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For example, the encoding of information vector <math>m = (1, 1, 1, 1, 1, 1, 1, 1)</math> gives the codeword <math>c = mG = (8, 8, 5, 9, 21, 3, 36, 31, 32, 12, 2, 20, 37, 33, 21)</math>.
Observe that we constructed an optimal LRC; therefore, using the Singleton bound, we have that the distance of this code is <math>d = n - k - \left\lceil\frac{k}{r}\right\rceil + 2 = 15 - 8 - 2 + 2 = 7</math>. Thus, we can recover any 6 erasures from our codeword by looking at no more than 8 other components.
==Locally Recoverable Codes with Availability==
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