But in Wotters and Guruswami's repair schemes,code length is limited by alphabet sizes.For this reason,Jin et al.solved this problem by using algebraic geometric code repair schemes.But they studied the repair for single failure node.In fact,there are often have multiple nodes failures at once in the process of com-munication.Mardia et al.extended Wotters' and Guruswam's repair schemes and gave multiple repair scheme for scalar MDS codes.As with Wotters and Guruswami's repair scheme,the problem that code length is limited by alphabet sizes still unsolved.To generalize the polynomial P(l,p)(x)constructed by Wotters and Guruswami to algebraic geometry code,the choice of the function h(i,u)is the key part.Inspired by the work of Jin et al.,we let h(a,u)=(?),then used function h(i,u)to construct matrix MI.Matrix MIis used to construct an invertible mapping ?,and invertible mapping ? is used to construct multiple repair algebraic geometric codes.It solves the problem that the code length limited by alphabet sizes.In particular,when the algebraic function field in is rational function field,our results generalize the results of Mardia et al.This paper consists of three parts:In Chanpter 1,we introduce the development of regenerating codes briefly,and give the main conclusions of this paper.In Chapter 2,we introduce some basic concepts and theorems in algebraic function fields and Algebraic Geometric Codes used in this paper.In Chapter 3,we give the construction of multiple repair algebraic geometric codes.Then,we select the appropriate parameters for algebraic geometric codes on rational functions,and get the results are the same as of Wotters and Guruswam.