New results on Erasure Combinatorial Batch Codes

We investigate in this work the problem of Erasure Combinatorial Batch Codes, in which $n$ files are stored on $m$ servers so that every set of $n-r$ servers allows a client to retrieve at most $k$ distinct files by downloading at most $t$ files from each server. Previous studies have solved this problem for the special case of $t=1$ using Combinatorial Batch Codes. We tackle the general case $t \geq 1$ using a generalization of Hall's theorem. Additionally, we address a realistic scenario in which the retrieved files are consecutive according to some order and provide a simple and optimal solution for this case.