Optimal combinatorial batch codes derived from dual systems

Combinatorial batch codes with parameters n, k, m, and t D 1 may be viewed as set systems F consisting of n subsets over an m-element set (repetitions allowed), satisfying the following restricted version of Hall’s Condition: for every 1 i k, the union of any i members of F has cardinality at least i . An optimization problem is to determine N.n;k;m/, the minimum total size P F 2F jF j in such systems. Beside its theoretical interest, the problem has strong practical motivation, too, concerning distributed storage and retrieval of data in a database. Already the case n D mC 2 turns out to be somewhat complicated. Here we give explicit optimal constructions and prove the following formulae: in the range k m kC p k