Research On Several Problems Of Algebraic Geometry Code

Regenerating code applied to distributed storage systems,distributed storage system refers to using certain technical means to store original data on several independent devices(nodes),often providing reliable access by adding redundancy to a single unreliable data node.The application scenarios include data centers,point-to-point storage systems,and storage in wireless networks.In the early stage,people used replication to maintain data stability,and then considered repairing the information on a failed node accurately,which is the exact repair problem,because generate excessive storage costs through replicating the overall data.When a node fails,the repair task performed by the system depends on the communication between the individual nodes,which is a new challenge for the design of the code.Especially a new parameter related to the overall system efficiency,i.e.repair bandwidth,that is,the amount of data communicated between nodes in the process of repairing the failed node.The first coding technique is to use the erasure code to repair the failed node information.The process is only to generate a coded block,but it needs to reconstruct the entire encoded data of the new node,so this method is obviously sub-optimal.In 2010,Dimakis et al.first proposed the concept of regenerating code,which is a coding solution that can provide effective repair.The regenerating code has been a focus of current research since the concept of regenerating code has been proposed.The regenerating code allows the new node to repair the failed node by communicating between the surviving node's stored information,which plays a crucial role in reducing the repair bandwidth and can achieve a benign trade-off between storage and repair bandwidth.In 2016,Venkatesan Guruswami et al.solved the problem of exact repair of RS codes in distributed storage systems,but the code length in the linear repair scheme is limited by the alphabet.In order to improve this problem,Jin Lingfei et al.constructed a regenerating code by algebraic geometry codes,which can be able to improve the repair efficiency and solves the node repair problem.For the problem that the regenerating code constructed by RS code or algebraic geometry code,it is impossible to determine whether it can achieve cut set,this paper is inspired by the research results of Itzhak Tamo et al.,constructing a repair scheme of rational algebraic geometry codes through a special subfamily of this rational algebraic geometry code over E,so that it can repair a single failed node while achieving the cut set bound.However,the node that can be repaired by this scheme is limited to a certain range.In order to improve this problem,the other repair scheme in this paper is that constructing rational algebraic geometry regenerating codes over a tower of field extentions enables it to repair any single failed node and achieve the cut set bound.In addition to constructing the optimal repairing rational algebraic geometry regenerating code,this paper also improv some general algebraic geometric reproduction codes constructed by Jin Lingfei et al.,through the adjustment of the parameters in a function of the repair scheme,breaking the limitation that the characteristic of the finite field is only allow to be even in the original scheme,and it is proved that the bandwidth of the repair scheme is still optimal under the new function.The several issues studied in this paper are:(1)Solving the problem uncertianly whether the regenerating code constructed by the algebraic geometry code can achieve the cut set bound or not.This paper does not completely solve this problem,but in this paper,the algebraic geometry regenerative code repair scheme on the rational function field is constructed,which can repair a single failed node and reach the cut set bound.It is mainly to construct a rational algebraic geometry code repair scheme by a special subfamily of rational algebraic geometric codes,that is,the repair scheme is constructed over some subfield Fi,of E,so that when repairing a certain failed node,only d,elements of F,needs to be accessed,and each element only needs to access 1/pi of the content.So as to reduce the repair bandwidth and reach the cut set bound.(2)For the problem that the repair node is limited to a certain range in(1),constructing a repair scheme can not only repair a single failed node and achieve the cut set bound,but also repair any information of a failed node.The main method is to construct a basic field tower and a subspace S;over an intermediate field,so that any d helper nodes downloads 1/s of pi symbols to repair any single failed node.s(3)For the problem that the characteristic of finite field of the algebraic geometry code repair scheme of Jin Lingfei et al.is limited to an even number,this paper corrects the parameter value of a function to make the repair scheme is also possible to repair a single failed node and achieve the optimal repair bandwidth as well under the premise of the chracteristic of the finite field being arbitrary.The article is divided into three chapters:(1)The first chapter mainly introduces the origin and concept of the regenerating code,not only using general language but also mathematical language to interpret the definition of the regenerating code.(2)The first section of the second chapter introduces the basic concepts of algebraic function fields and algebraic geometry codes for later use.The second section constructs a repair scheme of rational algebraic geometric codes,and Further adjustments are made based on the limitations of the failed nodes repaired in this scheme,giving another repair scheme can repair any single node.(3)The first section of the third chapter improves the regenerating code repair scheme constructed by Jin Lingfei et al.After improvement,it can be proved that the characteristics of the finite field can be either even prime or odd prime,and the repair scheme can also achieve optimal repair bandwidth.The second section is based on the repair scheme of the general algebraic geometry regenerative code in the first section,consructing a regenerating code by the Hermite code,and it is proved that the optimal repair bandwidth can be achieved.