Skip to main content

Linear Batch Codes

  • Conference paper
Coding Theory and Applications

Part of the book series: CIM Series in Mathematical Sciences ((CIMSMS,volume 3))


In an application, where a client wants to obtain many symbols from a large database, it is often desirable to balance the load. Batch codes (introduced by Ishai et al. in STOC 2004) do exactly that: the large database is divided between many servers, so that the client has to only make a small number of queries to every server to obtain sufficient information to reconstruct all desired symbols.

In this work, we formalize the study of linear batch codes. These codes, in particular, are of potential use in distributed storage systems. We show that a generator matrix of a binary linear batch code is also a generator matrix of classical binary linear error-correcting code. This immediately yields that a variety of upper bounds, which were developed for error-correcting codes, are applicable also to binary linear batch codes. We also propose new methods for constructing large linear batch codes from the smaller ones.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. Bar-Yossef, Z., Birk, Y., Jayram, T.S., Kol, T.: Index coding with side information. IEEE Trans. Inf. Theory 57(3), 1479–1494 (2011)

    Article  MathSciNet  Google Scholar 

  2. Bhattacharya, S., Ruj, S., Roy, B.: Combinatorial batch codes: a lower bound and optimal constructions. Adv. Math. Commun. 6(2), 165–174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brualdi, R.A., Kiernan, K., Meyer, S.A., Schroeder, M.W.: Combinatorial batch codes and transversal matroids. Adv. Math. Commun. 4(3), 419–431 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bujtás, C., Tuza, Z.: Batch codes and their applications. Electron. Notes Discret. Math. 38, 201–206 (2011)

    Article  Google Scholar 

  5. Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private information retrieval. In: Proceedings of the 36th Symposium on Foundations of Computer Science (FOCS), Milwaukee, Wisconsin, USA pp. 41–50 (1995)

    Google Scholar 

  6. Dimakis, A.G., Godfrey, P.B., Wu, Y., Wainwright, M.J., Ramchandran, K.: Network coding for distributed storage systems. IEEE Trans. Inf. Theory 59(9), 4539–4551 (2010)

    Article  Google Scholar 

  7. Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Batch codes and their applications. In: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago (2004)

    Google Scholar 

  8. Kushilevitz, E., Ostrovsky, R.: Replication is NOT needed: SINGLE database, computationally-private information retrieval. In: Proceedings of the 38th Symposium on Foundations of Computer Science (FOCS), Miami Beach, Florida, USA pp. 364–373 (1997)

    Google Scholar 

  9. Lipmaa, H.: First CPIR protocol with data-dependent computation. In: Proceedings of the International Conference on Information Security and Cryptology (ICISC), Seoul, South Korea pp. 193–210 (2009)

    Google Scholar 

  10. Lipmaa, H., Skachek, V.: Linear batch codes. Available online

  11. MacWilliams, F.J.: A theorem on the distribution of weights in a systematic code. Bell Syst. Tech. J. 42, 79–94 (1963)

    Article  MathSciNet  Google Scholar 

  12. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1978)

    Google Scholar 

  13. McEliece, R.J., Rodemich, E.R., Rumsey, H., Welch, L.R.: New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inf. Theory IT-23, 157–166 (1997)

    MathSciNet  Google Scholar 

  14. Roth, R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  15. Silberstein, N., Gál, A.: Optimal combinatorial batch codes based on block designs. Available online

  16. Stinson, D., Wei, R., Paterson, M.: Combinatorial batch codes. Adv. Math. Commun. 3(1), 13–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references


We thank Dominique Unruh for helpful discussions. The work of the authors is supported in part by the research grants PUT405 and IUT2-1 from the Estonian Research Council and by the European Regional Development Fund through the Estonian Center of Excellence in Computer Science, EXCS. The work of V. Skachek is also supported in part by the EU COST Action IC1104.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Vitaly Skachek .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Lipmaa, H., Skachek, V. (2015). Linear Batch Codes. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol 3. Springer, Cham.

Download citation

Publish with us

Policies and ethics