TY - JOUR
T1 - Optimal combinatorial batch codes based on block designs
AU - Silberstein, Natalia
AU - Gál, Anna
N1 - Funding Information:
The authors thank the anonymous referees for their valuable comments that helped to improve the presentation of the paper. N. Silberstein was supported in part at the Technion by a Fine Fellowship. A. Gál was supported in part by NSF Grant CCF-1018060. A preliminary version of the paper is available at http://arxiv.org/abs/1312.5505 .
Publisher Copyright:
© 2014, Springer Science+Business Media New York.
PY - 2016/2/1
Y1 - 2016/2/1
N2 - 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 {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 (Formula presented.). Our constructions are based on affine planes and transversal designs.
AB - 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 {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 (Formula presented.). Our constructions are based on affine planes and transversal designs.
KW - Affine planes
KW - Batch codes
KW - Transversal designs
UR - http://www.scopus.com/inward/record.url?scp=84955285875&partnerID=8YFLogxK
U2 - 10.1007/s10623-014-0007-9
DO - 10.1007/s10623-014-0007-9
M3 - Article
AN - SCOPUS:84955285875
SN - 0925-1022
VL - 78
SP - 409
EP - 424
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 2
ER -