Research Of Bounds On The Dimension And Constructions Of Locally Repairable Codes

Locally Repairable Codes(LRCs)are Erasure Codes(ECs)widely used in distributed storage systems.When an error occurs in any encoded symbol of a local repairable code,it can be recovered by accessing up to r other symbols.Because of this feature,local repairable codes have broad prospects in distributed storage systems.This thesis mainly studies the bounds on the dimension and constructions of LRCs,mainly in the following two aspects:Starting from the bounds on dimension of LRCs,the C-M bound is used as the basis.Firstly,this thesis generalizes the existing bounds on dimension,and obtains the bound on dimension for LRC codes with unequal localities and disjoint repair groups when the minimum distance d (?)2t(10)2.Secondly,this thesis uses the optimization methods to analyze the obtained bound on dimension,and puts forward a sufficient condition to reach the new bound under the conditions of a given code length n,minimum distance d and parameter s.Thirdly,through simulation analysis,under the condition of given a code length n and a minimum distance d,the variation of the maximum bound on dimension with the parameter s is obtained.Fourthly,the relationship between the new bound and existing bounds is analyzed based on software simulation.Starting from the constructions of LRCs,combined with the bound proposed in Chapter 3,construct LRCs with unequal localities and disjoint repair groups whose dimension reaches the bound proposed in Chapter 3.Firstly,we propose and prove a necessary and sufficient condition that the check matrix structure of the LRCs with unequal localities and disjoint repair groups must meet when the minimum distance d (?)2t(10)2.This conclusion generalizes the theorem in [17].Secondly,the construction algorithm in [42] is slightly modified,and the modified construction algorithm is used to construct the optimal LRCs of the given parameter dimension.Thirdly,design a new construction algorithm to construct the LRCs with unequal localities and disjoint repair groups with given parameters,and the codes dimension reach the bound on dimension in the previous chapter.Only on the basis of a given local parameter vector and minimum distance,the code length n,the optimal value of the dimension k and the size of the parameter s can be obtained by calculation,and then the code we need can be constructed through the new construction algorithm.After analysis and comparing the modified construction algorithm in literature [42],the construction algorithm newly designed in this thesis has a significant reduction in complexity.Fourthly,analyze the repair characteristics and error performance of the code we constructed through software simulation.Under the premise that the code length,dimension,and minimum distance are equal,we choose the code whose variable nodes have equal locality that has the same generalized locality parameters as the code we constructed.After comparison,it is found that although the LRCs with unequal localities we constructed has some loss in terms of dimension,in the process of recovering multiple failures,no matter whether the deletion decoding algorithm in literature [53] or the parallel distributed decoding algorithm is used,the codes we construct have fewer average access nodes;under the BEC channel,the codes we construct also have better error performance.