Locally Recoverable Codes Over Rational Function Fields

In the current era of big data and the development of cloud storage systems,distributed storage technology plays a vital role.When storing data in the storage system,in order to improve the data reliability of cloud storage,the system will store a large amount of redundant data.There are two common forms of redundancy: copy and erasure coding.Copying is the simplest form of redundancy,that is,storing the same information on multiple nodes;and erasure coding is to divide the original file into multiple information blocks,which are then encoded to generate several check blocks of the same size.The original file can be reconstructed by arbitrarily downloading the original file size information from all information blocks and parity blocks.Compared with copying,erasure coding can greatly improve data reliability.However,one of the challenges facing erasure coding is when an information block is lost,the bandwidth consumption required to repair the information block is too large.In order to reduce the repair cost of erasure codes,we introduce a new type of encoding method-locally recoverable codes.In a distributed storage system,if a node of the distributed storage system is lost or damaged,the information stored by that node can be immediately repaired by reading no more than other relevant nodes,which is called the localization parameter r of the code,here r is the smaller number.The recovery efficiency of the locally recoverable codes can be quantified by three different indexes.The three indexes are the recovery bandwidth,the number of read bits,and the localization parameter r.The main focus of this article is the localization parameter r.The locally recoverable codes is to reduce the repair overhead by limiting the number of nodes connected during data repair.First,in 2011,Itzhak Tamo constructed a family of optimal locally recoverable codes on a finite field.In order to overcome this limitation,Alexander barg and other expert algebraic curves constructed locally recoverable codes,The code length can break through the limitation of the character set size,but the minimum distance is not optimal,and the process is complicated.The locally recoverable codes are equivalently constructed on the algebraicfunction field,the description is more direct,and the code length can also break the limit of the size of the character set.Further,this article introduces the definition of algebraic function field into locally recoverable codes.It mainly studies the construction of locally recoverable codes on algebraic function field,and applies it to the field of rational functions to construct a locally recoverable code with optimal distance.Taking a specific example,after generalizing it,locally recoverable for multiple erasures codes are constructed on the rational function field,and the distance is optimized.This article is mainly divided into four chapters:The first chapter introduces the development and definition of locally recoverable codes.The second chapter mainly describes the algebraic background related to this article.The first section introduces some definitions and properties related to the algebraic function field;the second section introduces the construction of locally recoverable codes on the algebraic function field,and the algebraic function field Construction of multiple erasing locally recoverable codes.The third chapter mainly describes the construction and repair of the locally recoverable codes in the case of a symbol loss in the rational function field.The first section introduces the construction of the locally recoverable codes in the rational function field,and the second section introduces the locally recoverable codes.Repair and give specific examples.The fourth chapter is based on the third chapter to extend to the locally recoverable codes of multiple symbol loss.The first section introduces the construction of the locally recoverable codes for multiple erasures on the rational function field and obtains the optimal.The second section introduces Fixed a fix.