Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

Transcription

1 Bouds ad Costructios for Liear Locally Repairable Codes over Biary Fields Ayu Wag *, Zhifag Zhag, ad Dogdai Li * * State Key Laboratory of Iformatio Security, Istitute of Iformatio Egieerig, CAS, Beijig, Chia KLMM, NCMIS, Academy of Mathematics ad Systems Sciece, Uiversity of Chiese Academy of Scieces s: wagayu@iie.ac.c, zfz@amss.ac.c, ddli@iie.ac.c arxiv: v cs.it 4 Ja 07 Abstract For biary, k, d liear locally repairable codes (LRCs), two ew upper bouds o k are derived. The first oe applies to LRCs with disjoit local repair groups, for geeral values of, d ad locality r, cotaiig some previously kow bouds as special cases. The secod oe is based o solvig a optimizatio problem ad applies to LRCs with arbitrary structure of local repair groups. Particularly, a explicit boud is derived from the secod boud whe d 5. A specific compariso shows this explicit boud outperforms the Cadambe- Mazumdar boud for 5 d 8 ad large values of. Moreover, a costructio of biary liear LRCs with d 6 attaiig our secod boud is provided. I. INTRODUCTION Recetly, locally repairable codes (LRCs) have attracted a lot of attetio due to their applicatios i distributed storage systems. A, k, d liear code is called a LRC with locality r if the value at each coordiate ca be recovered by accessig at most r other coordiates. A LRC with small locality r is preferred i practice as it greatly reduces the disk I/O complexity i repairig ode failures. Meatime, large values of k ad d are also desirable to esure high level of storage efficiecy ad global fault tolerace ability respectively. Much work has bee doe toward explorig the relatioship betwee the parameters, k, d, r. The first trade-off is derived i 4, i.e., d k k +, () r which is also kow as the Sigleto-like boud for LRCs. The various methods are developed to costruct LRCs attaiig the boud (), e.g.,, 5, 9,. Tightess of the sigleto-like boud is discussed i 6, 7, ad some improved bouds are derived i 3, 3. It ca be see the boud () does ot care about the field size. However, i practice LRCs over small fiite fields, especially those over biary fields, are preferred due to their coveiece i implemetatio. The first trade-off takig ito cosideratio the field size is derived by Cadambe ad Mazumdar, i.e., k Mi t Z + tr + k (q) opt( (r + )t, d), () where k (q) opt(, d) is the largest possible dimesio of a, k, d liear code over F q. This trade-off is usually called the C-M boud, ad is proved achievable by the biary Simplex codes. Aother class of biary LRCs with r =, 3 costructed via aticodes i 6 also meets () with equality. Although these codes are optimal with respect to the C-M boud, their code legth icreases expoetially as the dimesio k grows, implyig poor performace i iformatio rate. Alteatively, cyclic codes provides more desirable cadidates for LRCs over small fields. I 5, biary LRCs with r = ad d =, 6, 0 are costructed from primitive cyclic codes. These codes do ot attai the C-M boud, but are show to be optimal uder a structural assumptio that the codeword coordiates are divided ito disjoit local repair groups. The same method is adopted i 4 to geerate biary LRCs with r =, d = 0 from oprimitive cyclic codes. I, BCHtype biary LRCs are costructed as the subfield subcodes of optimal Reed-Solomo-Type LRCs. Besides, some other approaches for costructig LRCs that attai the Sigletolike boud () over small fields are also developed i 7, 4. Recetly, two upper bouds takig the field size ito accout are derived i 8 for (r, δ)-lrcs. Sice (r, δ)-lrcs cotai LRCs as the special case of δ =, these two bouds apply to LRCs as well. For δ =, the first boud is equivalet to the Sigleto-like boud (), while the secod oe is a liear programmig boud for LRCs with disjoit repair groups. O the other had, asymptotic bouds o the parameters of LRCs are studied i, 0. Overall, most of the bouds derived so far for LRCs over particular fiite fields either deped o udetermied parameters i codig theory, e.g., the C-M boud, or rely o solvig optimizatio problems uder cocrete code parameters, e.g., the LP boud i 8. Ad the costructios of biary LRCs with good parameters mostly restrict to specific values of d ad r. Much work remais udoe for LRCs over particular fiite fields. I this work, we focus o liear LRCs over biary fields. A. Mai Idea ad Cotributio For ay, k, d biary liear LRC C, our mai idea for derivig upper bouds o k is to cosider a related sphere packig problem i a particular space, amely, the L-space. Specifically, a L-space of C is defied to be the dual of the liear space spaed by a miimum set of local parity checks with overall supports coverig all coordiates. Actually, the L-space ca be viewed as a LRC which cotais C as a

2 subcode. The by applyig the sphere packig boud i the L-space, two upper bouds for C are derived. Firstly, assumig the code C has disjoit local repair groups, we get a explicit boud (i.e. Corollary 3) o k for geeral values of, d, r. Note that for r = ad special forms of, d, upper bouds were also derived i 5, 4. It tus out our boud cotais their results as special cases. Secodly, for liear biary LRCs with arbitrary local repair groups, we derive a upper boud (i.e. Theorem 5) o k based o solvig a optimizatio problem. Although it is geerally difficult to solve this optimizatio problem, simplificatio ca be doe for d 5, ad thus a explicit upper boud (i.e. Theorem 6) is derived. Through a specific compariso, we show this boud ca outperform the C-M boud for 5 d 8 ad large values of. Moreover, a class of biary liear LRCs with d 6 attaiig this explicit boud is costructed. B. Orgaizatio Sectio II defies the L-space, ad derives our first boud (i.e. Corollary 3). Sectio III presets our secod boud (Theorem 6). Sectio IV gives the costructio attaiig our secod boud. Sectio V cocludes the paper. II. THE L-SPACE FOR LRCS For ay vector v = (v,..., v ) F, let Supp(v) = {i : v i 0} ad wt(v) = Supp(v), where = {,,..., }. For ay two vectors u, v F, dist(u, v) deotes the hammig distace of u ad v. Deote by Spa (u,..., u l ) the liear space spaed by a set of vectors {u,..., u l } over F. Let C be a, k, d biary liear LRC with locality r. The for each coordiate i, there is a local parity check h i C such that i Supp(h i ) ad wt(h i ) r +. Note by local parity checks we mea the codewords i the dual code C with weight at most r +. Defiitio. Let H {h,..., h } be a set of local parity checks of C such that h H Supp(h) =, ad h H Supp(h) for ay H H. We call H a L- cover of C. Deote H = Spa (H), the the dual space of H, i.e., V = {v F v h = 0, h H}, is called a L-space of C. Obviously, a L-cover H cotais the miimum umber of local parity checks guarateeig the locality r for all coordiates. H eeds ot be uique, either does the L-space V. Our proofs i this paper oly deped o their existece which is esured by the defiitio of LRCs. Sice H oly cotais partial parity checks of C, it follows that V also defies a LRC cotaiig C as a subcode. Ivestigatig the structure of V may help us to study the code C. I the followig, we use the sphere-packig boud i the L-space V, ad obtai a coectio betwee k, d ad V. Propositio. For a, k, d biary LRC C with locality r, it holds ( k dim(v) log BV ( d ) ), (3) where B V ( ) = {v V : wt(v) }, ad V is a L-space of C. Proof: For ay codeword c C, cosider the ball of radius aroud c i V. Sice C has miimum distace d, the these balls are o-overlappig. It follows that c C B V (c, d ) V, (4) where B V (c, ) = {v V : dist(v, c) }. Note that C V, so we have B V (c, ) = BV ( ), c C. Therefore (4) ca be writte as C B V ( d ) V. (5) Sice log C = k ad log V = dim(v), the the Theorem follows directly from (5). The right had side of (3) depeds o the locality space V, so explicit boud ca be derived from (3) if V is kow. I the followig, we apply Propositio to a special class of biary liear LRCs which has a clear L-space. A. Boud for LRCs with Disjoit Local Repair Groups Assume C has local parity checks h i, h i,..., h il C satisfyig l j= Supp(h i j ) =, wt(h ij ) = r + ad Supp(h ij ) Supp(h ij ) = for j j l. Obviously, r + ad l =. Such a LRC is usually said to have disjoit local repair groups, which is widely adopted i costructios of LRCs, e.g.,, 5, 9,. Uder this assumptio, the structure of the L-space V becomes quite simple. The based o Propositio, we derive the followig upper boud. Corollary 3. For ay, k, d biary LRC C with locality r that has disjoit local repair groups, it holds k r + log ( 0 i + +i l r + ). (6) Proof: Note that H = {h i, h i,..., h il } is a L- cover of C, the V = Spa (H) is a L-space of C. By Propositio, it suffices to determie dim(v) ad B V ( ). Clearly it has dim(v) =. Note that the liear space Spa (H) has weight eumerator polyomial W H (x, y) = (x + y ) l. The by the MacWilliams equality, (see e.g., 9), the weight eumerator polyomial of V is W V (x, y) = l W H(x + y, x y) i j = l ((x + y) + (x y) ) l = ( r + x i y i) l i i 0 = A u x u y u, 0 u

3 where A u = i + +i l =u l j= ( i j ). Thus we have B V ( d ) = A0 + + A 4 = 0 i + +i l r + ), ad (6) follows directly. The sphere packig approach was also used i 5, 4 to derive upper bouds o k for biary liear LRCs with disjoit local repair groups. However, their approach oly applies to the case of r = because it relies o a map from biary liear LRCs with r = to additive F 4 -codes. Our boud works for geeral values of, d, r, especially cotaiig the bouds i 5, 4 as special cases. For example, suppose = m, d = 6 ad r =, the Corollary 3 implies that k 3 log ( r + r + ) + l 0 = 3 log ( + ) = 3 (m ) m, which coicides with the Theorem i 5. Aother boud for LRCs with disjoit repair groups is the LP boud derived i 8. Table lists a compariso of the boud (7), the LP boud i 8 ad the C-M boud () for 3 r 0, = 3, d = 5. From the table we ca see the boud (6) is slightly weaker tha the LP boud but tighter tha the C-M boud (). Nevertheless, the boud (6) has a explicit form ad ca be more easily implemeted tha the other two bouds. i j r Our boud (6) The C-M boud () The LP boud Table III. NEW UPPER BOUND FOR BINARY LINEAR LRCS I this sectio we will remove the assumptio of disjoit local repair groups, ad derive parameter bouds for liear biary LRCs with arbitrary local repair groups. Suppose C is a, k, d biary liear LRC with locality r. Let H = {h i,..., h il } C be a L-cover of C. By shorteig at the coordiates that appear more tha oce i the supports of h i,..., h il, we ca derive from C a shorteed code C which has disjoit local repair groups. Thus the problem is reduced to that we discussed i last sectio. Specifically, defie L F l to be a matrix whose rows are the l local parity checks i H. Deote by N the umber of colums i L that have weight. Let L be the matrix obtaied from L by deletig the N colums of L that have weight greater tha. The takig L as the parity check matrix defies a biary code C. It ca be proved C has the followig properties. Lemma 4. The shorteed code C is a N, K, D biary liear LRC satisfyig (i) N l(r + ), K N ( k), D d; (ii) C has a L-cover H = {h i,..., h i l } such that wt(h i j ) r + ad Supp(h i j ) Supp(h i j ) = for all j j. Proof: Sice the shorteig operatio either icreases the redudacy or decreases the miimum distace, (see e.g., 9), the it has K N ( k) ad D d. To show the other statemets, we suppose without loss of geerality that L = ( L, L ), where L F l N cosists of the N colums that have weight, ad L F l ( N) cosists of the other ( N) colums that have weight. By coutig the umber of s i L, we have l(r + ) the umber of s i L N + ( N). Thus l(r + ) N. Lastly, deote h,..., h l to be the rows of L, the clearly {h,..., h l} is a set of parity checks of C such that wt(h i) r +. Note that each colum of L has exactly oe, therefore l i= Supp(h i) = N ad Supp(h i) Supp(h j) = for all i j, which completes the proof. Let V = Spa (H ) be a L-space of C, ad deote wt(h i j ) = r j + for j l. The it follows dim(v ) = N l, ad by a deductio similar to that i Corollary 3 it has B V ( D ) = 0 i + +i l D rj +. i j Applyig Propositio to the shorteed LRC C, we get ( rj + ). K (N l) log (7) i j 0 i + +i l D The combiig with Lemma 4, we get the followig theorem. Theorem 5. For ay, k, d biary liear LRC with locality r, it holds k Mi l + log (Φ l (r,..., r l )), (8) where Φ l (r,..., r l )= 0 i + +i l rj + ad the Mi is take over all itegers l, r,..., r l such that l r+ ; 0 r,..., r l r; (9) r + + r l = l(r + ). i j

4 Proof: From Lemma 4 it has K N ( k), D d. The k K N + (a) l log ( 0 i + +i l D (b) l log (Φ l (r,..., r l )), rj + ) where (a) is from (7) ad (b) holds because D d. Note that the itegers l, r,..., r l satisfies l; 0 r,..., r l r; l j= r j l(r + ). There are two cases. Case : l > r+. O the oe had, we have k l < r+. O the other had, with the restrictio (9), it has Mi l + log (Φ l (r,..., r l )) ( ) r + + log Φ (0,..., 0) r+ = r +. So iequality (8) holds. Case : l r+. I this case it has l r+ ; 0 r,..., r l r; l j= r j l(r + ). i j So we have k Mi l + log (Φ l (r,..., r l )),. (0) where the Mi is take over (0). Note that l(r+) 0, so a ecessary coditio for optimizig (8) is that l j= r j = l(r + ). The the optimizatio ca be restricted to the coditio (9), ad thus the theorem holds. For ay give, d, r, Theorem 5 gives a upper boud o the dimesio k based o solvig a optimizatio problem. However, solvig the optimizatio problem is very difficult i geeral sice the objective fuctio i (8) is oliear. Nevertheless, it is still possible to simplify the boud (8) i some special cases. Next, we will derive a explicit upper boud from Theorem 5 for d 5. A. Explicit Boud for d 5 Theorem 6. For ay, k, d biary liear LRC with locality r such that d 5 ad r, it holds k r + mi{log ( + ), }. () (r + )(r + ) Proof: Whe d 5, it has 4 ad therefore rj + Φ l (r,..., r l ) i 0 i + +i l j= j r + rl + = Note that x+ = x(x + ) is a covex real-valued fuctio ad it is required i (9) that r + + r l = l(r + ), so l Φ l (r,..., r l ) + l( l j= (r ) j + ) = + ( l(r + )) ( l(r + ))). l It follows from Theorem 5 that k Mi l + log (Φ l (r,..., r l )) ( ( l(r + ))( l(r + )) ) Mi l+log +, l l where the iteger l satisfies Let l r+ accordig to (9). f(l) = l + log ( + ( l(r + ))( l(r + ))) l be a fuctio defied for itegers l, r+. We claim f(l) r + + mi{log ( + ), (r + )(r + ) } for r, the the theorem follows directly. Firstly, we show that f (l) 0 for r. Note that f (l)= 804 (l (r +3r+) 8 ) 6l( 3 (3+r) ) l (4 +l ( + 3r + r ) + l( (3 + r))) l, the it suffices to prove g(l) 0 for r, where g(l) = 80 4 (l (r +3r+) 8 ) 6l( 3 (3+r) ). Sice g (l) = 4l(r + 3r + ) < 0, it has g (l) is a cocave fuctio with g ( ) = 4 (4r+4 r ) < 0 ad g ( r+ ) = 6 (+r+r) r+ > 0. It follows that g(l) max{g, g( r + r + )} = max{ 3 (6(r + ) (r + ) ) (r + ), 63 ((r + ) ) (r + ) } 0 for all r. Accordig to f (l) 0, f(l) is a cocave fuctio, the we have f(l) mi{f, f( r + r + )} = mi{ r + + log ( + ), r + } = r + + mi{log ( + ), ad therefore the theorem follows. (r + )(r + ) },

5 Sice we focus o liear codes, the k is actually upper bouded by the largest iteger o more tha the right had side of the iequality. For sufficietly large, it always holds ()(r+) log ( + ) <. More specifically, this iequality holds wheever 5(r + )(r + ). Therefore, the boud () ca be further simplified as k log ( + ). Next, we give a compariso betwee the boud () ad the C-M boud (), where the upper boud o k () opt(, d) is computed by usig SageMath 8 ad the web database 0. For 5 d 8, accordig to our computatio, the boud () ca always outperform the C-M boud for large values of. Specifically, Fig. ad Fig. display comparisos of the two bouds for r = 3, d = 5, 0 60 ad r =, d = 8, 60 0 respectively. Moreover i Table, based o a detailed calculatio of the two bouds for r 5 ad 50, we list the tippig poits of s that the boud () is tighter tha the C-M boud thereafter. r = r = 3 r = 4 r = 5 d = d = d = d = Table For d 9, it ca be checked that the boud () is iferior to the C-M boud. A reaso causig this disadvatage is that i this case it has 4 while the boud () is derived by lower-boudig Φ l by its value i the case 4 =. If we use a better lower boud of Φ l (r,..., r l ) istead of that used i the proof of Theorem 6, a upper boud tighter tha () could be expected. However, we ca ot get a explicit boud i this case yet sice the correspodig optimizatio problem is still very complicated. IV. CONSTRUCTION ATTAINING THE UPPER BOUND I this sectio, we give a ew costructio of biary liear LRCs. The code has miimum distace d 6, ad attais the upper boud () i Theorem 6. The costructio relies o two matrices A ad B defied as follows. Suppose s ad t are two positive itegers such that t s ad s t. Let A be a biary matrix of size t t such that ay 4 colums of A are liearly idepedet. For t, A ca be chose as the idetity matrix. For t 3, A is a parity check matrix of a t, t t, 5 biary code which ca be costructed from oprimitive cyclic codes of legth t + (see e.g., 3). We give a detailed costructio of A i Appedix A. Defie B to be a matrix whose colums are all ozero s t -tuples from F t with first ozero etry equal to. The B is actually a parity check matrix of a t -ary Hammig code, ad the size of B is s t s t. By fixig a basis of F t over F, each vector i F t ca be writte as a elemet i F t ad vice versa. We deote by a,..., a t F t the t elemets correspodig to the colums of A, ad deote by a vector β i F s t the ith colum t k 40 The C-M boud Our ew boud (0) Fig.. A compariso of the C-M boud ad the boud i Theorem 6 for r = 3, d = 5, k 65 The C-M boud 60 Our ew boud (0) Fig.. A compariso of the C-M boud ad the boud i Theorem 6 for r =, d = 8, of B for i costructed below. s t. The the biary liear LRC is Costructio. Defie C to be a biary liear code with the parity check matrix H = L L... L l, H H... H l where l = s t, ad for i l, L i is a l ( t + ) matrix whose i-th row is the all-oe vector ad the other rows are all-zero vectors, H i is a s ( t + ) matrix over F whose colums are biary expasios of the vectors {0, a β i, a β i,..., a tβ i }. Example. Suppose s = 4 ad t =. The we ca choose A = 0 F 0, B = 0 ω F 5 ω 0 4, where ω is a primitive elemet i F 4 such that ω +ω+ = 0. Fixig a basis {, ω}, the two colums of A ca be writte as two elemets i F 4, i.e., a = (, ω) ( 0 ) =, a =

6 (, ω) 0 = ω. Note that β = ( ), the ω ω α β = ω, α β =. By expadig {0, α β, α β } F 4 ito biary vectors with respect to the basis {, ω}, we get 0 0 H = The other H i s ca be computed similarly, so we have H = It ca be verified that ay 5 colums of H are liearly idepedet. So H defies a = 5, k = 6, d 6 biary LRC with locality r =. Substitutig = 5, d = 6, r = ito the C-M boud () yields k 6, so this biary liear LRC is optimal with respect to the C-M boud. Theorem 7. The code C obtaied from Costructio is a biary liear LRC with = s t, k s, d 6 ad r = t. Moreover, C attais the upper boud () for all positive itegers s, t satisfyig t s ad s t except the case s = 4, t =. Proof: Sice the values of, k, r ca be determied easily, we focus o provig d 6. Note that the sum of the first l rows of H is a all-oe vector, so the miimum distace of C must be eve. Therefore it suffices to show that d 5. Suppose to the cotrary that there exists a codeword c C such that Hc τ = 0, wt(c) 4. Deote c = (c,..., c l ), where c i F t + for i l. It ca be deduced from the defiitio of L i that wt(c i ) is eve, i l. So there are at most two ozero vectors i c,..., c l. Without loss of geerality, we suppose c 3 = =c l =0 ad wt(c, c ) 4. The by Hc τ = 0 it has H c τ + H c τ = 0. Deote c = (x 0, x,..., x t) ad c = (y 0, y,..., y t), we have (x a + + x ta t)β + (y a + + y ta t)β = 0. Sice β ad β are liearly idepedet over F t, it must has (x a + + x ta t) = (y a + + y ta t) = 0, which cotradicts to the fact that ay 4 out of a,..., a t F t are liearly idepedet over F. It remais to show C is optimal with respect to (). Settig = s t ad r = t, it has (r + ), the it follows from () that k r + mi{ log ( + ), }. (r + )(r + ) We claim log ( + ) = s ad ()(r+) s for all s s, t satisfyig t s, t except s = 4, t =. The the claim implies that k s, ad therefore C is optimal. To show log ( + ) = s, ote that Sice < log ( + ) = log ( + t t s ). t t, we have s < + t t s s. It follows that log ( + ) = s. It remais to show ()(r+) s. Whe t =, it has s > 4 ad (r + )(r + ) = t ( t + )( t + ) s t Whe t, it has = 6 (s ) s. (r + )(r + ) = t ( t + )( t + ) s t (a) t ( t + )( t + ) t s(4t ) 4t t + t + s 4t = t (b) s, where (a) holds sice s s of s 4t, ad (b) holds sice t. 4t 4t t, which is a cosequece t + ad t + 8t for V. CONCLUSIONS We itroduce the cocepts of L-covers ad L-spaces for LRCs. By usig the sphere-packig boud i the L-spaces, we derive ew upper bouds o the dimesio k for biary liear LRCs. Two explicit bouds are give respectively for LRCs with ad without the assumptio of disjoit local repair groups. Comparig with previously kow bouds for LRCs over particular fiite fields, our bouds preset a explicit form, geeralize previous results to geeral cases, ad outperform previous bouds i some cases. Moreover, a class of biary codes attaiig our secod boud are also desiged. A further ivestigatio ito the L-spaces are expected to brig more results i studyig biary LRCs. REFERENCES A. Agarwal ad A. Mazumdar. Bouds o the rate of liear locally repairable codes over small alphabets. arxiv preprit arxiv: , 06. V. Cadambe ad A. Mazumdar. Bouds o the size of locally recoverable codes. IEEE trasactios o iformatio theory, 6: , C. L. Che. Costructio of some biary liear codes of miimum distace five. IEEE trasactios o iformatio theory, 37:49 43, 99.

7 4 P. Gopala, C. Huag, H. Simitci, ad S. Yekhai. O the locality of codeword symbols. IEEE Trasactios o Iformatio Theory, 58: , 0. 5 S. Goparaju ad R. Calderbak. Biary cyclic codes that are locally repairable. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages IEEE, J. Hao ad S. Xia. Bouds ad costructios of locally repairable codes: Parity-check matrix approach. arxiv preprit arxiv: , J. Hao, S. Xia, ad B. Che. Some results o optimal locally repairable codes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages IEEE, S. Hu, I. Tamo, ad A. Barg. Combiatorial ad lp bouds for lrc codes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages IEEE, F. J. MacWilliams ad N. J. A. Sloae. The theory of error correctig codes. Amsterdam, The Netherlads: North-Hollad, G. Markus. Bouds o the miimum distace of liear codes ad quatum codes. Olie available at Accessed o D. S. Papailiopoulos ad A. G. Dimakis. Locally repairable codes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages IEEE, 0. N. Prakash, G. M. Kamath, V. Lalitha, ad P. V. Kumar. Optimal liear codes with a local-error-correctio property. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages , 0. 3 N. Prakash, V. Lalitha, ad P. V. Kumar. Codes with locality for two erasures. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages , M. Shahabiejad, M. Khabbazia, ad M. Ardakai. A class of biary locally repairable codes. IEEE Trasactios o Commuicatios, 64:38 393, N. Silberstei, A. S. Rawat, O. O. Koyluoglu, ad S. Vishwaath. Optimal locally repairable codes via rak-metric codes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages 89 83, N. Silberstei ad A. Zeh. Optimal biary locally repairable codes via aticodes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages IEEE, W. Sog, S. Dau, C. Yue, ad T. Li. Optimal locally repairable liear codes. IEEE Joual o Selected Areas i Commuicatios, 3: , May W. A. Stei et al. Sage Mathematics Software (Versio 7.0). The Sage Developmet Team, I. Tamo ad A. Barg. A family of optimal locally recoverable codes. IEEE Trasactios o Iformatio Theory, 60: , I. Tamo, A. Barg, ad A. Frolov. Bouds o the parameters of locally recoverable codes. IEEE Trasactios o Iformatio Theory, 6: , 06. I. Tamo, A. Barg, S. Goparaju, ad R. Calderbak. Cyclic lrc codes ad their subfield subcodes. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages 6 66, 05. I. Tamo, D. S. Papailiopoulos, ad A. G. Dimakis. Optimal locally repairable codes ad coectios to matroid theory. I IEEE Iteatioal Symposium o Iformatio Theory (ISIT), pages 84 88, A. Wag ad Z. Zhag. A iteger programmig-based boud for locally repairable codes. IEEE Trasactios o Iformatio Theory, 6: , A. Zeh ad E. Yaakobi. Optimal liear ad cyclic locally repairable codes over small fields. I Iformatio Theory Workshop (ITW), pages 5, 05. APPENDIX A Let β be the primitive root of x t +, ad let M(x) deote the miimum polyimial of β, the deg(m(x)) = t. Defie A to be the biary cyclic code of legth ( t +) which is geerated (x )M(x). The it ca be checked that {β i : i =,, 0,,, } forms a subset of the roots of (x )M(x). It follows from the BCH boud that A is a t +, t t, 6 biary liear code. The a t, t t, 5 puctured code ca be obtaied by deletig oe coordiate of A, ad thus the matrix A ca be just chose as the parity check matrix of the puctured code.