Construction Of LRC Codes Based On Algebraic Function Field And Its Automorphic Groups

Today's distributed storage systems have developed to a large scale.Even when the system fails,it has become the norm.Therefore,the storage system must introduce redundancy and encoding techniques to recover the lost data.The simplest technique for recovering data is replication,but the disadvantage is that it consumes a lot of storage space.While a class of classic error-correcting codes,although they can accurately detect the lost data and recover in time,the recovery efficiency is low.So experts put forward more advanced coding technology —— locally recoverable code(LRC).The code that produces n symbol number word from k information symbol.If any symbol of the code word can be recovered from up to r(r =n)other symbols.Compared with error-correction codes,it is easier to access the values of n-1 other symbols.The efficiency is naturally much improved.Although the LRC codes constructed on finite field,algebraic curve and algebraic surface at present have good locality and achieve low recovery costs,their construction and decoding are too complicated.Therefore,this paper construct LRC codes and explore its recoverability on algebraic function field.The first part of this paper describes the research background,significance and existing research results of locally recoverable codes.The second part explains the theoretical background of this paper.The third part constructs the LRC codes by using a trinomial on the algebraic function fields,and the recovery process of such codes only needs to go through one linear operation.Then a specific trinomial+1-+qqyxx is used in the Hermite function fields to verify the sharp recovery method and process of this kind of code.And the fourth part studies the construction of LRC codes with multi-covering on the self-isomorphism group of algebraic function fields.Then,according to the construction and properties of locally recoverable code with double recovery sets,the conclusion of LRC(2)code is extended to LRC(t)code.