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An <math>[n, k, d, r]_{q}</math>-LRC <math>C</math> is said to be optimal if the minimum [[Hamming distance|distance]] of <math>C</math> satisfies <div style="text-align: center;"><math>d = n - k - \left\lceil\frac{k}{r}\right\rceil + 2</math></div>
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Let <math>f</math> ∈ <math>\mathbb F_{q} [x]</math> be a [[polynomial]] and let <math>\ell</math> be a positive [[integer]]. Then <math>f</math> is said to be (<math>r</math>, <math>\ell</math>)-good if
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|arxiv=1806.11492 }}</ref>
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The
|first1=I.|last1=Tamo |first2=A. |last2=Barg |title="A family of optimal locally recoverable code" |chapter=A family of optimal locally recoverable codes |pages=686–690 |___location=Honolulu, HI, USA |publisher=IEEE International Symposium on Information Theory |date=2014 |doi=10.1109/ISIT.2014.6874920|isbn=978-1-4799-5186-4 }}</ref>
:• Suppose that a <math>(r, \ell)</math>-good polynomial <math>f(x)</math> over <math>\mathbb F_{q}</math> is given with splitting covering <math>i \in \{1, \ldots, \ell\}</math>.
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:• The code <math> \{ ev_{T}(g):g \in V \}</math> is an <math>((r+1)\ell,(s+1)r,d,r)</math> optimal locally coverable code, where <math>ev_{T}</math> denotes evaluation of g at all points in the [[Set (mathematics)|set]] <math>T</math>.
=== Parameters of
:• '''Length.''' The length is the number of evaluation points. Because the [[Set (mathematics)|sets]] <math>A_i</math> are [[Disjoint sets|disjoint]] for <math>i \in \{1, \ldots, \ell\}</math>, the length of the code is <math>|T| = (r+1)\ell</math>.
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