Optimal LRC[1,2] 

Description

An LRC whose parameters saturate a generalized Singleton bound.

A \((n,k,r)\) LRC with distance \(d\) is optimal if its parameters \(n\), \(k\), \(d\), and \(q\) are such that the generalized Singleton bound \begin{align} \label{eq:gen-singleton} d\leq n-k-\left \lceil\frac{k}{r}\right \rceil+2 \tag*{(1)}\end{align} becomes an equality. When \(k=r\), the generalized Singleton bound becomes the Singleton bound.

The generalized Singleton bound (1) does not account for \(q\)-ary alphabet size. A more general bound (including the nonlinear case) is given in Ref. [3].

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References

[1]
P. Gopalan et al., “On the Locality of Codeword Symbols”, (2011) arXiv:1106.3625
[2]
D. S. Papailiopoulos and A. G. Dimakis, “Locally Repairable Codes”, (2014) arXiv:1206.3804
[3]
V. R. Cadambe and A. Mazumdar, “Bounds on the Size of Locally Recoverable Codes”, IEEE Transactions on Information Theory 61, 5787 (2015) DOI
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Zoo Code ID: optimal_lrc

Cite as:
“Optimal LRC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/optimal_lrc
BibTeX:
@incollection{eczoo_optimal_lrc, title={Optimal LRC}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/optimal_lrc} }
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“Optimal LRC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/optimal_lrc

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/q-ary_digits/distributed_storage/lrc/optimal_lrc.yml.