Abstract
To store data reliably, a number of coding schemes including Exact-Minimum Bandwidth Regenerating codes (exact-MBR) and Homomorphic Self Repairing Codes (HSRC) exist. Exact-MBR offers minimum bandwidth usage whereas HSRC has low computational overhead in node repair. We propose a new hybrid scheme, Homomorphic Minimum Bandwidth Repairing Codes, derived from the above coding schemes. Our coding scheme provides two options for node repair operation. The first option offers to repair a node using minimum bandwidth and higher computational complexity while the second one repairs a node using fewer nodes, lower computational complexity and higher bandwidth. In addition, our scheme introduces a basic integrity checking mechanism.
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Notes
- 1.
Field size of \(2^d\) is enough for finding \(d\) linearly independent symbols (and also linearly independent rows). Also encoding matrix requires a finite field with order at least \(n+1\) and for using the homomorphism property the field size must be a power of \(2\).
- 2.
Notice that, \([n,k,d]\) HMBR encodes and stores more amount of data than \([n,k]\) HSRC, when they use the same field size.
- 3.
Notice that, \([n,k,d=k\ge 2]\) HMBR encodes and stores more data than \([n,k]\) HSRC, when they use the same field size.
- 4.
Here, d denotes the symbol size in bits and \(R(x, d, r)\) function counts the number of \(x \times d\) binary sub-matrices having rank \(r\) [4]. In HMBR, \(R(x, d, r)\) can be used for counting all possible live node permutations having at least k linearly independent polynomial inputs.
- 5.
If the same finite field size is used in all coding schemes, while comparing the complexities of the coding schemes, we can ignore the time taken by the finite field operations shown in Table 1.
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Acknowledgments
We would like to thank Frédérique Oggier and Rashmi K. Vinayak for kindly answering our many questions. This study was supported by ÖYP research fund of Turkish Government No:05-DPT-003/35.
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Haytaoglu, E., Dalkilic, M.E. (2013). Homomorphic Minimum Bandwidth Repairing Codes. In: Gelenbe, E., Lent, R. (eds) Information Sciences and Systems 2013. Lecture Notes in Electrical Engineering, vol 264. Springer, Cham. https://doi.org/10.1007/978-3-319-01604-7_33
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DOI: https://doi.org/10.1007/978-3-319-01604-7_33
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