# American Institute of Mathematical Sciences

February  2018, 12(1): 49-65. doi: 10.3934/amc.2018003

## On erasure combinatorial batch codes

 Department of Mathematics, Marshall University, Huntington, WV 25755, USA

Received  November 2015 Revised  July 2017 Published  March 2018

Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers, each of which stores a subset of the items. A batch code is an arrangement for storing items on servers so that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Silberstein defined an erasure batch code (with redundancy $r$) as a batch code in which any $k$ items can be retrieved by reading at most $t$ items from each server, while any $r$ servers are unavailable (failed).

In this paper, we investigate erasure batch codes with $t = 1$ (each server can read at most one item) in a combinatorial manner. We determine the optimal (minimum) total storage of an erasure batch code for several ranges of parameters. Additionally, we relate optimal erasure batch codes to maximum packings. We also identify a necessary lower bound for the total storage of an erasure batch code, and we relate parameters for which this trivial lower bound is achieved to the existence of graphs with appropriate girth.

Citation: 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
##### References:
 [1] E. Abajo and A. Diánez, Graphs with maximum size and lower bounded girth, Appl. Math. Lett., 25 (2012), 575-579.  doi: 10.1016/j.aml.2011.09.062.  Google Scholar [2] 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.  Google Scholar [3] C. Bujtás and Z. Tuza, Optimal batch codes: many items or low retrieval requirement, Adv. Math. Commun., 5 (2011), 529-541.  doi: 10.3934/amc.2011.5.529.  Google Scholar [4] C. Bujtás and Z. Tuza, Optimal combinatorial batch codes derived from dual systems, Miskolc Math. Notes, 12 (2011), 11-23.   Google Scholar [5] C. Bujtás and Z. Tuza, Relaxations of Hall's condition: optimal batch codes with multiple queries, Appl. Anal. Discrete Math., 6 (2012), 72-81.  doi: 10.2298/AADM111130024B.  Google Scholar [6] Y. Ishai, E. Kushilevitz, R. Ostrovsky and A. Sahai, Batch codes and their applications, in Proc. 36th Ann. ACM Symp. Theory Comp., ACM, New York, 2004,262–271. doi: 10.1145/1007352.1007396.  Google Scholar [7] G. B. Khosrovshahi and R. Laue, t-Designs with t ≥ 3, in Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), 2nd edition, Chapman & Hall/CRC, Boca Raton, 2007, 79–100. Google Scholar [8] W. H. Mills and R. C. Mullin, Coverings and packings, in Contemporary Design Theory, Wiley, New York, 1992,371–399.  Google Scholar [9] M. B. Paterson, D. R. Stinson and R. Wei, Combinatorial batch codes, Adv. Math. Commun., 3 (2009), 13-27.  doi: 10.3934/amc.2009.3.13.  Google Scholar [10] N. Silberstein, Fractional repetition and erasure batch codes, in Coding Theory and Applications (eds. R. Pinto, P. Rocha Malonek and P. Vettori), Springer, 2015,335–343. doi: 10.1007/978-3-319-17296-5_36.  Google Scholar [11] H. Zhang, E. Yaakobi and N. Silberstein, Multiset combinatorial batch codes, in 2017 IEEE Int. Symp. Inf. Theory (ISIT), IEEE, 2017. doi: 10.1109/ISIT.2017.8006916.  Google Scholar

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##### References:
 [1] E. Abajo and A. Diánez, Graphs with maximum size and lower bounded girth, Appl. Math. Lett., 25 (2012), 575-579.  doi: 10.1016/j.aml.2011.09.062.  Google Scholar [2] 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.  Google Scholar [3] C. Bujtás and Z. Tuza, Optimal batch codes: many items or low retrieval requirement, Adv. Math. Commun., 5 (2011), 529-541.  doi: 10.3934/amc.2011.5.529.  Google Scholar [4] C. Bujtás and Z. Tuza, Optimal combinatorial batch codes derived from dual systems, Miskolc Math. Notes, 12 (2011), 11-23.   Google Scholar [5] C. Bujtás and Z. Tuza, Relaxations of Hall's condition: optimal batch codes with multiple queries, Appl. Anal. Discrete Math., 6 (2012), 72-81.  doi: 10.2298/AADM111130024B.  Google Scholar [6] Y. Ishai, E. Kushilevitz, R. Ostrovsky and A. Sahai, Batch codes and their applications, in Proc. 36th Ann. ACM Symp. Theory Comp., ACM, New York, 2004,262–271. doi: 10.1145/1007352.1007396.  Google Scholar [7] G. B. Khosrovshahi and R. Laue, t-Designs with t ≥ 3, in Handbook of Combinatorial Designs (eds. C. J. Colbourn and J. H. Dinitz), 2nd edition, Chapman & Hall/CRC, Boca Raton, 2007, 79–100. Google Scholar [8] W. H. Mills and R. C. Mullin, Coverings and packings, in Contemporary Design Theory, Wiley, New York, 1992,371–399.  Google Scholar [9] M. B. Paterson, D. R. Stinson and R. Wei, Combinatorial batch codes, Adv. Math. Commun., 3 (2009), 13-27.  doi: 10.3934/amc.2009.3.13.  Google Scholar [10] N. Silberstein, Fractional repetition and erasure batch codes, in Coding Theory and Applications (eds. R. Pinto, P. Rocha Malonek and P. Vettori), Springer, 2015,335–343. doi: 10.1007/978-3-319-17296-5_36.  Google Scholar [11] H. Zhang, E. Yaakobi and N. Silberstein, Multiset combinatorial batch codes, in 2017 IEEE Int. Symp. Inf. Theory (ISIT), IEEE, 2017. doi: 10.1109/ISIT.2017.8006916.  Google Scholar
Ranges of the parameter $n$ addressed here, in terms of $k$, $m$, and $r$, always assuming $t = 1$ and that the conditions of Lemma 3 are met. Theorem 8 applies when $k \leq n \leq m$, and Theorem 9 applies when $n > m$ and $k = m-r$. Theorem 13 applies to certain values of $n$ less than $(k-1)\binom{m}{r+k-1}$, while Theorem 12 applies when $(k-1)\binom{m}{r+k-1} \leq n$. When $n = (k-1)\binom{m}{r+k-1}$, the constructions in the latter two theorems are the same
The matrix $A$ constructed as in the proof of Theorem 8 for $m = 6, n = 4, r = 3,$ and $k\leq 3$
The matrix $A$ constructed as in the proof of Theorem 9 when $m = 6, n = 7, r = 3,$ and $k = 3$
The matrix $A$ constructed as in the proof of Theorem 12 when $m = 4, k = 2, r = 1,$ and $n = 8\geq (k-1)\binom{m}{r+k-1}$
A $1-\text{ECBC}(6, 3, 5)$ illustrating that $F(3, 5, 1)\geq 6$. See Example 18
A maximal 2-(5, 3, 2) packing design
(a) a 1-ECBC(7, 3, 6) achieving minimal weight 14 and (b) its corresponding graph outlined in Theorem 28

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