# American Institute of Mathematical Sciences

May  2012, 6(2): 165-174. doi: 10.3934/amc.2012.6.165

## Combinatorial batch codes: A lower bound and optimal constructions

 1 Applied Statistics Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India, India 2 School of Information Technology and Engineering, University of Ottawa, Ottawa, K1N6N5, Canada

Received  February 2011 Revised  December 2011 Published  April 2012

Batch codes, introduced by Ishai et al. in [11], are methods for solving the following data storage problem: $n$ data items are to be stored in $m$ servers in such a way that any $k$ of the $n$ items can be retrieved by reading at most $t$ items from each server, and that the total number of items stored in $m$ servers is $N$. A combinatorial batch code (CBC) is a batch code where each data item is stored without change, i.e., each stored data item is a copy of one of the $n$ data items.
One of the basic yet challenging problems is to find optimal CBCs, i.e., CBCs for which total storage ($N$) is minimal for given values of $n$, $m$, $k$, and $t$. In [13], Paterson et al. exclusively studied CBCs and gave constructions of some optimal CBCs.
In this article, we give a lower bound on the total storage ($N$) for CBCs. We give explicit construction of optimal CBCs for a range of values of $n$. For a different range of values of $n$, we give explicit construction of optimal and almost optimal CBCs. Our results partly settle an open problem of [13].
Citation: 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
##### References:
 [1] E. Agrell, A. Vardy and K. Zeger, Upper bounds for constant-weight codes, IEEE Trans. Inform. Theory, 46 (2000), 2373-2395. doi: 10.1109/18.887851. [2] B. Bollobas, "Combinatorics,'' Cambridge University Press, 1986. [3] A. E. Brouwer and T. Etzion, Some new distance-4 constant weight codes, Adv. Math. Commun., 5 (2011), 417-424. doi: 10.3934/amc.2011.5.417. [4] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory, 36 (1990), 1334-1380. doi: 10.1109/18.59932. [5] R. A. Brualdi, K. P. Kiernan, S. A. Meyer and M. W. Schroeder, Combinatorial batch codes and transversal matroids, Adv. Math. Commun., 4 (2010), 419-431. doi: 10.3934/amc.2010.4.419. [6] R. A. Brualdi, K. P. Kiernan, S. A. Meyer and M. W. Schroeder, Erratum: Combinatorial batch codes and transversal matriods, Adv. Math. Commun., 4 (2010), 597-597. doi: 10.3934/amc.2010.4.597. [7] C. Bujtás and Z. Tuza, Optimal combinatorial batch codes derived from dual systems, Miskolc Math. Notes, 12 (2011), 11-23. [8] C. Bujtás and Z. Tuza, Optimal batch codes: many items or low retrieval requirement, Adv. Math. Commnun., 5 (2011), 529-541. doi: 10.3934/amc.2011.5.529. [9] R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43. doi: 10.1109/TIT.1980.1056141. [10] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, 2003. [11] Y. Ishai, E. Kushilevitz, R. Ostrovsky and A. Sahai, Batch codes and their applications, in "Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing,'' 36 (2004), 262-271. doi: 10.1145/1007352.1007396. [12] T. Klove, A lower bound for $A(n, 4, w)$, IEEE Trans. Inform. Theory, 27 (1981), 257-258. doi: 10.1109/TIT.1981.1056318. [13] M. B. Patterson, D. R. Stinson and R. Wei, Combinatorial batch codes, Adv. Math. Commun., 3 (2009), 13-27. doi: 10.3934/amc.2009.3.13. [14] D. H. Smith, L. A. Hughes and S. Perkins, A new table of constant weight codes of length greater than $28$, Electr. J. Combin., 13 (2006), Article A2. [15] C. L. M. Van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 35 (1989), 1324-1329. doi: 10.1109/18.45293.

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##### References:
 [1] E. Agrell, A. Vardy and K. Zeger, Upper bounds for constant-weight codes, IEEE Trans. Inform. Theory, 46 (2000), 2373-2395. doi: 10.1109/18.887851. [2] B. Bollobas, "Combinatorics,'' Cambridge University Press, 1986. [3] A. E. Brouwer and T. Etzion, Some new distance-4 constant weight codes, Adv. Math. Commun., 5 (2011), 417-424. doi: 10.3934/amc.2011.5.417. [4] A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory, 36 (1990), 1334-1380. doi: 10.1109/18.59932. [5] R. A. Brualdi, K. P. Kiernan, S. A. Meyer and M. W. Schroeder, Combinatorial batch codes and transversal matroids, Adv. Math. Commun., 4 (2010), 419-431. doi: 10.3934/amc.2010.4.419. [6] R. A. Brualdi, K. P. Kiernan, S. A. Meyer and M. W. Schroeder, Erratum: Combinatorial batch codes and transversal matriods, Adv. Math. Commun., 4 (2010), 597-597. doi: 10.3934/amc.2010.4.597. [7] C. Bujtás and Z. Tuza, Optimal combinatorial batch codes derived from dual systems, Miskolc Math. Notes, 12 (2011), 11-23. [8] C. Bujtás and Z. Tuza, Optimal batch codes: many items or low retrieval requirement, Adv. Math. Commnun., 5 (2011), 529-541. doi: 10.3934/amc.2011.5.529. [9] R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43. doi: 10.1109/TIT.1980.1056141. [10] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, 2003. [11] Y. Ishai, E. Kushilevitz, R. Ostrovsky and A. Sahai, Batch codes and their applications, in "Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing,'' 36 (2004), 262-271. doi: 10.1145/1007352.1007396. [12] T. Klove, A lower bound for $A(n, 4, w)$, IEEE Trans. Inform. Theory, 27 (1981), 257-258. doi: 10.1109/TIT.1981.1056318. [13] M. B. Patterson, D. R. Stinson and R. Wei, Combinatorial batch codes, Adv. Math. Commun., 3 (2009), 13-27. doi: 10.3934/amc.2009.3.13. [14] D. H. Smith, L. A. Hughes and S. Perkins, A new table of constant weight codes of length greater than $28$, Electr. J. Combin., 13 (2006), Article A2. [15] C. L. M. Van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 35 (1989), 1324-1329. doi: 10.1109/18.45293.
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