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Let <math>C</math> be an <math>[n, k, d]_{q}</math>-locally recoverable code. Then a deleted component can be recovered linearly, 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}, \ldots, x_{i_{r}})</math>, where <math>i_{j} \neq i</math>.
==Optimal Locally Recoverable Codes==
'''Theorem 1.3''' Let <math>n = (r+1)s</math> and let <math>C</math> be an <math>[n, k, d]_{q}</math>-locally recoverable code having <math>s</math> disjoint locality sets of size <math>r+1</math>. Then, <div style="text-align: center;"><math>d \leq n - k - \lceil\frac{k}{r}\rceil + 2</math></div>
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