Linear locally repairable codes with availability

In this work, we present a new upper bound on the minimum distance d of linear locally repairable codes (LRCs) with information locality and availability. The bound takes into account the code length n, dimension k, locality r, availability t, and field size q. We use tensor product codes to construct several families of LRCs with information locality, and then we extend the construction to design LRCs with information locality and availability. Some of these codes are shown to be optimal with respect to their minimum distance, achieving the new bound. Finally, we study the all-symbol locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and 4-cycle free regular low-density parity-check (LDPC) codes. We also investigate their optimality using the new bound.

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

[2]  Jehoshua Bruck,et al.  EVENODD: An Efficient Scheme for Tolerating Double Disk Failures in RAID Architectures , 1995, IEEE Trans. Computers.

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

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

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

[6]  A. Robert Calderbank,et al.  Binary cyclic codes that are locally repairable , 2014, 2014 IEEE International Symposium on Information Theory.

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

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

[9]  Alexandros G. Dimakis,et al.  Repairable Fountain Codes , 2014, IEEE J. Sel. Areas Commun..

[10]  Jack K. Wolf,et al.  On codes derivable from the tensor product of check matrices , 1965, IEEE Trans. Inf. Theory.

[11]  Hideki Imai,et al.  Generalized tensor product codes , 1981, IEEE Trans. Inf. Theory.

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

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

[14]  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).

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

[16]  Camilla Hollanti,et al.  Constructions of optimal and almost optimal locally repairable codes , 2014, 2014 4th International Conference on Wireless Communications, Vehicular Technology, Information Theory and Aerospace & Electronic Systems (VITAE).

[17]  Anyu Wang,et al.  Achieving arbitrary locality and availability in binary codes , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[18]  L. Litwin,et al.  Error control coding , 2001 .

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

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