American Institute of Mathematical Sciences

Advanced
February  2009, 3(1): 13-27. doi: 10.3934/amc.2009.3.13

Combinatorial batch codes

M. B. Paterson 1, , D. R. Stinson 2, and  R. Wei 3,

1. 

Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
2. 

David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
3. 

Department of Computer Science, Lakehead University, hunder Bay, ON, P7B 5E1, Canada

Received  July 2008 Revised  December 2008 Published  January 2009

In this paper, we study batch codes, which were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [4]. A batch code specifies a method to distribute a database of $n$ items among $m$ devices (servers) in such a way that any $k$ items can be retrieved by reading at most $t$ items from each of the servers. It is of interest to devise batch codes that minimize the total storage, denoted by $N$, over all $m$ servers.
We restrict out attention to batch codes in which every server stores a subset of the items. This is purely a combinatorial problem, so we call this kind of batch code a ''combinatorial batch code''. We only study the special case $t=1$, where, for various parameter situations, we are able to present batch codes that are optimal with respect to the storage requirement, $N$. We also study uniform codes, where every item is stored in precisely $c$ of the $m$ servers (such a code is said to have rate $1/c$). Interesting new results are presented in the cases $c = 2, k-2$ and $k-1$. In addition, we obtain improved existence results for arbitrary fixed $c$ using the probabilistic method.
Keywords: Batch codes, combinatorial set system..
Mathematics Subject Classification: Primary: 05B30, 05D05; Secondary: 68P20, 68R0.
Citation: M. B. Paterson, D. R. Stinson, R. Wei. Combinatorial batch codes. Advances in Mathematics of Communications, 2009, 3 (1) : 13-27. doi: 10.3934/amc.2009.3.13
[1]

JiYoon Jung, Carl Mummert, Elizabeth Niese, Michael Schroeder. On erasure combinatorial batch codes. Advances in Mathematics of Communications, 2018, 12 (1) : 49-65. doi: 10.3934/amc.2018003
[2]

Richard A. Brualdi, Kathleen P. Kiernan, Seth A. Meyer, Michael W. Schroeder. Combinatorial batch codes and transversal matroids. Advances in Mathematics of Communications, 2010, 4 (3) : 419-431. doi: 10.3934/amc.2010.4.419
[3]

Yuebo Shen, Dongdong Jia, Gengsheng Zhang. The results on optimal values of some combinatorial batch codes. Advances in Mathematics of Communications, 2018, 12 (4) : 681-690. doi: 10.3934/amc.2018040
[4]

Srimanta Bhattacharya, Sushmita Ruj, Bimal Roy. Combinatorial batch codes: A lower bound and optimal constructions. Advances in Mathematics of Communications, 2012, 6 (2) : 165-174. doi: 10.3934/amc.2012.6.165
[5]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53
[6]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[7]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83
[8]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59
[9]

Csilla Bujtás, Zsolt Tuza. Optimal batch codes: Many items or low retrieval requirement. Advances in Mathematics of Communications, 2011, 5 (3) : 529-541. doi: 10.3934/amc.2011.5.529
[10]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[11]

Pradeep Kumar Mishra, Vassil Dimitrov. A combinatorial interpretation of double base number system and some consequences. Advances in Mathematics of Communications, 2008, 2 (2) : 159-173. doi: 10.3934/amc.2008.2.159
[12]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191
[13]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[14]

Irene Márquez-Corbella, Edgar Martínez-Moro. Algebraic structure of the minimal support codewords set of some linear codes. Advances in Mathematics of Communications, 2011, 5 (2) : 233-244. doi: 10.3934/amc.2011.5.233
[15]

Finley Freibert. The classification of complementary information set codes of lengths $14$ and $16$. Advances in Mathematics of Communications, 2013, 7 (3) : 267-278. doi: 10.3934/amc.2013.7.267
[16]

Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial & Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073
[17]

Chongyang Liu, Zhaohua Gong, Enmin Feng, Hongchao Yin. Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture. Journal of Industrial & Management Optimization, 2009, 5 (4) : 835-850. doi: 10.3934/jimo.2009.5.835
[18]

Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007
[19]

Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369
[20]

Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (137)
  • HTML views (0)
  • Cited by (27)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences