Locally Recoverable Codes are a family of error correction codes that were introduced first by D. S. Papailiopoulos and A. G. Dimakis and have been widely studied in Information theory due to their applications related to Distributive and Cloud Storage Systems.
==Definition & Basic Properties==
'''Definition 1.1''' Let <math>C</math> be a <math>[n, k, d]_{q}</math> linear code. For <math>i \in \{1, \ldots, n\}</math>, let us denote by <math>r_{i}</math> the minimum number of other coordinates we have to look at to recover an erasure in coordinate <math>i</math>. The number <math>r_{i}</math> is said to be the ''locality of the <math>i</math>-th coordinate'' of the code. The ''locality'' of the code is defined as <div style="text-align: center;"><math>r = max\{r_{i}|i \in \{1, \ldots, n\}\}</math></div>
'''Definition 1.2''' An <math>[n, k, d, r]_{q}</math> ''locally recovorablerecoverable code'' (LRC) is an <math>[n, k, d]_{q}</math> linear code <math>C \in F_q^n</math> with locality <math>r</math>.
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'''Definition 1.4''' An <math>[n, k, d, r]_{q}</math>-LRC <math>C</math> is sai to be optimal if the minimum distance of <math>C</math> satisfies <div style="text-align: center;"><math>d = n - k - \lceil\frac{k}{r}\rceil + 2</math></div> By rewriting this new bound as <div style="text-align: center;"><math>d \leq n - k + 1 - ( \lceil\frac{k}{r}\rceil - 1)</math></div> we can see that some of the maximum possible minimum distance is sacrificed to obtain the locality <math>r</math> in our code.
A locally recoverable code is a linear code such that there is a function that takes set of coordinates of a codeword and some specific coordinte and outputs an appropriate coordinate.
