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Locally recoverable code
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<h2 id="overview">Overview</h2> <p><a href="/facts/Error_correction_code/cLUTUJrA">Erasure-correcting codes</a>, or simply <a href="/facts/Error_correction_code/cLUTUJrA">erasure codes</a>, for distributed and <a href="/facts/Cloud_storage/cNw9IK3Q">cloud storage</a> systems, are becoming more and more popular as a result of the present spike in demand for <a href="/facts/Cloud_computing/guhg0FGV">cloud computing</a> and storage services. This has inspired researchers in the fields of <a href="/facts/Information_theory/qNLH3U5P">information</a> and <a href="/facts/Coding_theory/SBqStoGL">coding theory</a> to investigate new facets of codes that are specifically suited for use with storage systems. </p><p>It is well-known that LRC is a <a href="/facts/Linear_code/HoFI3eyF">code</a> that needs only a limited <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of other symbols to be accessed in order to restore every symbol in a codeword. This idea is very important for distributed and <a href="/facts/Cloud_storage/cNw9IK3Q">cloud storage</a> systems since the most common error case is when one storage node fails (erasure). The main objective is to recover as much <a href="/facts/Data/AUBFN97o">data</a> as possible from the fewest additional storage nodes in order to restore the node. Hence, Locally Recoverable Codes are crucial for such systems. </p><p>The following <a href="/facts/Definition/5SlGKKDJ">definition</a> of the LRC follows from the description above: an [ n , k , r ] {\displaystyle [n,k,r]} -Locally Recoverable Code (LRC) of length n {\displaystyle n} is a <a href="/facts/Linear_code/HoFI3eyF">code</a> that produces an n {\displaystyle n} -symbol codeword from k {\displaystyle k} information symbols, and for any symbol of the codeword, there exist at most r {\displaystyle r} other symbols such that the value of the symbol can be recovered from them. The locality <a href="/facts/Parameter/zFvVFMv9">parameter</a> satisfies 1 ≤ r ≤ k {\displaystyle 1\leq r\leq k} because the entire codeword can be found by accessing k {\displaystyle k} symbols other than the erased symbol. Furthermore, Locally Recoverable Codes, having the minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> d {\displaystyle d} , can recover d − 1 {\displaystyle d-1} erasures. </p> <h2 id="definition">Definition</h2><p> Let C {\displaystyle C} be a [ n , k , d ] q {\displaystyle [n,k,d]_{q}} <a href="/facts/Linear_code/HoFI3eyF">linear code</a>. For i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , let us denote by r i {\displaystyle r_{i}} the minimum number of other <a href="/facts/Coordinate_system/Mtd0q4SP">coordinates</a> we have to look at to recover an erasure in <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate</a> i {\displaystyle i} . The number r i {\displaystyle r_{i}} is said to be the <i>locality of the i {\displaystyle i} -th <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate</a></i> of the code. The <i>locality</i> of the code is defined as </p> r = max { r i ∣ i ∈ { 1 , … , n } } . {\displaystyle r=\max\{r_{i}\mid i\in \{1,\ldots ,n\}\}.} <p>An [ n , k , d , r ] q {\displaystyle [n,k,d,r]_{q}} <i>locally recoverable code</i> (LRC) is an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} <a href="/facts/Linear_code/HoFI3eyF">linear code</a> C ∈ F q n {\displaystyle C\in \mathbb {F} _{q}^{n}} with locality r {\displaystyle r} . </p><p>Let C {\displaystyle C} be an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} -locally recoverable code. Then an erased component can be recovered linearly,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> i.e. for every i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , the space of <a href="/facts/Linear_equation/JfCgALwf">linear equations</a> of the code contains <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> of the form x i = f ( x i 1 , … , x i r ) {\displaystyle x_{i}=f(x_{i_{1}},\ldots ,x_{i_{r}})} , where i j ≠ i {\displaystyle i_{j}\neq i} . </p> <h2 id="optimal-locally-recoverable-codes">Optimal locally recoverable codes</h2> <p>Theorem<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> Let n = ( r + 1 ) s {\displaystyle n=(r+1)s} and let C {\displaystyle C} be an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} -locally recoverable code having s {\displaystyle s} disjoint locality <a href="/facts/Disjoint_sets/nLr8XRVd">sets</a> of size r + 1 {\displaystyle r+1} . Then </p> d ≤ n − k − ⌈ k r ⌉ + 2. {\displaystyle d\leq n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.} <p> An [ n , k , d , r ] q {\displaystyle [n,k,d,r]_{q}} -LRC C {\displaystyle C} is said to be optimal if the minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> of C {\displaystyle C} satisfies </p> d = n − k − ⌈ k r ⌉ + 2. {\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2.} <h2 id="tamobarg-codes">Tamo–Barg codes</h2> <p>Let f ∈ F q [ x ] {\displaystyle f\in \mathbb {F} _{q}[x]} be a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> and let ℓ {\displaystyle \ell } be a positive <a href="/facts/Integer/PUwwrawx">integer</a>. Then f {\displaystyle f} is said to be ( r {\displaystyle r} , ℓ {\displaystyle \ell } )-good if </p> • f {\displaystyle f} has <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> r + 1 {\displaystyle r+1} , • there exist distinct <a href="/facts/Subset/SJGERmbA">subsets</a> A 1 , … , A ℓ {\displaystyle A_{1},\ldots ,A_{\ell }} of F q {\displaystyle \mathbb {F} _{q}} such that – for any i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} , f ( A i ) = { t i } {\displaystyle f(A_{i})=\{t_{i}\}} for some t i ∈ F q {\displaystyle t_{i}\in \mathbb {F} _{q}} , i.e., f {\displaystyle f} is <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a> on A i {\displaystyle A_{i}} , – # A i = r + 1 {\displaystyle \#A_{i}=r+1} , – A i ∩ A j = ∅ {\displaystyle A_{i}\cap A_{j}=\varnothing } for any i ≠ j {\displaystyle i\neq j} . <p>We say that { A 1 , … , A ℓ {\displaystyle A_{1},\ldots ,A_{\ell }} } is a splitting covering for f {\displaystyle f} .<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>Tamo–Barg construction</h3> <p>The Tamo–Barg construction utilizes good polynomials.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> • Suppose that a ( r , ℓ ) {\displaystyle (r,\ell )} -good polynomial f ( x ) {\displaystyle f(x)} over F q {\displaystyle \mathbb {F} _{q}} is given with splitting covering i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} . • Let s ≤ ℓ − 1 {\displaystyle s\leq \ell -1} be a positive <a href="/facts/Integer/PUwwrawx">integer</a>. • Consider the following F q {\displaystyle \mathbb {F} _{q}} -<a href="/facts/Vector_space/M1pxMLx2">vector space</a> of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> V = { ∑ i = 0 s g i ( x ) f ( x ) i : deg ( g i ( x ) ) ≤ deg ( f ( x ) ) − 2 } . {\displaystyle V=\left\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\right\}.} • Let T = ⋃ i = 1 ℓ A i {\textstyle T=\bigcup _{i=1}^{\ell }A_{i}} . • The code { ev T ( g ) : g ∈ V } {\displaystyle \{\operatorname {ev} _{T}(g):g\in V\}} is an ( ( r + 1 ) ℓ , ( s + 1 ) r , d , r ) {\displaystyle ((r+1)\ell ,(s+1)r,d,r)} -optimal locally coverable code, where ev T {\displaystyle \operatorname {ev} _{T}} denotes evaluation of g {\displaystyle g} at all points in the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> T {\displaystyle T} . <h3>Parameters of Tamo–Barg codes</h3> • Length. The length is the number of evaluation points. Because the <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> A i {\displaystyle A_{i}} are <a href="/facts/Disjoint_sets/nLr8XRVd">disjoint</a> for i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} , the length of the code is | T | = ( r + 1 ) ℓ {\displaystyle |T|=(r+1)\ell } . • Dimension. The <a href="/facts/Dimension/0ktmUyac">dimension</a> of the code is ( s + 1 ) r {\displaystyle (s+1)r} , for s {\displaystyle s} ≤ ℓ − 1 {\displaystyle \ell -1} , as each g i {\displaystyle g_{i}} has <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> at most deg ( f ( x ) ) − 2 {\displaystyle \deg(f(x))-2} , covering a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of <a href="/facts/Dimension/0ktmUyac">dimension</a> deg ( f ( x ) ) − 1 = r {\displaystyle \deg(f(x))-1=r} , and by the construction of V {\displaystyle V} , there are s + 1 {\displaystyle s+1} distinct g i {\displaystyle g_{i}} . • Distance. The <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> is given by the fact that V ⊆ F q [ x ] ≤ k {\displaystyle V\subseteq \mathbb {F} _{q}[x]_{\leq k}} , where k = r + 1 − 2 + s ( r + 1 ) {\displaystyle k=r+1-2+s(r+1)} , and the obtained code is the <a href="/facts/Reed%E2%80%93Solomon_error_correction/HPc3Qj6N">Reed-Solomon code</a> of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> at most k {\displaystyle k} , so the minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> equals ( r + 1 ) ℓ − ( ( r + 1 ) − 2 + s ( r + 1 ) ) {\displaystyle (r+1)\ell -((r+1)-2+s(r+1))} . • Locality. After the erasure of the single component, the evaluation at a i ∈ A i {\displaystyle a_{i}\in A_{i}} , where | A i | = r + 1 {\displaystyle |A_{i}|=r+1} , is unknown, but the evaluations for all other a ∈ A i {\displaystyle a\in A_{i}} are known, so at most r {\displaystyle r} evaluations are needed to uniquely determine the erased component, which gives us the locality of r {\displaystyle r} . To see this, g {\displaystyle g} restricted to A j {\displaystyle A_{j}} can be described by a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> h {\displaystyle h} of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> at most deg ( f ( x ) ) − 2 = r + 1 − 2 = r − 1 {\displaystyle \deg(f(x))-2=r+1-2=r-1} thanks to the form of the <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> in V {\displaystyle V} (i.e., thanks to the fact that f {\displaystyle f} is <a href="/facts/Constant_function/BmPx6dn7">constant</a> on A j {\displaystyle A_{j}} , and the g i {\displaystyle g_{i}} 's have <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> at most deg ( f ( x ) ) − 2 {\displaystyle \deg(f(x))-2} ). On the other hand | A j ∖ { a j } | = r {\displaystyle |A_{j}\backslash \{a_{j}\}|=r} , and r {\displaystyle r} evaluations uniquely determine a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> r − 1 {\displaystyle r-1} . Therefore h {\displaystyle h} can be constructed and evaluated at a j {\displaystyle a_{j}} to recover g ( a j ) {\displaystyle g(a_{j})} . <h3>Example of Tamo–Barg construction</h3> <p>We will use x 5 ∈ F 41 [ x ] {\displaystyle x^{5}\in \mathbb {F} _{41}[x]} to construct [ 15 , 8 , 6 , 4 ] {\displaystyle [15,8,6,4]} -LRC. Notice that the <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> of this <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> is 5, and it is <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a> on A i {\displaystyle A_{i}} for i ∈ { 1 , … , 8 } {\displaystyle i\in \{1,\ldots ,8\}} , where A 1 = { 1 , 10 , 16 , 18 , 37 } {\displaystyle A_{1}=\{1,10,16,18,37\}} , A 2 = 2 A 1 {\displaystyle A_{2}=2A_{1}} , A 3 = 3 A 1 {\displaystyle A_{3}=3A_{1}} , A 4 = 4 A 1 {\displaystyle A_{4}=4A_{1}} , A 5 = 5 A 1 {\displaystyle A_{5}=5A_{1}} , A 6 = 6 A 1 {\displaystyle A_{6}=6A_{1}} , A 7 = 11 A 1 {\displaystyle A_{7}=11A_{1}} , and A 8 = 15 A 1 {\displaystyle A_{8}=15A_{1}} : A 1 5 = { 1 } {\displaystyle A_{1}^{5}=\{1\}} , A 2 5 = { 32 } {\displaystyle A_{2}^{5}=\{32\}} , A 3 5 = { 38 } {\displaystyle A_{3}^{5}=\{38\}} , A 4 5 = { 40 } {\displaystyle A_{4}^{5}=\{40\}} , A 5 5 = { 9 } {\displaystyle A_{5}^{5}=\{9\}} , A 6 5 = { 27 } {\displaystyle A_{6}^{5}=\{27\}} , A 7 5 = { 3 } {\displaystyle A_{7}^{5}=\{3\}} , A 8 5 = { 14 } {\displaystyle A_{8}^{5}=\{14\}} . Hence, x 5 {\displaystyle x^{5}} is a ( 4 , 8 ) {\displaystyle (4,8)} -good polynomial over F 41 {\displaystyle \mathbb {F} _{41}} by the definition. Now, we will use this <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> to construct a code of <a href="/facts/Dimension/0ktmUyac">dimension</a> k = 8 {\displaystyle k=8} and length n = 15 {\displaystyle n=15} over F 41 {\displaystyle \mathbb {F} _{41}} . The locality of this code is 4, which will allow us to recover a single <a href="/facts/Server_(computing)/dEaYFD7r">server</a> failure by looking at the information contained in at most 4 other <a href="/facts/Server_(computing)/dEaYFD7r">servers</a>. </p><p>Next, let us define the encoding <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>: f a ( x ) = ∑ i = 0 r − 1 f i ( x ) x i {\displaystyle f_{a}(x)=\sum _{i=0}^{r-1}f_{i}(x)x^{i}} , where f i ( x ) = ∑ i = 0 k r − 1 a i , j g ( x ) j {\displaystyle f_{i}(x)=\sum _{i=0}^{{\frac {k}{r}}-1}a_{i,j}g(x)^{j}} . So, f a ( x ) = {\displaystyle f_{a}(x)=} a 0 , 0 + {\displaystyle a_{0,0}+} a 0 , 1 x 5 + {\displaystyle a_{0,1}x^{5}+} a 1 , 0 x + {\displaystyle a_{1,0}x+} a 1 , 1 x 6 + {\displaystyle a_{1,1}x^{6}+} a 2 , 0 x 2 + {\displaystyle a_{2,0}x^{2}+} a 2 , 1 x 7 + {\displaystyle a_{2,1}x^{7}+} a 3 , 0 x 3 + {\displaystyle a_{3,0}x^{3}+} a 3 , 1 x 8 {\displaystyle a_{3,1}x^{8}} . </p><p>Thus, we can use the obtained encoding <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> if we take our <a href="/facts/Data/AUBFN97o">data</a> to encode as the <a href="/facts/Row_and_column_vectors/pPuRr9Xk">row vector</a> a = {\displaystyle a=} ( a 0 , 0 , a 0 , 1 , a 1 , 0 , a 1 , 1 , a 2 , 0 , a 2 , 1 , a 3 , 0 , a 3 , 1 ) {\displaystyle (a_{0,0},a_{0,1},a_{1,0},a_{1,1},a_{2,0},a_{2,1},a_{3,0},a_{3,1})} . Encoding the <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> m {\displaystyle m} to a length 15 message <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> c {\displaystyle c} by multiplying m {\displaystyle m} by the <a href="/facts/Generator_matrix/V76ugWLN">generator matrix</a> </p> G = ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 32 32 32 32 32 38 38 38 38 38 1 10 16 18 37 2 20 32 33 36 3 7 13 29 30 1 10 16 18 37 23 25 40 31 4 32 20 2 36 33 1 18 10 37 16 4 31 40 23 25 9 8 5 21 39 1 18 10 37 16 5 8 9 39 21 14 17 26 19 6 1 16 37 10 18 8 5 9 21 39 27 15 24 35 22 1 16 37 10 18 10 37 1 16 18 1 37 10 18 16 ) . {\displaystyle G={\begin{pmatrix}1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\1&1&1&1&1&32&32&32&32&32&38&38&38&38&38\\1&10&16&18&37&2&20&32&33&36&3&7&13&29&30\\1&10&16&18&37&23&25&40&31&4&32&20&2&36&33\\1&18&10&37&16&4&31&40&23&25&9&8&5&21&39\\1&18&10&37&16&5&8&9&39&21&14&17&26&19&6\\1&16&37&10&18&8&5&9&21&39&27&15&24&35&22\\1&16&37&10&18&10&37&1&16&18&1&37&10&18&16\end{pmatrix}}.} <p>For example, the encoding of information <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> m = ( 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 ) {\displaystyle m=(1,1,1,1,1,1,1,1)} gives the codeword c = m G = ( 8 , 8 , 5 , 9 , 21 , 3 , 36 , 31 , 32 , 12 , 2 , 20 , 37 , 33 , 21 ) {\displaystyle c=mG=(8,8,5,9,21,3,36,31,32,12,2,20,37,33,21)} . </p><p>Observe that we constructed an optimal LRC; therefore, using the <a href="/facts/Singleton_bound/FEeBetsk">Singleton bound</a>, we have that the <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> of this code is d = n − k − ⌈ k r ⌉ + 2 = 15 − 8 − 2 + 2 = 7 {\displaystyle d=n-k-\left\lceil {\frac {k}{r}}\right\rceil +2=15-8-2+2=7} . Thus, we can recover any 6 erasures from our codeword by looking at no more than 8 other components. </p> <h2 id="locally-recoverable-codes-with-availability">Locally recoverable codes with availability</h2> <p>A code C {\displaystyle C} has all-symbol locality r {\displaystyle r} and availability t {\displaystyle t} if every code symbol can be recovered from t {\displaystyle t} disjoint repair sets of other symbols, each <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Cardinality/m6VPojwT">size</a> at most r {\displaystyle r} symbols. Such codes are called ( r , t ) a {\displaystyle (r,t)_{a}} -LRC.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p><p>Theorem The minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> of [ n , k , d ] q {\displaystyle [n,k,d]_{q}} -LRC having locality r {\displaystyle r} and availability t {\displaystyle t} satisfies the <a href="/facts/Upper_and_lower_bounds/YJje9oC2">upper bound</a> </p> d ≤ n − ∑ i = 0 t ⌊ k − 1 r i ⌋ . {\displaystyle d\leq n-\sum _{i=0}^{t}\left\lfloor {\frac {k-1}{r^{i}}}\right\rfloor .} <p>If the code is <a href="/facts/Systematic_code/DM3P6SDT">systematic</a> and locality and availability apply only to its information symbols, then the code has information locality r {\displaystyle r} and availability t {\displaystyle t} , and is called ( r , t ) i {\displaystyle (r,t)_{i}} -LRC.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>Theorem<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> The minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> d {\displaystyle d} of an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} linear ( r , t ) i {\displaystyle (r,t)_{i}} -LRC satisfies the <a href="/facts/Upper_and_lower_bounds/YJje9oC2">upper bound</a> </p> d ≤ n − k − ⌈ t ( k − 1 ) + 1 t ( r − 1 ) + 1 ⌉ + 2. {\displaystyle d\leq n-k-\left\lceil {\frac {t(k-1)+1}{t(r-1)+1}}\right\rceil +2.} <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Papailiopoulos, Dimitris S.; Dimakis, Alexandros G. (2012), "Locally repairable codes", 2012 IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, USA: IEEE, pp. 2771–2775, arXiv:1206.3804, doi:10.1109/ISIT.2012.6284027, ISBN 978-1-4673-2579-0 <a href="978-1-4673-2579-0" target="_blank">978-1-4673-2579-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Barg, A.; Tamo, I.; Vlăduţ, S. (2015), "Locally recoverable codes on algebraic curves", 2015 IEEE International Symposium on Information Theory, Hong Kong, China: IEEE, pp. 1252–1256, arXiv:1603.08876, doi:10.1109/ISIT.2015.7282656, ISBN 978-1-4673-7704-1 <a href="978-1-4673-7704-1" target="_blank">978-1-4673-7704-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Cadambe, V. R.; Mazumdar, A. (2015), "Bounds on the Size of Locally Recoverable Codes", IEEE Transactions on Information Theory, 61 (11), IEEE: 5787–5794, doi:10.1109/TIT.2015.2477406 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Dukes, A.; Ferraguti, A.; Micheli, G. (2022), "Optimal selection for good polynomials of degree up to five", Designs, Codes and Cryptography, 90 (6), IEEE: 1427–1436, arXiv:2104.01434, doi:10.1007/s10623-022-01046-y <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Haymaker, K.; Malmskog, B.; Matthews, G. (2022), Locally Recoverable Codes with Availability t≥2 from Fiber Products of Curves, doi:10.3934/amc.2018020 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Papailiopoulos, Dimitris S.; Dimakis, Alexandros G. (2012), "Locally repairable codes", 2012 IEEE International Symposium on Information Theory, Cambridge, MA, USA, pp. 2771–2775, arXiv:1206.3804, doi:10.1109/ISIT.2012.6284027, ISBN 978-1-4673-2579-0{{citation}}: CS1 maint: location missing publisher (link) <a href="978-1-4673-2579-0" target="_blank">978-1-4673-2579-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Cadambe, V.; Mazumdar, A. 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