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: To see this, <math>g</math> restricted to <math>A_j</math> can be described by a [[polynomial]] <math>h</math> of [[Degree of a polynomial|degree]] at most <math>{\text{deg}}(f(x))-2 = r + 1 - 2 = r - 1</math> thanks to the form of the [[Element (mathematics)|elements]] in <math>V</math> (i.e. thanks to the fact that <math>f</math> is [[Constant (mathematics)|constant]] on <math>A_j</math>, and the <math>g_i</math>'s have [[Degree of a polynomial|degree]] at most <math>{\text{deg}}(f(x))-2</math>). On the other hand <math>|A_j \backslash \{a_j\}| = r</math>, and <math>r</math> evaluations uniquely determine a [[polynomial]] of [[Degree of a polynomial|degree]] <math>r-1</math>. Therefore <math>h</math> can be constructed and evaluated at <math>a_j</math> to recover <math>g(a_j)</math>.
=== Example of
We will use <math>x^5 \in \mathbb F_{41} [x]</math> to construct <math>[15, 8, 6, 4]</math>-LRC. Notice that the [[Degree of a polynomial|degree]] of this [[polynomial]] is 5, and it is [[Constant (mathematics)|constant]] on <math>A_i</math> for <math>i \in \{1, \ldots, 8\}</math>, where <math>A_1 = \{1, 10, 16, 18, 37\}</math>, <math>A_2 = 2A_1</math>, <math>A_3 = 3A_1</math>, <math>A_4 = 4A_1</math>, <math>A_5 = 5A_1</math>, <math>A_6 = 6A_1</math>, <math>A_7 = 11A_1</math>, and <math>A_8 = 15A_1</math>: <math>A_1^5 = \{1\}</math>, <math>A_2^5 = \{32\}</math>, <math>A_3^5 = \{38\}</math>, <math>A_4^5 = \{40\}</math>, <math>A_5^5 = \{9\}</math>, <math>A_6^5 = \{27\}</math>, <math>A_7^5 = \{3\}</math>, <math>A_8^5 = \{14\}</math>. Hence, <math>x^5</math> is a <math>(4,8)</math>-good polynomial over <math>\mathbb F_{41}</math> by the definition. Now, we will use this [[polynomial]] to construct a code of [[dimension]] <math>k = 8</math> and length <math>n = 15</math> over <math>\mathbb F_{41}</math>. The locality of this code is 4, which will allow us to recover a single [[Server (computing)|server]] failure by looking at the information contained in at most 4 other [[Server (computing)|servers]].
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