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'''Locally Recoverable Codes''' are a family of [[error correction code]]s that were introduced first by D. S. Papailiopoulos and A. G. Dimakis<ref>{{Citation
|first1=Dimitris S.|last1=Papailiopoulos |first2=Alexandros G. |last2=Dimakis |title="Locally Repairable Codes" |chapter=Locally repairable codes |pages=2771–2775 |___location=Cambridge, MA, USA |publisher=IEEE International Symposium on Information Theory |date=2012 |doi=10.1109/ISIT.2012.6284027|isbn=978-1-4673-2579-0 |arxiv=1206.3804 }}</ref> and have been widely studied in [[Information theory]] due to their applications related to Distributive and [[Cloud storage|Cloud Storage]] Systems.<ref>{{Citation
|first1=A.
|last1=Barg
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|date=2022
|doi=10.3934/amc.2018020
|doi-access=free
}}</ref>
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Let <math>C</math> be an <math>[n, k, d]_{q}</math>-locally recoverable code. Then an erased component can be recovered linearly,<ref>{{Citation
|first1=Dimitris S.|last1=Papailiopoulos |first2=Alexandros G. |last2=Dimakis |title="Locally Repairable Codes" |chapter=Locally repairable codes |pages=2771–2775 |___location=Cambridge, MA, USA |publisher=IEEE International Symposium on Information Theory |date=2012 |doi=10.1109/ISIT.2012.6284027|isbn=978-1-4673-2579-0 |arxiv=1206.3804 }}</ref> i.e. for every <math>i \in \{1, \ldots, n\}</math>, the space of linear equations of the code contains elements of the form <math> x_{i} = f(x_{i_{1}}, \ldots, x_{i_{r}})</math>, where <math>i_{j} \neq i</math>.
==Optimal Locally Recoverable Codes==
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