Algebraic Method For Reproducing Codes With Multiple Node Failures

Distributed storage systems spread data across different nodes through storage devices on different machines in the network.Because system storage nodes usually need to store a large amount of old data,multi-node systems often have node failures.In the case of node failure,the system needs to use the pre-stored data to repair the failed node.As a new data redundancy technology,the regenerating code can effectively reduce the bandwidth resources occupied by the failed node storage data in the distributed storage system.This also makes regenerating code technology to be a hot topic for the repair of multi-node failures.In recent years,Venkatesan Guruswami et al.used RS codes to construct a regenerating code that achieves optimal bandwidth over a finite fieldF_q,but its solution requires a code lengthn?q,That is,the code length is limited to a certain range.Later,the regenerating code constructed by algebraic geometry code by Jin Lingfei et al.breaks through this limitation,although the RS code and algebraic geometry code constructed on the same field are compared,the latter repair bandwidth is larger than the former,but since the code length of the algebraic geometry regenerating code is not limited,so the stored and the bandwidth have reached a better balance.At present,the research of regenerating code has been perfected,but most of the existing researches discuss the single-node failure when discussing the regenerating code,while due to the instability of the system,there is often a problem of multi-node failure.Only a few experts such as Hoang Dau have done research in this area.The main research in this paper is the repair of multi-node failure.The algebraic methods used in this paper are:algebraic structure of finite field,trace mapping over finite field,algebraic function field of univariate and algebraic geometry code constructed by algebraic function field on finite field.This paper mainly uses algebraic geometry code to construct regenerating code.On the one hand,we discuss the repair scheme of multi-node,especially the rational algebraic geometric code of two failed nodes.We use two algorithms are respectively the distributed repair scheme and the centralized repair scheme to repair two failed nodes.On the other hand,this paper discusses the multi-node repair scheme of general algebraic geometric codes,especially the repair scheme of algebraic geometric regenerating codes with two nodes failed,can also be extended to the repair problem of multi-node failure,and the repair scheme is illustrated by an example.This article is divided into four chapters,which are:（1）The first chapter mainly describes the origin and development status of the regenerating code.（2）The second chapter introduces the related algebra background,which is divided into three parts.The first section is about the related concepts and properties of the finite field and its trace mapping;the second section is mainly related to the background of the algebraic function field.The third section is the definition of algebraic geometry codes and their dual codes and some propositions that need to be used later.（3）The third chapter introduces the regenerating code and linear repair scheme.The first section introduces the overall development process of the regenerating code and its definition in mathematical sense.The second section mainly introduces the linear repair scheme of the trace mapping,including linear repair scheme for general linear codes and linear repair scheme for RS codes.（4）The fourth chapter is about multi-node repair of algebraic geometry code,which is mainly divided into two parts.The first section is the repair scheme of rational algebraic geometric code.It mainly gives two repair schemes,one is distributed repair,the scheme is another centralized repair scheme.The second section is the repair scheme of general algebraic geometry code,which is mainly inspired by the repair scheme of Jin Lingfei et al.,constructing a functionh_?,usubject toh_?,u（7）P_?（8）（28）?_u,in which???i,i^（9）?,to repair the failed information f（7）_iP（8）andf（7）_iP_（9）（8）reach the optimal bandwidth,and then the algebraic geometry regenerating code on the Hermite function field is given according to the repair scheme of this general algebraic geometric regenerating code.Finally,the paper summarizes the practice of this paper and puts forward someprospects for multi-node repair regeneration code.