Existence Analysis And Construction Of LRC Codes Based On Algebraic Structure

Modern communication and data processing require high reliability of large-capacity data operation,which requires rapid recovery of error information(such as node damage or node congestion)of some actual nodes.At present,there are three methods to repair,that is,bandwidth repair method,disk I/O bit reading method and local recovery method.This paper focuses on the local recovery method.Local recovery code(LRC)is one of the most widely used linear codes in modern communication.Especially in the network coding of large data and distributed storage systems,LRC code has more obvious advantages.Its advantage is that the value of any coordinate position of the codeword can be recovered by a small number of r(called local parameter,r is far less than the code length n)values of other positions,which can reduce storage overhead and improve operation efficiency.The commonly used method to study LRC codes is based on the structure of finite fields,and the method to construct good polynomials,such as the use of finite field addition and multiplication subgroup structure.However,based on these structures,there are some limitations.For the desired local parameter r,it may not be possible to use the above method to construct a good polynomial on the finite field_qF.In order to solve this problem,we use the method of algebraic combination to prove the existence of a class of good polynomials,and give sufficient conditions for the existence,and then give a concrete example to illustrate its existence.In addition,in order to solve the problem of node access congestion in communication,we need to construct LRC codes with multiple recovery sets.In this paper,based on the automorphism group on the algebraic function field,we use the inner direct product of the automorphism subgroup of the function field to construct multiple recovery sets by expanding the algebraic function field.The assignment space for coding is the intersection of multiple function spaces.Furthermore,this method is further improved by using the theory of algebraic function field,that is,an explicit set of functions is used to construct the assignment space.This method makes it easy to calculate all parameters of the constructed LRC code.