### Abstract

Combinatorial batch codes model the storage of a database on a given number of servers such that any k or fewer items can be retrieved by reading at most t items from each server. A combinatorial batch code with parameters n, k,m, t can be represented by a system F of n (not necessarily distinct) sets over an m-element underlying set X, such that for any k or fewer members of F there exists a system of representatives in which each element of X occurs with multiplicity at most t. The main purpose is to determine the minimum N(n, k,m, t) of total data storage ∑ _{F∈F} |F| over all combinatorial batch codes F with given parameters. Previous papers concentrated on the case t = 1. Here we obtain the first nontrivial results on combinatorial batch codes with t > 1. We determine N(n, k,m, t) for all cases with k ≤ 3t, and also for all cases where n ≥. Our results can be considered equivalently as minimum total size ∑ _{F∈F} |F| over all set systems F of given order m and size n, which satisfy a relaxed version of Hall's Condition; that is, |∪?'| ≥ |F'|/t holds for every subsystem F' ⊆ F of size at most k.

Original language | English |
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Pages (from-to) | 72-81 |

Number of pages | 10 |

Journal | Applicable Analysis and Discrete Mathematics |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 2012 |

### Keywords

- Combinatorial batch code
- Dual system
- Hall-type condition
- System of representatives

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics