Locally recoverable code (LRC)

Description

Any code for which, given a codeword \(x\) and coordinate \(i\), \(x_i\) can be recovered from (at most \(r\)) other coordinates of \(x\). An \(r\)-locally recoverable code of length \(n\) and dimension \(k\) is denoted as an \((n,k,r)\) LRC code.

More technically, a \(q\)-ary code \(C\) with length \(n\) is \(r\)-locally recoverable, or has locality \(r\), if \(\forall i \in [n]\), there exists \(I_i \subset [n]\setminus i\) such that \(|I_i|\leq r\), and the projection of the set \(\mathcal{C}(i,a)=\{x\in C : x_i=a\}\) on to the coordinates in \(I_i\), i.e., \(\mathcal{C}_{I_i}(i,a)\) is disjoint from \(\mathcal{C}_{I_i}(i,a^\prime)\) where \(a\neq a^\prime\).

The definition can be generalized to \(t\)-LRC code, if every coordinate is recoverable from \(t\) disjoint subsets of coordinates. A study on the parameters of \(t\)-LRC codes, together with known bounds, can be found in Ref. [1].

Protection

The distance, \(d\), of an \((n,k,r)\) LRC code satisfies \begin{align} d\leq n-k-\left \lceil\frac{k}{r}\right \rceil+2~,\label{eq:gen-singleton} \tag*{(1)}\end{align} where \(r\leq k\). When \(k=r\), the bound on the distance gives the Singleton bound. The generalized Singleton bound (1) does not account for \(q\)-ary alphabet size. A more generalized bound (including the non-linear case) is given in Ref. [2].

Rate

The rate \(r\) of an \((n,k,r)\) LRC code satisfies \begin{align} \frac{k}{n}\leq\frac{r}{r+1}~. \tag*{(2)}\end{align}

Realizations

An \((18,14,7)\) LRC code has beed used in the Windows Azure cloud storage system [3]; see also Sec. 31.3.1.2 in Ref. [4].

Parents

Children

  • Batch code — A systematic batch code with restricted size of reconstruction sets can recover any query of \(t\) information symbols with recovery sets of size \(r\) [5][6].
  • Maximum distance separable (MDS) code — MDS codes are most efficient in terms of minimizing storage overhead for handling erasures. They are locally recoverable with locality \(k\).
  • Sequential-recovery code

References

[1]
I. Tamo, A. Barg, and A. Frolov, “Bounds on the Parameters of Locally Recoverable Codes”, IEEE Transactions on Information Theory 62, 3070 (2016) DOI
[2]
V. R. Cadambe and A. Mazumdar, “Bounds on the Size of Locally Recoverable Codes”, IEEE Transactions on Information Theory 61, 5787 (2015) DOI
[3]
C. Huang, H. Simitci, Y. Xu, A. Ogus, B. Calder, P. Gopalan, J. Li, and S. Yekhanin. Erasure coding in Windows Azure Storage. In Proc. USENIX Annual Technical Conference (ATC), pgs. 15-26, Boston, Massachusetts, June 2012.
[4]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]
V. Skachek, “Batch and PIR Codes and Their Connections to Locally Repairable Codes”, Network Coding and Subspace Designs 427 (2018) DOI
[6]
A.-E. Riet, V. Skachek, and E. K. Thomas, “Batch Codes for Asynchronous Recovery of Data”, IEEE Transactions on Information Theory 68, 1545 (2022) DOI
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Zoo Code ID: locally_recoverable

Cite as:
“Locally recoverable code (LRC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/locally_recoverable
BibTeX:
@incollection{eczoo_locally_recoverable, title={Locally recoverable code (LRC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/locally_recoverable} }
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“Locally recoverable code (LRC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/locally_recoverable

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/distributed_storage/locally_recoverable.yml.