Multiset Erasure Combinatorial Batch Codes

In 2004,the batch codes were first proposed by Ishai Y et al..In 2009,Pett-erson M B et al.put forward combinatorial batch codes from the perspective of pure combination.That is t.o put n data directly int.o m servers,and any k data can be retrieved by reading at most t data from each server.On t.he basis of combinatorial batch codes.Silberstein N proposed erasure combinatorial batch codes in 2015 and Zhang H et al.defined multiset combinatorial bat.ch codes in 2018.In this article,we propose a new class of codes by combining multiset combinatorial batch codes and erasure combinatorial batch codes,which we call the multiset erasure combinatorial batch codes.In this codes,for redundancy rany multiset requirement of k order can be retrieved by taking at mostt data from each server of any m-r servers.Each element of the multiset requirement is a data and is repeated at most s times and s stands for repeat number.The parameter t guarantees that the load on the servers is balanced.We want t to be as small as possible,while reducing load.We only consider the case t?1 in this paper.The central content of this paper is to determine the minimum value N of the total amount of data stored in m servers in the multiset erasure combinatorial batch codes,namely the optimal value N(n,k,m.:r;s).It not only has theoretical significance in combinatorial mathematics.but also has important use value in computer information storage.Based on this,we make some optimal constructions and get the optimal values corresponding to specific parameters.