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Fractional Repetition and Erasure Batch Codes

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


Batch codes are a family of codes that represent a distributed storage system (DSS) of n nodes so that any batch of t data symbols can be retrieved by reading at most one symbol from each node. Fractional repetition codes are a family of codes for DSS that enable efficient uncoded repairs of failed nodes. In this work these two families of codes are combined to obtain fractional repetition batch (FRB) codes which provide both uncoded repairs and parallel reads of subsets of stored symbols. In addition, new batch codes which can tolerate node failures are considered. This new family of batch codes is called erasure combinatorial batch codes (ECBCs). Some properties of FRB codes and ECBCs and examples of their constructions based on transversal designs and affine planes are presented.


  • Fractional repetition codes
  • Batch codes
  • Transversal designs
  • Affine planes

This research was supported in part by the Fine Fellowship and by the Israeli Science Foundation (ISF), Jerusalem, Israel, under Grant 10/12.

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  1. Anderson, I.: Combinatorial Designs and Tournaments. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  2. Bhattacharya, S., Ruj, S., Roy, B.K.: Combinatorial batch codes: a lower bound and optimal constructions. Adv. Math. Commun. 3(1), 165–174 (2012). doi:10.3934/amc.2012.6.165

    MathSciNet  CrossRef  Google Scholar 

  3. Bujtás, C., Tuza, Z.: Optimal batch codes: Many items or low retrieval requirement. Adv. Math. Commun. 5(3), 529–541 (2011). doi:10.3934/amc.2011.5.529

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Dimakis, A., Godfrey, P., Wu, Y., Wainwright, M., Ramchandran, K.: Network coding for distributed storage systems. IEEE Trans. Inf. Theory 56(9), 4539–4551 (2010). doi:10.1109/TIT.2010.2054295

    CrossRef  Google Scholar 

  5. Dimakis, A.G., Ramchandran, K., Wu, Y., Suh, C.: A survey on network codes for distributed storage. Proc. IEEE 99, 476–489 (2011)

    CrossRef  Google Scholar 

  6. El Rouayheb, S., Ramchandran, K.: Fractional repetition codes for repair in distributed storage systems. In: Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Urbana-Champaign, pp. 1510–1517 (2010)

    Google Scholar 

  7. Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Batch codes and their applications. In: Proceedings of the 36th Annual ACM Symposium on the Theory of computing (STOC’04), Chicago, pp. 262–271 (2004). doi:10.1145/1007352.1007396

    Google Scholar 

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

    Google Scholar 

  9. Olmez, O., Ramamoorthy, A.: Repairable replication-based storage systems using resolvable designs. In: Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, pp. 1174–1181 (2012). doi:10.1109/Allerton.2012.6483351

    Google Scholar 

  10. Paterson, M.B., Stinson, D.R., Wei, R.: Combinatorial batch codes. Adv. Math. Commun. 3(1), 13–27 (2009). doi:10.3934/amc.2009.3.13

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Pawar, S., Noorshams, N., El Rouayheb, S., Ramchandran, K.: Dress codes for the storage cloud: simple randomized constructions. In: Proceedings of the 2011 IEEE International Symposium on Information Theory (ISIT 2011), St. Petersburg, pp. 2338–2342 (2011). doi:10.1109/ISIT.2011.6033980

  12. Rashmi, K.V., Shah, N., Kumar, P.: Optimal exact-regenerating codes for distributed storage at the MSR and MBR points via a product-matrix construction. IEEE Trans. Inf Theory 57(8), 5227–5239 (2011). doi:10.1109/TIT.2011.2159049

    MathSciNet  CrossRef  Google Scholar 

  13. Shah, N., Rashmi, K.V., Kumar, P., Ramchandran, K.: Distributed storage codes with repair-by-transfer and nonachievability of interior points on the storage-bandwidth tradeoff. IEEE Trans. Inf. Theory 58(3), 1837–1852 (2012). doi:10.1109/TIT.2011.2173792

    MathSciNet  CrossRef  Google Scholar 

  14. Silberstein, N., Etzion, T.: Optimal fractional repetition codes (2014). arXiv:1401.4734

  15. Silberstein, N., Gál, A.: Optimal combinatorial batch codes based on block designs (2013). Accepted for publication in Designs, Codes and Cryptography (Springer) http://dx.doi:10.1007/s10623-014-0007-9

    Google Scholar 

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The author thanks Tuvi Etzion and Mark Silberstein for the valuable discussions. The author also wishes to thank COST Action IC1104 “Random Network Coding and Designs over GF(q)” on travel support to present this work.

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Correspondence to Natalia Silberstein .

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Silberstein, N. (2015). Fractional Repetition and Erasure Batch Codes. In: Pinto, R., Rocha Malonek, P., Vettori, P. (eds) Coding Theory and Applications. CIM Series in Mathematical Sciences, vol 3. Springer, Cham.

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