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:• <math>f</math> has degree <math>r+1</math>,
:• there exist <math>A_{1}</math> , . . ., <math>A_{\ell}</math> distinct [[Subset|subsets]] of <math>\mathbb F_{q}</math> such that
::– for any <math>i \in \{1, \ldots, \ell\}</math>, <math>f</math> (<math>A_{i}</math> ) = {<math>t_{i}</math>} for some <math>t_{i}</math> ∈ <math>\mathbb F_{q}</math> , i.e. f is [[Constant (mathematics)|constant]] on <math>A_{i}</math>,
::– #<math>A_{i}</math> = <math>r + 1</math>,
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<div style="text-align: center;"><math>V = \{\sum_{i=0}^s g_{i}(x)f(x)^i:{\text{deg}}(g_{i}(x)) \leq {\text{deg}}(f(x))-2\}</math></div>
:• Let <math>T</math> = <math display="inline">\bigcup_{i=1}^\ell A_i</math>
:• 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 Tamo—Barg Codes ===
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:• '''Length.''' The length is the number of evaluation points. Because the sets <math>A_i</math> are disjoint for <math>i \in \{1, \ldots, \ell\}</math>, the length of the code is <math>|T| = (r+1)\ell</math>.
:• '''Dimension.''' The [[dimension]] of the code is <math>(s+1)r</math>, for <math>s</math> ≤ <math>\ell-1</math>, as each <math>g_i</math> has degree at most <math>{\text{deg}}(f(x))-2</math>, covering a [[vector space]] of dimension <math>{\text{deg}}(f(x))-1=r</math>, and by the construction of <math>V</math>, there are <math>s+1</math> distinct <math>g_i</math>.
:• '''Distance.''' The distance is given by the fact that <math>V \subseteq \mathbb F_q [x]_{\leq k}</math>, where <math>k = r + 1 - 2 + s(r+1)</math>, and the obtained code is the [[Reed–Solomon error correction|Reed-Solomon code]] of degree at most <math>k</math>, so the minimum distance equals <math>(r+1)\ell-((r+1)-2+s(r+1))</math>.
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