Optimal combinatorial batch codes based on block designs

Abstract

Batch codes, introduced by Ishai, Kushilevitz, Ostrovsky and Sahai, represent the distributed storage of an n-element data set on m servers in such a way that any batch of k data items can be retrieved by reading at most one (or more generally, t) items from each server, while keeping the total storage over m servers equal to N . This paper considers a class of batch codes (for t = 1), called combinatorial batch codes (CBCs), where each server stores a subset of a database. A CBC is called optimal if the total storage N is minimal for given n, m, and k. A c-uniform CBC is a combinatorial batch code where each item is stored in exactly c servers. A c-uniform CBC is called optimal if its parameter n has maximum value for given m and k. Optimal c-uniform CBCs have been known only for c ∈ {2, k − 1, k − 2}. In this paper we present new constructions of optimal CBCs in both the uniform and general settings, for values of the parameters where tight bounds have not been established previously. In the uniform setting, we provide constructions of two new families of optimal uniform codes with c ∼ √ k. Our constructions are based on affine planes and transversal designs.

Multiset combinatorial batch codes

“…Several works have considered codes under this setup; see e.g. [2,3,4,5,6,7,8,9,16,18,19]. However, note that combinatorial batch codes are not multiset batch codes and don't allow to request an item more than once.…”

“…Several constructions of optimal uniform CBCs were given in [2,3,16,19]. In this paper we consider a slightly different class of MCBCs, in which each server stores the same number of items, and call these codes regular multiset combinatorial batch codes (regular MCBCs).…”

“…Example 2. The following is a (20,80,16,16)-CBC given in [19] based on an affine plane of order 4. Here, V = [20], each column contains the indices of items stored in a server C i ∈ C and also forms a block of the set system (V, C).…”

“…It is well known that an affine plane exists for any prime power q [10]. The next result of CBCs based upon affine planes was given in [19].…”

Fractional Repetition and Erasure Batch Codes

“…it maximizes M = M(n, k, α, ρ) (see [6,14] for details). We say that a uniform combinatorial batch code is an optimal code if it stores the maximum number of symbols, i.e., it maximizes θ = θ (n, ρ,t) (see [2,10,15]). It was proved recently [15] that combinatorial batch codes based on some transversal designs are (near) optimal CBCs.…”