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## 3 Explicit Maximally Recoverable Codes with Locality

### Citations

2647 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ... )⌉ ) . We also obtain further improvements in the special cases of h = 3 and h = 4. The only lower bound for the alphabet size known currently comes from results on the main conjecture for MDS codes =-=[14]-=- and is Ω(k). One way to construct maximally-recoverable local codes is by picking coefficients in heavy parities at random from a large enough finite field. In order to compare our constructions with... |

815 | Finite Fields - Lidl, Niederreiter - 1997 |

333 | Network Coding for Distributed Storage Systems
- Dimakis, Godfrey, et al.
- 2007
(Show Context)
Citation Context ...ions. There are several other models of codes with locality in the literature. The ones most closely related to our work include SD codes [3, 17], locally decodable codes [20], and regenerating codes =-=[6]-=-. 1.2 Organization In section 2 we formally define data-local and local (k, r, h)-codes. We introduce the notion of maximal recoverability, and show that maximally-recoverable local codes yield maxima... |

81 | Pyramid Codes: Flexible Schemes to Trade Space for Access Efficiency in Reliable Data Storage Systems
- Huang, Chen, et al.
- 2007
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Citation Context ...y) parity symbols, may depend on all k data symbols. In what follows we refer to a group of r data symbols and their local parity defined above as a local group. Data-local codes have been studied in =-=[11, 10, 16, 18, 19, 9]-=-. The importance of this topology was partially explained in [10, Theorem 9]. There it has been shown that in case h < r + 1, any systematic [n, k] code that corrects all patterns of (h + 1) erasures,... |

64 | Locally Repairable Codes
- Papailiopoulos, Dimakis
- 2012
(Show Context)
Citation Context ...y) parity symbols, may depend on all k data symbols. In what follows we refer to a group of r data symbols and their local parity defined above as a local group. Data-local codes have been studied in =-=[11, 10, 16, 18, 19, 9]-=-. The importance of this topology was partially explained in [10, Theorem 9]. There it has been shown that in case h < r + 1, any systematic [n, k] code that corrects all patterns of (h + 1) erasures,... |

59 | XORing Elephants: Novel Erasure Codes for Big Data
- Sathiamoorthy, Asteris, et al.
- 2013
(Show Context)
Citation Context ...y) parity symbols, may depend on all k data symbols. In what follows we refer to a group of r data symbols and their local parity defined above as a local group. Data-local codes have been studied in =-=[11, 10, 16, 18, 19, 9]-=-. The importance of this topology was partially explained in [10, Theorem 9]. There it has been shown that in case h < r + 1, any systematic [n, k] code that corrects all patterns of (h + 1) erasures,... |

29 | Optimal Linear Codes with a Local-Error-Correction Property
- Prakash, Kamath, et al.
- 2012
(Show Context)
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15 | Recursive constructions for large caps,”
- Edel, Bierbrauer
- 1999
(Show Context)
Citation Context ... of the parity check matrix of the code C ′ from [21, Theorem 5], where we set q = 2r and d = 4. S ′2 naturally defines a set S2 ⊆ F2t that is 3-independent over F2r . We remark that using results in =-=[8]-=- one can get further small improvements upon the theorem above. Theorem 18. Let positive integers k, r, h = 4 be such that ℓ = k+h r is an integer. Let m be the smallest integer such that 3 | (m− 1) a... |

15 |
Locally decodable codes,” Foundations and trends in theoretical computer science,
- Yekhanin
- 2012
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Citation Context ...ts {αi} used in our constructions. There are several other models of codes with locality in the literature. The ones most closely related to our work include SD codes [3, 17], locally decodable codes =-=[20]-=-, and regenerating codes [6]. 1.2 Organization In section 2 we formally define data-local and local (k, r, h)-codes. We introduce the notion of maximal recoverability, and show that maximally-recovera... |

11 |
PartailMDS codes and their application to RAID type of architectures
- Blaum, Hafner, et al.
- 2012
(Show Context)
Citation Context ...ng the XOR of respective data symbols. There also are h heavy parities where each heavy parity depends on all k data symbols. In what follows we refer to codes above as data-local (k, r, h)-codes. In =-=[4]-=- motivated by applications to data storage on SSDs the authors studied systematic linear [n, k] codes with two extra parameters r and h, where r | (k + h). In codes of [4] there are k data symbols and... |

10 | SD Codes: Erasure Codes Designed for How Storage Systems Really Fail”
- Plank, Blaum, et al.
- 2012
(Show Context)
Citation Context ...that we explicitly specify the sets {αi} used in our constructions. There are several other models of codes with locality in the literature. The ones most closely related to our work include SD codes =-=[3, 17]-=-, locally decodable codes [20], and regenerating codes [6]. 1.2 Organization In section 2 we formally define data-local and local (k, r, h)-codes. We introduce the notion of maximal recoverability, an... |

7 |
On sets of vectors of a finite vector space in which every subset of basis size is a basis II, Des. Codes Cryptogrophy.
- Ball, Beule
- 2012
(Show Context)
Citation Context ...r the h× (k+ h) parity check matrix of the code C ′ with entries in Fq. By [1, Lemma 1.2], k + h 6 q + h− 1. Details regarding the recent progress on the main conjecture for MDS codes can be found in =-=[1, 2]-=-. In particular, results there allow one to get small non-asymptotic improvements upon Theorem 19. 4.1 Random codes One way to construct maximally-recoverable local codes is by picking coefficients in... |

6 | On maximally recoverable property for multi-protection group codes,” - Chen, Huang, et al. - 2007 |

6 | On the locality of codeword symbols in non-linear codes,” Arxiv 1303.3921,
- Forbes, Yekhanin
- 2013
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4 |
Sergey Yekhanin. Erasure coding in Windows Azure Storage
- Huang, Simitci, et al.
- 2012
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Citation Context ...t this coordinate can be recovered by accessing at most r other coordinates. Recently there has been two lines of work on codes with locality. In [10] motivated by applications to distributed storage =-=[12]-=- the authors studied systematic linear [n, k] codes that tolerate up to h+ 1 erasures, but also have locality r for all information coordinates. In canonical codes of [10], r divides k and n = k r +h.... |

3 |
Nonbinary double-error-correcting codes designed by means of algebraic varieties,”
- Dumer
- 1995
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Citation Context ...tomorphisms (6) implies ∑ i∈S γiλ j i , for all 0 6 j 6 h− 1 which contradicts the proprieties of the Vandermonde determinant. Example 16. Instantiating Theorem 15 with k = 60, r = h = 4, we obtain a =-=[80, 60, 7]-=- maximally recoverable (60, 4, 4) local code over the field F216 . Prior to our work [4, Theorem 4.2] a code with such parameters was not known to exist over any field of size below 280. In the proof ... |

2 |
Huseyin Simitci, and Sergey Yekhanin. On the locality of codeword symbols
- Gopalan, Huang
- 2012
(Show Context)
Citation Context ...cting code has locality r if, when erased, the value at this coordinate can be recovered by accessing at most r other coordinates. Recently there has been two lines of work on codes with locality. In =-=[10]-=- motivated by applications to distributed storage [12] the authors studied systematic linear [n, k] codes that tolerate up to h+ 1 erasures, but also have locality r for all information coordinates. I... |