### Abstract

Combinatorial batch codes were introduced by Ishai et al. [36^{th} ACM STOC (2004), 262-271] and studied in detail by Paterson et al. [Adv. Math. Commun., 3 (2009), 13-27] for the purpose of distributed storage and retrieval of items of a database on a given number of servers in an economical way. A combinatorial batch code with parameters n, k, m, t means that n items are stored on m servers such that any k different items can be retrieved by reading out at most t items from each server. If t = 1, this can equivalently be represented with a family F of n not necessarily distinct sets over an m-element underlying set, such that the union of any i members of F has cardinality at least i, for every 1 ≤ P i ≤ k. The goal is to determine the minimum N(n, k,m) of ∑ F ∈ F |F| over all combinatorial batch codes F with given parameters n; k;m and t = 1. Together with the results of Paterson et al. for n larger, this completes the determination of N(n, 3,m). We also compute N(n, 4,m) in the entire range n ≥ m ≥ 4. Several types of code transformations keeping optimality are described, too.

Original language | English |
---|---|

Pages (from-to) | 529-541 |

Number of pages | 13 |

Journal | Advances in Mathematics of Communications |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 2011 |

### Fingerprint

### Keywords

- Batch code
- Dual system
- Hypergraph
- System of distinct representatives

### ASJC Scopus subject areas

- Computer Networks and Communications
- Algebra and Number Theory
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*Advances in Mathematics of Communications*,

*5*(3), 529-541. https://doi.org/10.3934/amc.2011.5.529

**Optimal batch codes : Many items or low retrieval requirement.** / Bujtás, Csilla; Tuza, Z.

Research output: Contribution to journal › Article

*Advances in Mathematics of Communications*, vol. 5, no. 3, pp. 529-541. https://doi.org/10.3934/amc.2011.5.529

}

TY - JOUR

T1 - Optimal batch codes

T2 - Many items or low retrieval requirement

AU - Bujtás, Csilla

AU - Tuza, Z.

PY - 2011/8

Y1 - 2011/8

N2 - Combinatorial batch codes were introduced by Ishai et al. [36th ACM STOC (2004), 262-271] and studied in detail by Paterson et al. [Adv. Math. Commun., 3 (2009), 13-27] for the purpose of distributed storage and retrieval of items of a database on a given number of servers in an economical way. A combinatorial batch code with parameters n, k, m, t means that n items are stored on m servers such that any k different items can be retrieved by reading out at most t items from each server. If t = 1, this can equivalently be represented with a family F of n not necessarily distinct sets over an m-element underlying set, such that the union of any i members of F has cardinality at least i, for every 1 ≤ P i ≤ k. The goal is to determine the minimum N(n, k,m) of ∑ F ∈ F |F| over all combinatorial batch codes F with given parameters n; k;m and t = 1. Together with the results of Paterson et al. for n larger, this completes the determination of N(n, 3,m). We also compute N(n, 4,m) in the entire range n ≥ m ≥ 4. Several types of code transformations keeping optimality are described, too.

AB - Combinatorial batch codes were introduced by Ishai et al. [36th ACM STOC (2004), 262-271] and studied in detail by Paterson et al. [Adv. Math. Commun., 3 (2009), 13-27] for the purpose of distributed storage and retrieval of items of a database on a given number of servers in an economical way. A combinatorial batch code with parameters n, k, m, t means that n items are stored on m servers such that any k different items can be retrieved by reading out at most t items from each server. If t = 1, this can equivalently be represented with a family F of n not necessarily distinct sets over an m-element underlying set, such that the union of any i members of F has cardinality at least i, for every 1 ≤ P i ≤ k. The goal is to determine the minimum N(n, k,m) of ∑ F ∈ F |F| over all combinatorial batch codes F with given parameters n; k;m and t = 1. Together with the results of Paterson et al. for n larger, this completes the determination of N(n, 3,m). We also compute N(n, 4,m) in the entire range n ≥ m ≥ 4. Several types of code transformations keeping optimality are described, too.

KW - Batch code

KW - Dual system

KW - Hypergraph

KW - System of distinct representatives

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U2 - 10.3934/amc.2011.5.529

DO - 10.3934/amc.2011.5.529

M3 - Article

VL - 5

SP - 529

EP - 541

JO - Advances in Mathematics of Communications

JF - Advances in Mathematics of Communications

SN - 1930-5346

IS - 3

ER -