An Integer Programming-Based Bound for Locally Repairable Codes

The locally repairable code (LRC) studied in this paper is an [n, k] linear code of which the value at each coordinate can be recovered by a linear combination of at most r other coordinates. The central problem in this paper is to determine the largest possible minimum distance for LRCs. First, an integer programming-based upper bound is derived for any LRC. Then, by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters n<sub>1</sub> > n<sub>2</sub>, where n<sub>1</sub> = ⌈(n/r + 1)⌉ and n2 = n<sub>1</sub>(r +1)-n. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field F<sub>2</sub><sub>m</sub>,where m ≥ n<sub>1</sub>r. Based on these r ≤ √n - 1 has been definitely determined, which is of great results, the largest possible minimum distance for all LRCs with significance in practical use.

[1]  Itzhak Tamo,et al.  Bounds on locally recoverable codes with multiple recovering sets , 2014, 2014 IEEE International Symposium on Information Theory.

[2]  Frédérique Oggier,et al.  Self-repairing homomorphic codes for distributed storage systems , 2010, 2011 Proceedings IEEE INFOCOM.

[3]  Chau Yuen,et al.  Optimal Locally Repairable Linear Codes , 2014, IEEE Journal on Selected Areas in Communications.

[4]  Zhifang Zhang,et al.  Repair locality from a combinatorial perspective , 2014, 2014 IEEE International Symposium on Information Theory.

[5]  Dimitris S. Papailiopoulos,et al.  Optimal locally repairable codes and connections to matroid theory , 2013, 2013 IEEE International Symposium on Information Theory.

[6]  P. Vijay Kumar,et al.  Optimal linear codes with a local-error-correction property , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[7]  Cheng Huang,et al.  Erasure Coding in Windows Azure Storage , 2012, USENIX Annual Technical Conference.

[8]  Frédérique E. Oggier,et al.  Locally repairable codes with multiple repair alternatives , 2013, 2013 IEEE International Symposium on Information Theory.

[9]  Zhifang Zhang,et al.  Repair Locality With Multiple Erasure Tolerance , 2014, IEEE Transactions on Information Theory.

[10]  Sriram Vishwanath,et al.  Optimal locally repairable codes via rank-metric codes , 2013, 2013 IEEE International Symposium on Information Theory.

[11]  Sergey Yekhanin,et al.  On the locality of codeword symbols in non-linear codes , 2013, Discret. Math..

[12]  Dimitris S. Papailiopoulos,et al.  Simple regenerating codes: Network coding for cloud storage , 2011, 2012 Proceedings IEEE INFOCOM.

[13]  Arya Mazumdar,et al.  An upper bound on the size of locally recoverable codes , 2013, 2013 International Symposium on Network Coding (NetCod).

[14]  P. Vijay Kumar,et al.  Codes with locality for two erasures , 2014, 2014 IEEE International Symposium on Information Theory.

[15]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[16]  Cheng Huang,et al.  On the Locality of Codeword Symbols , 2011, IEEE Transactions on Information Theory.

[17]  Dimitris S. Papailiopoulos,et al.  Locality and Availability in Distributed Storage , 2014, IEEE Transactions on Information Theory.

[18]  Dimitris S. Papailiopoulos,et al.  XORing Elephants: Novel Erasure Codes for Big Data , 2013, Proc. VLDB Endow..

[19]  Dimitris S. Papailiopoulos,et al.  Locally Repairable Codes , 2012, IEEE Transactions on Information Theory.

[20]  Minghua Chen,et al.  Pyramid Codes: Flexible Schemes to Trade Space for Access Efficiency in Reliable Data Storage Systems , 2007, Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007).

[21]  Itzhak Tamo,et al.  A Family of Optimal Locally Recoverable Codes , 2013, IEEE Transactions on Information Theory.