Locally recoverable code: Difference between revisions

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:• 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>.

:• Let <math>s \leq \ell-1</math> be a positive [[integer]].

:• Consider the following <math>\mathbb F_{q}F_q</math>-[[vector space]] of [[polynomial]]s <math display="block">V = \left\{\sum_{i=0}^s g_{i}g_i(x)f(x)^i: \deg(g_{i}g_i(x)) \leq \deg(f(x))-2 \right\}.</math>

:• Let <math display="inline">T = \bigcup_{i=1}^\ell A_i</math>.

:• The code <math> \{ ev_\operatorname{Tev}_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_\operatorname{Tev}_T</math> denotes evaluation of <math>g</math> at all points in the [[Set (mathematics)|set]] <math>T</math>.

=== Parameters of Tamo–Barg codes ===