On The Construction Of Locally Recoverable Codes

The traditional large-scale storage system improves the reliability of data through replication,which has the defect of high storage overhead.Locally recoverable codes achieve high data reliability with relatively small storage overhead.For this reason locally recoverable codes are applied to a variety of storage systemsThe recovery efficiency of locally recoverable codes can be quantified by three different indexes,which are the repair-bandwidth,the number of bits read,and the number of symbols that participate in the recoverable process,namely,the localiza-tion parameter r,we focus on the localization parameter r.In recent years,Itzhak Tamo et al.has constructed a class of distal-optimal locally recoverable codes over finite field Fq.The minimum distance of the locally recoverable codes can reach the generalized singleton bound.However,the code length of this construction scheme is limited by the size of the alphabet sets.namely n≤q.Later,Alexander Barg et al.constructed locally recoverable codes by using rational smooth absolute irreducible curves,code length can exceed the limitation of alphabet set,but the minimum distance is not optimal.At present,some experts have studied the locally recoverable codes on the algebraic curves,but the structure of locally recoverable codes on the algebraic curves is more complicated.This paper is based on the construction of locally recoverable codes on algebraic function fields This construction is easy to understandThe optimal locally recoverable codes over the finite field Fq is obtained by means of constructing critical polynomial and effective partition.The general method of constructing critical polynomial and effective partition is given.And expand it,locally recoverable codes with double recovery sets over finite fields Fq are obtained and a concrete example is given.At last,we construct locally recoverable codes with optimal distance which can recover the loss of multiple symbols in a finite field IFq and an example is givenThen the locally recoverable codes on algebraic function fields is constructed,the locally recoverable codes on the Hermite function field by using the construction of locally recoverable codes on the algebraic function fields,the lower bound of the minimum distance of locally recoverable codes in the generalized Hermite function field is improved by constructing subcodes.Of course,the method of this construc-tion can be applied to rational function fields with good parameters.Finally,a class of locally recoverable codes with double recovery sets are constructed by using the Hermite function field.Compared with using Hermite curve to construct locally recoverable codes,the lower bound of minimum distance is obviously improved.The content of this paper is divided into four chapters,respectively:The first chapter is the origin and development of locally recoverable codes.The second chapter mainly describes the algebraic background of this paper.The first section introduces some definitions and properties of groups.The second section is some definitions and properties of finite field.The third section introduces algebraic function fields.The fourth section mainly describes RS codes and algebraic geometry codes.The fifth section introduces the Hermite function fields.The third chapter mainly discusses the construction of locally recoverable codes on finite fields.In the first section,we discuss method of the construction of locally recoverable codes with optimal distance over finite field Fq in case of a symbolloss,and a concrete example is given.The second section discusses the construction of locally recoverable codes with double recovery sets on finite fields Fq and gives an example.In the third section,we mainly discuss how to construct the locally recoverable codes with the optimal distance in the case of loss of multiple symbols.The fourth chapter mainly discusses the locally recoverable codes on algebraic function fields.The first section introduces method of the construction of locally recoverable codes on algebraic function fields.The second section introduces the locally recoverable codes on the generalized Hermite function field,and the lower bound of minimum distance is improved by constructing subcode.In the third sec-tion,the locally recoverable codes with double recovery sets on the Hermite function field are discussed.