Elsevier

Discrete Applied Mathematics

Volume 262, 15 June 2019, Pages 127-137
Discrete Applied Mathematics

Some optimal combinatorial batch codes with k=5

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Abstract

A combinatorial batch code was defined by Paterson, Stinson and Wei as a purely combinatorial description of batch codes, which were first introduced by Ishai, Kushilevitz, Ostrovsky and Sahai. It is a system consisting of m subsets of an n-element set such that any k distinct elements can be retrieved by reading at most one (or in general, t) elements from each subset and the number of total elements in m subsets is N. For given parameters n,k,m, the goal is to determine the minimum N, denoted by N(n,k,m). So far, for k5 and m+3n<mk2, precise values of N(n,k,m) have not been established except for some special parameters. In this paper, we determine the N(n,5,m) for nm24+m23+m23, and give a bound on N(n,5,m) for m24+m23+m23<nm3, which is tight for some m,n.

Keywords

Combinatorial batch code
Dual set system
k-Hall Condition
Bipartite graph

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