An Integer Programming Based Bound for Locally Repairable Codes

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1 An Intege Pogamming Based Bound fo Locally Repaiable Codes Anyu Wang and Zhifang Zhang Key Laboatoy of Mathematics Mechanization NCMIS Academy of Mathematics and Systems Science CAS Beijing China {wanganyu Abstact axiv: v1 [cs.it] 3 Sep 2014 The locally epaiable code (LRC) studied in this pape is an [n k] linea code of which the value at each coodinate can be ecoveed by a linea combination of at most othe coodinates. The cental poblem in this wok is to detemine the lagest possible minimum distance fo LRCs. Fist an intege pogamming based uppe bound is deived fo any LRC. Then by solving the pogamming poblem unde cetain conditions an eplicit uppe bound is obtained fo LRCs with paametes n 1 > n 2 whee n 1 = and n 2 = n 1 ( + 1) n. Finally an eplicit constuction fo LRCs attaining this uppe bound is pesented ove the finite field F 2 m whee m n 1. Based on these esults the lagest possible minimum distance fo all LRCs with n 1 has been definitely detemined which is of geat significance in pactical use. I. INTRODUCTION In distibuted stoage systems edundancy must be intoduced to potect data against device failues. The simplest fom of edundancy is eplication. But it is etemely inefficient due to its lage stoage ovehead namely c copies of the data have to be stoed to guaantee (c 1)-easue toleance. To impove the stoage efficiency easue codes ae employed in distibuted stoage systems such as Windows Azue [5] Facebook s Hadoop cluste [14] etc whee the oiginal data ae divided into k equal-size fagments and then encoded into n fagments (n > k) stoed in n diffeent nodes. The fault toleance popety of the easue code ensues that the system can toleate up to d 1 node failues whee d is the minimum distance of the easue code. Paticulaly the MDS code is a kind of easue code that attains the optimal minimum distance with espect to the Singleton bound and thus povides the highest level of fault toleance fo given stoage ovehead. But the MDS code is still inefficient fo distibuted stoage systems because of the disk I/O compleity it causes in the node epai issue. Specifically when an [n k] MDS code is employed epaiing a failed node usually needs the access of k othe suvival nodes which entails too much compleity in contast with the amount of data to be epaied. To impove this Gopalan et al. [3] Oggie et al. [7] and Papailiopoulos et al. [10] intoduced epai locality fo easue codes. The ith coodinate of a code has epai locality if the value at this coodinate can be ecoveed by accessing at most othe coodinates. In moe detail a code is said to have infomation locality if the locality is ensued fo each coodinate in an infomation set containing infomation symbols e.g. systematic coodinates in a linea systematic code. Altenatively a code is said to have all symbol locality if the locality is ensued fo all coodinates. In this pape we call an [n k] linea code with all symbol locality as a locally epaiable code (LRC). When k it geatly educes the disk I/O compleity fo epai. Consideing the fault toleance level the minimum distance is also an impotant metic fo LRCs. Gopalan et al. [3] fist deived the following uppe bound fo codes with infomation locality: k d n k + 1 ( 1) (1) which is a tight bound by the constuction of pyamid codes [4]. Although the bound (1) cetainly holds fo LRCs it is not tight in many cases. The esults in [3] pointed out that when ( + 1) n and k the n +1

2 2 bound (1) cannot be attained fo codes with all symbol locality and fo those attaining this bound only the eistence esult was given fo the case ( + 1) n and the finite field needs to be lage enough. Late in pape [9] and [2] the bound (1) was genealized to vecto codes and nonlinea codes. The impact of field size on the minimum distance of LRCs was consideed in [1]. The esult povides an impoved uppe bound but elies on a paamete elated to anothe open poblem in coding theoy. In ode to deal with multiple easues in local epai Pakash et al [11] poposed the locality ( δ) associating the coodinate with an inne-eo-coecting code with length less than + δ 1 and minimum distance at least δ. It is evident that the locality ( δ) degeneates into the locality when δ = 2. An uppe bound was deived in [11] fo codes with infomation locality ( δ) which coincides with the bound (1) at δ = 2 and an eplicit code attaining this bound was given fo a specific value of the length n = k ( + 1). Fo simplicity the LRC that achieves the bound (1) with equality is usually called an optimal LRC. The fist eplicit optimal LRCs fo the case ( + 1) n wee constucted in [18] and [15] by using Reed-Solomon codes and Gabidulin codes espectively. Both constuctions wee built ove a finite field of size eponential in the code length n. Moeove it was poved in [15] that the constuction also induces an optimal LRC when n mod ( + 1) > k mod > 0. Then in [19] fo the same case ( + 1) n the authos constucted an optimal code ove a finite field of size compaable to n by using specially designed polynomials. This constuction can be etended to the case (+1) n with the minimum distance d n k k + 1 which is at most one less than the uppe bound defined in (1). Recently Song et al. [16] obtained moe esults about tightness of the bound (1). Specifically they deived a new case whee thee ae no optimal LRCs and two new cases whee thee eist optimal LRCs ove sufficiently lage fields leaving only two cases in which tightness of the bound (1) is unknown. Anothe ecent impovement was in [12] whee Pakash et al. showed a new uppe bound on the minimum distance fo LRCs. This bound elies on a sequence of ecusively defined paametes and is tighte than the bound (1). But no geneal constuctions attaining this new bound was pesented. Thee ae lots of othe wok devoted to the locality in the handling of multiple node failues such as [13] [17] [19] [20] consideing LRCs which pemit paallel access of hot data the papes [8] [20] studying LRCs with geneal local epai goups and the wok [12] which poposed sequential local epai. In a wod moe and moe eseach wok have concened about codes with the local epai popety especially those codes attaining the lagest possible minimum distance. A. Ou Contibution Since the bound (1) is not tight fo LRCs in many cases the cental poblem in this wok is detemining the lagest possible minimum distance of an [n k] LRC. Ou fist esult is an intege pogamming based uppe bound d n k + 1 η whee η = ma{ : Ψ() < k} and the function Ψ() elies on an intege pogamming poblem defined below l 1 n Ψ() = Ma Min ( + 1 (a hi t hi )) 1 st 1...t s lh 1...h l + 1 a 1...a s whee the Ma is subject to and the Min is subject to t t s = n 1 ; a a s = n 2 ; a i t i 1 i [s]; s 1; t i 1 i [s] t h1 + + t hl 1 < t h1 + + t hl.

3 3 By solving the intege pogamming poblem when n 1 > n 2 we get the second esult of this pape: an eplicit uppe bound on the minimum distance (Theoem 14) whee n 1 = n +1 and n2 = n 1 ( + 1) n. This uppe bound stands fo all possible values of k while most pevious esults (e.g. [15] [16]) that depend on the value of k in addition to the paametes n and which means ou bound sometimes coves wide paamete egion. Additionally in Section IV-B we show by compaisons that this eplicit bound can give shape desciption of the lagest possible minimum distance than pevious esults (i.e. the esults in [3] [12] [16]) in many cases. The thid esult concens the constuction of LRCs. Specifically when n 1 > n 2 we give an eplicit constuction (Constuction 1) of the [n k] LRC attaining the bound in Theoem 14 ove the finite field F 2 m whee m n 1. Theefoe we have definitely detemined the lagest possible minimum distance fo all [n k] LRCs unde the condition n 1 > n 2. Since the condition n ( + 1) 2 implies n 1 > n 2 we have completely obtained the lagest possible minimum distance fo LRCs with n 1 which is of geat significance in pactical use. B. Related Wok In [21] the authos developed the famewok of egeneating sets which detemines the uppe bound on the minimum distance fo any LRC by computing a function elated to the stuctue of local epai goups. The uppe bound deived in this wok can be viewed as an optimization based on this famewok. A bief intoduction of the famewok and the motivation fo optimization can be found in Section II. C. Oganization Section II intoduces the famewok of egeneating sets and shows the motivation of optimization. Section III deives an intege pogamming based uppe bound on the minimum distance fo LRCs. Then Section IV solves the intege pogamming poblem fo n 1 > n 2 and obtains an eplicit uppe bound. Section V pesents an eplicit constuction attaining this bound. Finally Section VI concludes the pape. II. REGENERATING SETS AND LOCALLY REPAIRABLE CODES Let C be an [n k d] q linea code with geneato mati G = (g 1... g n ) whee g i F k q fo 1 i n. Then the egeneating set intoduced in [21] can be defined as follows. Definition 1. Fo an [n k d] q linea code C a egeneating set of the ith coodinate 1 i n is a subset R [n] such that i R and g i is an F q -linea combination of {g j } j R\{i} whee [n] denotes the set of integes { n}. The collection of all egeneating sets of the ith coodinate is denoted by R i. Futhemoe a sequence of egeneating sets R 1 R 2... R m whee R i R li and l i [n] fo 1 i m is said to have a nontivial union if l j / j 1 R i fo 1 j m. Fo a linea code C define the function Φ() = min{ R i : R i R li and R 1... R have a nontivial union}. (2) In paticula it is assumed Φ(0) = 0. Then it was poved that the minimum distance is closely elated to the function Φ(). Theoem 2 ( [21]). Fo any [n k d] linea code d n k + 1 ρ whee ρ = ma{ : Φ() < k}. Remak. An eplicit bound fom Theoem 2 depends on computation of the function Φ() which is detemined by the specific geneato mati. Sometimes patial infomation of the geneato mati may help get a pecise estimate of Φ() which in tun gives a tight bound fo the minimum distance. An instance whee Theoem 2 deives a tight bound is the squae code poposed in [21]. In this pape we aim to tighten the minimum distance bound fo LRCs by estimating Φ() and then optimizing the value. The following two subsections eplain ou motivations though eamples.

4 4 A. Estimate of Φ() Fist we need to edefine the locality by using the concept of egeneating sets. Definition 3. Fo 1 i n the ith coodinate of an [n k] code C has locality if thee eists a egeneating set R R i with R + 1. We efe to an [n k] linea code of which each coodinate has locality as a locally epaiable code (LRC). Because = 1 implies epetition and fo k MDS code possess the optimal distance we assume 1 < < k thoughout the pape. Moeove because of the uppe bound on the infomation ate of LRCs [19] we assume that k fo any [n k] LRC. n +1 In [21] the authos estimated the function Φ() fo diffeent kinds of locality and epoved the minimum distance bounds that had been given in pevious liteatues. Fo eample it poved Φ() ( + 1) fo LRCs which induces the bound (1); Φ() δ 1 + fo codes with locality ( δ) and deived the uppe bound given in [11]; etc. In this pape we focus on LRCs. The following eample shows that when ( + 1) n one can estimate Φ() bette than Φ() ( + 1) and thus can deive a tighte bound. Eample 1. Let C be an [n k d] LRC with ( + 1) n. We claim that Φ() ( + 1) 1 fo 2. Fist the following algoithm geneates a sequence of egeneating sets R 1... R l that has a nontivial union and l R l = [n]. 1: Set i = 1 2: while i 1 j=1 R j [n] do 3: Pick i 0 [n] i 1 j=1 R j 4: Choose R i R i0 such that R i = + 1 5: Set i = i + 1 6: end while Because ( + 1) n and R i = + 1 fo 1 i l thee eist i 1 i 2 [l] such that R i1 R i2. By the definition of Φ() Φ() min{ i I R i : I [l] I = }. Theefoe { + 1 if = 1 Φ() ( + 1) 1 if 2. It follows that ρ k+1 1 and thus k + 1 d n k + 1 ( 1). (3) Obviously the bound (3) is tighte than the bound (1) fo the case (+1) n. Paticulaly the diffeence occus when k which also eplains a known fact (see [3] [16]) that the bound (1) is unachievable when ( + 1) n and k. Late in Section III we will give a shape estimate of Φ() and deive a tighte bound fo LRCs. B. Optimization of Φ() Fom Theoem 2 we obseve that fo a given LRC its minimum distance d is uppe bounded by n k + 1 ρ whee ρ depends on the function Φ() which is detemined by the code itself. Theefoe to uppe bound d fo all LRCs with paametes n k one needs to find the code which gives the minimum ρ o the maimum Φ(). Actually we find the stuctue of egeneating sets plays an impotant ole in detemining the function Φ() which in tun influence the minimum distance.

5 5 Eample 2. Conside LRCs with paametes n = 10 k = 5 and = 3. We constuct two such LRCs which have diffeent stuctue of egeneating sets. The fist code C 1 is constucted by using ank-metic codes [15]. Specifically let {α 1 α 2 α 3 α 5 α 6 α 7 α 9 } F 2 7 be a basis of F 2 7 ove F 2 and let α 4 = α 1 + α 2 + α 3 α 8 = α 5 + α 6 + α 7 α 10 = α 9. The geneato mati of C 1 is G 1 = (g 1 g 2... g 10 ) whee g i = (α i αi 2 αi 4 αi 8 αi 16 ) τ fo 1 i 10. It is easy to veify that C 1 is an LRC ove F 2 7 and a sequence of its egeneating sets is { } { } {9 10}. (4) Theefoe Φ() 4 2 fo 1 3 and ρ 2. By Theoem 2 we have d n k + 1 ρ 4. On the othe hand since any 7 columns of G 1 has full ank it implies d n 6 = 4. As a esult C 1 has minimum distance d = 4. The second code C 2 is an [n = 10 k = 5] linea code ove F 13 with geneato mati G = Obseve that C 2 has locality = 3 and a sequence of its egeneating sets is { } { } { }. (5) Futhemoe it can be veified that Φ(1) = 4 Φ(2) = 7 and Φ(3) = 10. Then ρ = 1 and d n k+1 ρ 5 fom Theoem 2. On the othe hand one can veify that C 2 has minimum distance d = 5. Fom (4) and (5) we can see that C 1 and C 2 have diffeent stuctue of egeneating sets. The fome has paiwise disjoint egeneating sets while the latte has ovelapped egeneating sets. This diffeence esults in that the Φ() of C 1 is no moe than that of C 2 theefoe the latte code has a highe uppe bound fom Theoem 2. III. UPPER BOUNDS ON THE MINIMUM DISTANCE Denote n 1 = n +1 and n2 = n +1 ( + 1) n. It follows that n = n1 ( + 1) n 2 and 0 n 2 < + 1. The intege pogamming based uppe bound is deived in thee steps as descibed in the following thee subsections espectively. A. Fom Φ() to a Set Cove Poblem Fist fo any [n k] LRC we convet the poblem of estimating the Φ() to a set cove poblem (Lemma 5 Lemma 6). To begin with we intoduce the concept of an ( + 1)-cove. Definition 4. Let S = {S 1... S t } be a collection of subsets of [n]. We call S an ( + 1)-cove ove [n] if the following conditions ae satisfied: (1) S i = + 1 fo 1 i t; (2) i [t] S i = [n] and i [t]\{j} S i [n] fo any j [t].

6 6 In the emainde of this pape we usually omit the phase ove [n] fo an ( + 1)-cove when it is evident fom the contet. Lemma 5. Fo a given [n k] locally epaiable code C it induces an ( + 1)-cove S = {S 1... S t } t n 1 satisfying Φ() Min i J S i J [t] J = fo 1 n 1 whee Φ() is defined as in (2). Poof: By using the algoithm in Eample 1 we get a sequence of egeneating sets R 1... R l which has a nontivial union. Then by deleting some R i s which lie in the union of the emaindes we can finally get an ( + 1)-cove {R i1... R it } as equied by the lemma. Lemma 6. Fo any ( + 1)-cove S = {S 1... S t } t > n 1 thee eists an ( + 1)-cove consisting of n 1 subsets denoted as T = {T 1... T n1 } which satisfies fo 1 n 1 Min J [t] J = i J S i Min i I T i. Poof: Since t > n 1 set T i = S i initially fo 1 i n 1. Due to the condition (2) in Definition 4 it obviously has n 1 T i [n]. Then we ecusively invoke the following Step 1 to Step 3 on the collection T = {T 1... T n1 } epanding n 1 T i by one element at each invocation until finally n 1 T i = [n]. Step 1. Pick T j T such that T j ( T T \{Tj }T ). Step 2. Choose a T j ( T T \{Tj }T ) and b [n] n 1 T i. Step 3. T j (T j {a}) {b}. Note that the subset T j eists in Step 1 because n 1 T i = n 1 ( + 1) n > n 1 T i. Afte the thee steps only one element in T j is eplaced by an outside element and all othe subsets emain unchanged. Theefoe n 1 T i is epanded by one element. Futhemoe the union size of any subsets 1 n 1 is unchanged o inceased by 1. Theefoe fo 1 n 1 Min J [t] J = I [n 1 ] I = i J S i Min i I S i Min i I T i. I [n 1 ] I = Moeove the condition i [t]\{j} S i [n] fo any j [t] implies that S j i [t]\{j} S i fo any j [t]. It is easy to veify that the popety T j i [n1 ]\{j}t i fo any j [n 1 ] still holds afte an invocation of Step 1 to Step 3. Thus we finally get an ( + 1)-cove T as the lemma equies. By Lemma 5 and Lemma 6 we have tansfomed the poblem of deiving an uppe bound fo Φ() into the poblem of estimating the set union size in an ( + 1)-cove consisting of n 1 subsets. In the sequel a futhe investigation into the ( + 1)-cove helps to finally deive an uppe bound of Φ(). B. Fom the Set Cove to an Intege Pogamming Poblem Then we tansfom the set cove poblem into an intege pogamming poblem (Lemma 11). The following definition comes fom the concept of connectivity in gaph theoy. Definition 7. Let S = {S 1... S t } be a collection of nonempty subsets of [n]. We say S is connected if fo any nonempty subset I [t] it has ( i I S i ) ( j [t]\i S j ). Paticulaly a collection containing only one subset i.e. t = 1 is also called connected. Remak. In fact a collection S defines a gaph G(V E) whee each vete v i V coesponds to a subset S i S and thee is an edge (v i v j ) E if and only if S i S j. Thus a connected collection in Definition 7 actually coesponds to a connected gaph. I [n 1 ] I =

7 7 Poposition 8. Fo a connected collection of subsets S = {S 1... S t } thee eists a pemutation of [t] say {i 1... i t } such that S ij ( j 1 h=1 S i h ) 2 j t. (6) Poof: In fact i 1... i t can be detemined by the following algoithm. 1: Pick i 1 [t] 2: fo h = 2 to l do 3: Pick i h [t] {i 1 i 2... i h 1 } such that S ih (S i1 S ih 1 ) 4: end fo Note that the i h at line 3 eists because the collection S is connected. Coollay 9. Fo a connected collection of subsets S = {S 1... S t } define an intege a = t S i t S i then a t 1. Poof: By Poposition 8 we can assume without loss of geneality that S satisfies the condition (6) i.e. S j ( j 1 h=1 S j) fo all 2 j t. Since t S i = S t S t ( t 1 S i) + t 1 S i = S t S t ( t 1 S i) + S t 1 S t 1 ( t 2 S i) + t 2 S i = t S i t i=2 We have a = t i=2 S i ( i 1 j=1 S j) t 1. S i ( i 1 j=1 S j) Remak. In the following we intoduce a set of integes to chaacteize the stuctue of an ( + 1)-cove. Fist fo an ( + 1)-cove S = {S 1... S n1 } we detemine a patition of [n 1 ] say [n 1 ] = I 1 I s such that (1) fo 1 i s the induced collection S Ii = {S j j I i } is connected; and (2) fo 1 i < j s ( h Ii S h ) ( h Ij S h ) =. In othe wods this patition of a collection S actually coesponds to splitting the gaph G(V E) into connected components whee the gaph G(V E) is detemined as in the emak afte Definition 7. Then fo 1 i s define integes t i = I i and a i = j I i S j j Ii S j. It is easy to deive the following lemma. Lemma 10. Fo an ( + 1)-cove S = {S 1... S n1 } define integes s t 1... t s a 1... a s as in the above emak. Then the following conditions must hold: t t s = n 1 ; a a s = n 2 ; (7) a i t i 1 1 i s; s 1; t i 1 1 i s. Poof: By using the notations in the emak I 1 I s is a patition of [n 1 ] theefoe a 1 + +a s = s ( j I i S j j Ii S j ) = n 1 S i n 1 S i = n 1 ( + 1) n = n 2. The othe conditions come fom Coollay 9 and the emak.

8 8 Lemma 11. Fo any ( + 1)-cove S = {S 1... S n1 } define integes s t 1... t s a 1... a s as befoe then fo 1 n 1 it holds Min I [n 1 ] I = i I S i Min lh 1...h l ( + 1 l 1 (a hi t hi )) whee the Min on the ight side is subject to all integes l h 1... h l satisfying t h1 + + t hl 1 < t h1 + + t hl. (8) Poof: Suppose l and h 1... h l ae integes satisfying (8). Then thee eists J I hl such that J = (t h1 + + t hl 1 ) and the collection S J is connected. It follows that l 1 Min i I S i i Ihj S i + i J S i I [n 1 ] I = j=1 l 1 = ( S i a hj ) + i J S i j=1 i I hj (a) l 1 ( S i a hj ) + S i ( J 1) j=1 i I hj i J l 1 = (t hj ( + 1) a hj ) + J ( + 1) ( J 1) j=1 l 1 (b) = + 1 (a hj t hj ) whee (a) is fom Coollay 9 and (b) is fom the equality that J = (t h1 + + t hl 1 ). C. An Intege Pogamming Based Bound j=1 In this subsection we deive an intege pogamming based bound on the minimum distance of any LRC (Theoem 12). Define Ψ() = Ma st 1...t s a 1...a s Min ( + 1 lh 1...h l l 1 (a hi t hi )) 1 n 1 (9) whee the Ma is subject to (7) and the Min is subject to (8). Then the value of Ψ() is detemined only by integes n 1 and n 2 o equivalently by n and. Theoem 12. Fo any [n k d] LRC it holds Φ() Ψ() fo 1 n 1 and whee η = ma{ : Ψ() < k}. d n k + 1 η (10) Poof: Fist we show that Φ() Ψ() 1 n 1. By Lemma 5 and Lemma 6 thee eists an ( + 1)-cove T consisting of n 1 subsets {T 1... T n1 } such that Φ() Min j J T j 0 n 1. J [n 1 ] J =

9 9 Define integes s t 1... t s a 1... a s as in the emak afte Coollay 9. By Lemma 11 we have Φ() Min ( + 1 lh 1...h l l 1 (a hi t hi )) 1 n 1 whee the minimum is subject to (8). Then it follows fom Lemma 10 that Φ() Ψ() fo 1 n 1. Theefoe k > Ψ(η) η Φ(η) η. We have η ρ and then by Theoem 2 the bound (10) is obtained. Remak. Diffeence between the bound (10) and the bound in Theoem 2. The two bounds ae of the same fom ecept that the fome is detemined by η and the function Ψ() while the latte is detemined by ρ and the function Φ(). But Ψ() is defined fo all integes n and while Φ() is defined with espect to specific egeneating set stuctue. In othe wods given paametes n and the bound (10) definitely povide an uppe bound fo any LRC with the paametes n and but Theoem 2 cannot give a specific bound due to the lack of infomation about egeneating set stuctue. Nevetheless no efficient algoithm has been established fo solving the intege pogamming poblem involved in the bound (10). But we can solve it by ehaustive seach fo small n and as in the eample below. Futhemoe we can detemine the solution fo a wide class of the values of n and which plays an impotant ole in pactical use. The details ae in the net section. Eample 3. Suppose n = 13 = 3 then n 1 = 4 and n 2 = 3. Because of the assumption 1 < < k and the uppe bound on the infomation ate of LRCs i.e. k we conside 4 k 9. n +1 Fist compute the value of Ψ() fo 1 4. Obseve that up to pemutation all possible integes s and {a i t i } i [s] satisfying (7) ae s = 1 s = 3 t a (t 1 t 2) (a 1 a 2) (1 3) (0 3) s = 2 (t 1 t 2 t 3) (a 1 a 2 a 3) (1 3) (1 2) (1 1 2) (0 0 3) (2 2) (1 2) (1 1 2) (0 1 2) (t 1 t 2 t 3 t 4) (a 1 a 2 a 3 a 4) (1 1 2) (0 2 1) s = 4 ( ) ( ) (1 1 2) (1 1 1) ( ) ( ) ( ) ( ) Then by an ehaustive seach we get Ψ(1) = 4 Ψ(2) = 7 Ψ(3) = 10 Ψ(4) = 13. Fo simplicity we can wite Ψ() = fo 1 4. Theefoe we have η = ma{ : Ψ() < k} = ma{ : < k} = k 3 2 fo 4 k 9. Thus by Theoem 12 k 3 d n k + 1. (11) 2 It gives an eplicit uppe bound. We compae it with the well known bound i.e. the bound (1) given by Gopolan et al. As displayed in Fig. 1 the bound (11) goes though thee points beneath the bound (1) i.e. k = 6 9 and 8 whee the fome two points have been epected by the impossible condition ( + 1) n and k (see Eample 1) but the point k = 8 is a new impossible esult (not included in the impossible esults in [16]). IV. EXPLICIT BOUND FOR THE CASE n 1 > n 2 In this section fo a wide class of paametes i.e. n 1 > n 2 we solve the intege pogamming poblem involved in Theoem 12 and then deive an eplicit uppe bound fo all LRCs satisfying n 1 > n 2. Since the condition n 1 > n 2 can be viewed as a esult of n 1 which is a natual constaint fo LRCs to be used in pactice the eplicit bound we obtain hee is sufficient to cove most pactical use. In the second pat of this section we make compaisons with all peviously known esults to show the impovements of ou eplicit bound. Actually in Section V we will show this bound is tight fo the case n 1 > n 2.

10 10 d 8 Gopalan et al.'s bound (1) The bound (11) k Fig. 1. Compaison of the two bounds fo n = 13 = 3. A. Bound fom Solution of the Intege Pogamming Poblem Fist Poposition 13 detemines the value of the function Ψ() unde the condition n 1 > n 2. Then Theoem 14 deives an eplicit uppe bound accodingly. Denote µ = n 1 n 2 and let λ ν be integes such that n 1 = λµ + ν and 0 ν < µ. Poposition 13. Fo 1 n 1 Ψ() = + ma{ Poof: The poof is given in Appendi A. λ + 1 ν Theoem 14. Fo any [n k d] LRC with n 1 > n 2 whee n 1 = n +1 and n2 = n 1 ( + 1) n it holds whee η = min{ (λ+1)(k 1)+1 (λ+1)( 1)+1 λ(k 1)+ν+1 λ( 1)+1 λ }. d n k + 1 η (12) } 1. Poof: We pove this by showing η = η whee η is defined in Theoem 12. Specifically Thus we have η = min{ Theoem 12. η = ma{ : Ψ() < k} ν = ma{ : ( 1) + ma{ } < k} λ + 1 λ = ma{ : ( 1) + ν k 1 and ( 1) + k 1} λ + 1 λ (λ + 1)(k 1) λ(k 1) + ν = ma{ : and (λ + 1)( 1) + 1 λ( 1) + 1 }. (λ+1)(k 1)+1 (λ+1)( 1)+1 λ(k 1)+ν+1 λ( 1)+1 } 1 = η and the statement follows diectly fom Eample 4. Let C be an [n k] LRC with ( + 1) n. We have n 1 = n n +1 2 = 0 and theefoe µ = n 1 ν = 0 λ = 1. Then it follows fom Theoem 14 that η = min{ 2k k } 1 = k 1 and k d n k + 1 ( 1) which coincides with the bound (1).

11 11 d d Gopalan et al.'s bound (1) 55 The bound (12) k Pakash et al.'s bound (1) 65 The bound (12) k (a) (b) Fig. 2. Compaison of the thee bounds fo n = 101 = 9. B. Impovements of the Bound Since the bound (12) in Theoem 14 holds fo n 1 > n 2 all the compaisons we make below ae unde the condition n 1 > n 2. 1) Compaison with Gopolan et al s Bound: The bound (1) given by Gopalan et al. [3] is the fist uppe bound on the minimum distance of LRCs. It states k d n k + 1 ( 1). Because n 1 > n 2 it follows λ 1 and ν 0. Then along with the assumption 1 < < k a detailed calculation shows that η k 1. Theefoe the bound (12) geneally povides a tighte uppe bound than the bound (1). Actually the fome bound is stictly tighte than the latte at many points. The left gaph of Fig. 2 gives a compaison of the two bounds fo n = 101 = 9. 2) Compaison with Pakash et al s Bound: Recently Pakash et al. [12] deived an impoved uppe bound on the minimum distance i.e. d n k + 1 l (13) whee l is the unique intege satisfying e l < k + l < e l+1 and {e m } m [n1 ] is defined ecusively as below 2em e n1 = n and e m 1 = e m + ( + 1) fo 2 m n 1. m It was poved in [12] that the bound (13) impoves the bound (1). We claim that the bound (12) in Theoem 14 futhe impoves the bound (13). Geneally obseve that η = ma{ : Ψ() < k} and the definition of l is equivalent to l = ma{m e m m < k}. Then the claim follows fom the fact that e m Ψ(m) 1 m n 1. (14) We pove (14) by induction on m. Fist fo m = n 1 Ψ(n 1 ) = n 1 + µ = n = e n1. Then suppose the agument holds fo m + 1 i.e.

12 12 e m+1 Ψ(m + 1) whee m < n 1. Thus 2em+1 e m = e m+1 + ( + 1) m + 1 m 1 = m + 1 e m+1 + ( + 1) m 1 Ψ(m + 1) + ( + 1) m + 1 m 1 m + 1 m + 1 ν = ((m + 1) + ma{ }) + ( + 1) m + 1 λ + 1 λ m 1 m + 1 m + 1 ν = m m + 1 ma{ } λ + 1 λ m ma{ m 1 λ + 1 m 1 ν } λ m m ν = m ma{ 1 1} λ + 1 λ = Ψ(m). The above poof shows that the bound (12) cannot go upon the bound (13). A detailed calculation with specific values of n k shows the fome bound does go beneath the latte bound at some points. As an illustation the ight gaph in Fig. 2 plots the two bounds fo n = 101 = 9. 3) Compaing with the Results of Song et al : In [16] Song et al. deived some conditions unde which thee eists no LRC attaining the bound (1) and also poved the eistence of LRCs attaining the bound (1) unde some conditions. Howeve they left some scope of paametes unde which it was unknown whethe thee eist LRCs attaining the bound (1). In Section V of this pape we will give an eplicit constuction of [n k] LRCs fo n 1 > n 2 attaining the bound (12) in Theoem 14. Theefoe ou bound (12) completely descibes the lagest possible minimum distance fo LRCs with n 1 > n 2. Fig. 3 illustates the coesponding esults fo n = k 17 and 2 9. In the tables Y means thee eist LRCs attaining the bound (1) N means thee is no LRC attaining the bound (1) and a blank means it is unknown whethe thee eist LRCs attaining the bound (1). Results of Song et al. k N N N N 3 Y N N Y N N N 4 Y Y Y Y Y Y Y Y 5 N Y Y Y Y N Y Y 6 Y YN Y Y 7 Y Y Y N Y Y Y 8 Y Y Y Y Y N Y 9 Y Y Y Y Y Y Y Y Results of this pape k N Y N Y N Y N Y 3 Y N N Y N N Y N 4 Y Y Y Y Y Y Y Y 5 N Y Y Y Y N Y Y 6 Y Y N Y Y Y Y N 7 Y Y Y Y N Y Y Y 8 Y Y Y Y Y Y N Y 9 Y Y Y Y Y Y Y Y Fig. 3. A compaison of Song et al. s esults and ou esults fo n = 50.

13 13 V. CODE CONSTRUCTION WHEN n 1 > n 2 In this section we pesent an eplicit constuction of LRCs attaining the bound (12). The constuction is based on lineaized polynomials. We stat this section with some basic facts about lineaized polynomials. A. The Lineaized Polynomial Definition 15. A polynomial of the fom f() = t i=0 a i qi with coefficients a i F q m fo 0 i t and a t 0 is called a lineaized polynomial of q-degee t ove the etension field F q m. A lineaized polynomial f() can be viewed as an F q -linea tansfomation fom F q m to itself i.e. fo any c 1 c 2 F q and ω 1 ω 2 F q m it holds f(c 1 ω 1 + c 2 ω 2 ) = c 1 f(ω 1 ) + c 2 f(ω 2 ). Futhemoe a standad esult of finite fields states that Poposition 16. [6] A lineaized polynomial f() of q-degee no moe than t can be uniquely detemined by the values of f(ω 1 )... f(ω t+1 ) whee ω 1... ω t+1 ae t + 1 elements in F q m that ae linealy independent ove F q. B. An Eplicit Code Constuction In this subsection we assume n 1 > n 2 and constuct an [n k] LRC ove F q m attaining the bound (12) in Theoem 14 whee F q m is an etension field of F q with m n 1. In a wod the codewods ae obtained as evaluations of a lineaized polynomial at n points in F q m. Because of the popety of lineaized polynomials intoduced in Poposition 16 the key point of the code constuction is the selection of the n evaluation points such that the esulting code has the lagest possible minimum distance. Denote the set of the n evaluation points by Ω. Since F q m can be viewed as an F q -linea space of dimension m by fiing a basis of F q m ove F q the n elements in Ω can be epessed as n vectos of length m ove F q. These n vectos ae detemined though the following thee steps. Fo simplicity we can set q = 2 and m = n 1 and the pocess below also woks fo othe values of q and m. Step 1. Let X = ( ) be the geneato mati of an [ + 1 ] 2 MDS code and let c = (1 c 1... c ) be one of its codewod whee i F2 fo 0 i. Fo eample we can choose X = and c = ( ) Step 2. Define vectos α 0 α ij F (λ+1) 2 fo 1 i λ + 1 and 1 j whee c j 0 0 α 0 = 0.. and α ij = j. 0 c j 0 that is α 0 consists of (λ + 1) 0 s and α ij is defined by eplacing the i-th c j 0 of c j α 0 with an j. Similaly define β 0 β ij F λ 2 1 i λ and 1 j such that β 0 consists of λ 0 s and β ij is defined by eplacing the i-th c j 0 of c j α 0 with an j. Fo eample let = 2 λ = 2 and X = ( ) c = (1 c 1 c 2 ) then we have (α 0 α 11 α 12 α 21 α 22 α 31 α 32 ) = c 1 0 c 2 0 c 1 0 c c 1 0 c c 1 0 c c 1 0 c 2 0 c 1 0 c

14 14 and (β 0 β 11 β 12 β 21 β 22 ) = ( ) c 1 0 c c 1 0 c The vectos α 0 α ij and β 0 β ij defined above have the following popeties. Lemma 17. Denote A i = {α 0 α i1... α i } fo 1 i λ + 1. Then we have (i) Fo 1 i λ + 1 each vecto contained in A i is an F q -linea combination of the othe vectos in A i. (ii) Fo any F λ+1 A i satisfying that thee eists a vecto α F such that (F {α}) A i 1 fo 1 i λ + 1 the vectos in F ae F q -linealy independent. Denote B j = {β 0 β j1... β j } fo 1 j λ. Then the same statements also hold fo B j fo 1 j λ. Poof: The poof is given in Appendi B. Step 3. Let A be the mati consisting of the ((λ + 1) + 1) column vectos in λ+1 A i and let B be the mati consisting of the (λ + 1) column vectos in λ B i. Define a block diagonal mati A... A W = B... B which is composed of ν A s and (µ ν) B s on the diagonal and zeos eleswhee. Note that A has ((λ + 1) + 1) columns and B has (λ + 1) columns then W has ((λ + 1) + 1)ν + (λ + 1)(µ ν) = n 1 ( + 1) n 2 = n columns. Similaly W has ν(λ + 1) + (µ ν)λ = (λµ + ν) = n 1 ows. Then the set of n vectos in Ω ae defined to be the n columns of W. We give a gaphical eplanation of linea dependences among the n vectos. Refe to Fig. 4 each point actually coesponds to a vecto. Then the left ν tees each composed of λ + 1 banches coesponds to the ν blocks of A in W and the ight µ ν tees each composed of λ banches coesponds to the µ ν blocks of B in W. In moe detail the set A i fo 1 i λ + 1 coesponds to a banch in the left tees and paticulaly the vecto α 0 coesponds to the oot point. The simila coespondence holds fo B i and the banches in the ight tees. 0 ω (ν) 0 ω (ν+1) 0 ω (µ) λ W (1) 1 W (1) 2 W (1) λ+1 λ+1 W (ν) 1 W (ν) 2 W (ν) λ+1 W (ν+1) 1 W (ν+1) 2 W (ν+1) λ W (µ) 1 W (µ) 2 W (µ) λ }{{} }{{} ν tees each composed of λ + 1 banches µ ν tees each composed of λ banches Fig. 4. The n points in Ω.

15 15 Fo convenience we denote the n points (o equivalently the n vectos in Ω) by {ω (l) 0 ω (l) ij l [µ] i [λ + 1] j []}1 whee the supescipt l denotes which tee it belongs to the subscipt i denotes which banch it lies in and j is the point inde in that banch. Moveove denote each banch by W (l) i = {ω (l) 0 ω (l) i1 ω(l) i2... ω(l) i } fo l [µ] and i [λ + 1]. Then by Lemma 17 (i) each vecto in W (l) i is an F q -linea combination of the othe vectos in W (l) i and by the constuction of the mati W the vectos in diffeent tees ae linealy independent. Constuction 1. Define an [n k] linea code C ove F q m as follows. Let Ω F q m be s set of the n vectos defined above i.e. Ω = {ω (l) 0 ω (l) ij l [µ] i [λ+1] j []}. Note that each vecto is of length n 1 = m ove F q and thus can be viewed as an element in F q m. C encodes a file (m 0... m k 1 ) F k q into (f(ω)) m ω Ω F n q m whee f() = k 1 i=0 m i qi. Denote the n coodinates of C by the coesponding element in Ω then W (l) i is a egeneating set of each coodinate contained in W (l) i. Theefoe C is an [n k] LRC with locality. Eample 5. We illustate the constuction though a specific eample. Suppose n = 8 k = 4 = 2 then it has n 1 = 3 n 2 = 1 and λ = 1 µ = 2 ν = 1. The constuction is ove the field F 2 6 = F 2 (θ) whee θ is a pimitive element of F 2 6 with minimal polynomial By fiing a basis {1 θ θ 2... θ 5 } the subset Ω F 2 6 is constucted as follows. Fist let ( ) X = ( ) = and c = (1 c c 2 ) = (1 0 1). Then Theefoe and thus A =(α 0 α 11 α 12 α 21 α 22 ) ( ) 0 = 1 2 c 1 0 c 2 0 = 0 c 1 0 c and B =(β 0 β 11 β 12 ) =( ) = ( ) ( A W = = B) = 1 + θ 2 11 = θ 12 = 1 + θ + θ 2 21 = θ 3 22 = 1 + θ 2 + θ 3 and Fig. 5 gives a gaphical illustation of the eight elements in Ω. ω (2) 0 = θ 4 ω (2) 11 = θ 5. ω (2) 12 = θ 4 + θ 5 1 A pecise desciption of the ange of l and i j is (l i) ([ν] [λ+1]) ([ν +1 µ] [λ]) j [] whee [ν +1 µ] = {ν +1 ν µ}.

16 16 0 ω (2) ω (2) W (1) 1 W (1) 2 ω (2) 12 W (2) 1 Fig. 5. The eight elements of Ω fo the [8 4] code. Then the [n = 8 k = 4] linea code C encodes a file (m 0 m 1 m 2 m 3 ) into (f(ω)) ω Ω whee f() = m 0 + m m m 3 8. A sequence of egeneating sets of the linea code C is { } { } {ω (2) 0 ω (2) 11 ω (2) 12} and it is easy to see that Φ(1) = 3 Φ(2) = 5 Φ(3) = 8 which coincides with the uppe bound defined by Ψ() (see Poposition 13). Moeove it can be veified that the minimum distance of C is d = 3 which is optimal with espect to the bound (12) in Theoem 14. Actually the following theoem states that the code C in Constuction 1 alsways attains the bound (12) in Theoem 14. Theoem 18. The [n k] LRC C obtained fom Constuction 1 has the minimum distance whee η = min{ (λ+1)(k 1)+1 (λ+1)( 1)+1 λ(k 1)+ν+1 λ( 1)+1 d = n k + 1 η } 1. Poof: Fist we claim that fo any V Ω with V = k + η thee eist subsets V 1... V µ V such that the following two conditions ae satisfied: (1) µ l=1 V l k; W (l) i and thee eists ω l V l such that (V l {ω l }) W (l) i 1 fo (2) Fo 1 l µ V l λ+1 all i [λ + 1]. The poof of the claim is given in Lemma 21 of Appendi C. Fom the claim and Lemma 17 (ii) we can deduce that fo 1 l µ the elements in V l ae linealy independent ove F q and thus the elements in V 1 V 2 V µ ae linealy independent ove F q. Then by Poposition 16 C can toleate any n (k + η) easues. Consequently the minimum distance of C satisfies d n k + 1 η and the equality actually holds because of Theoem 14. C. Influence of the Regeneating Set Stuctue As we have stated in Eample 2 and ealie sections the stuctue of egeneating sets can influence the value of the function Φ() which in tun influence the value of the minimum distance. In this section we will check the egeneating set stuctue of the code C in Constuction 1 to suppot its attaining the optimal minimum distance and also make a compaison with some peviously constucted codes. In Fig. 4 it gives a gaphical desciption of the egeneating sets fo C while each line (o a banch i.e. W (l) i ) epesents a egeneating set. Conside the collection of egeneating sets {W (l) i has a nontivial union with espect to any ode they ae aanged in. In fact it is easy to see that fo the code C Φ() = Min I {W (l) i } l [µ]i [λ+1] I = V I V. } l [µ]i [λ+1]. It

17 17 d 20 Constuction 1 Constuctionin [12] o [14] k Fig. 6. A compaison of the two LRCs fo n = 25 = 3 We can count fom Fig. 4 that { + λ+1 if ν(λ + 1) Min V I V = + ν + ν(λ+1) if > ν(λ + 1) I {W (l) i } li I = = + ma{ Theefoe the Φ() of C satisfies Φ() = + ma{ λ λ + 1 λ + 1 ν λ ν which attains the uppe bound defined by Ψ() (see Theoem 12 and Poposition 13). That is C achieves the maimum value of Φ() among all the LRCs with n 1 > n 2 which can be egaded as a suppot of the code C attaining the optimal minimum distance. On the othe hand we will see some peviously constucted codes have smalle minimum distance due to thei egeneating set stuctue. The code pesented by Silbestein et al. in [15] and that poposed by Tamo et al. in [19] ae both of paiwise disjoint egeneating sets. Namely patition the set [n] into n 1 subsets I 1 I 2... I n1 such that I j = + 1 fo 1 j n 1 1 and I n1 = + 1 n 2 then I 1 I 2... I n1 fom a sequence of egeneating sets that has a nontivial union. Clealy the Φ() satisfies Φ() ( + 1) n 2 1 n 1. Then by Theoem 2 ρ = ma{ : Φ() < k} k+n 2 1 and the minimum distance satisfies k + n2 d n k + 1 ( 1). Thus it cannot attain the bound (1) when k+n 2 > k i.e. k mod n mod ( +1) > 0. In fact the minimum distance sometimes goes beneath the bound (12) of Theoem 14 that is the optimal minimum distance cannot be attained unde this kind of egeneating set stuctue. Fig. 6 gives a compaison between the minimum distance of C and that of the codes in [15] [19] fo n = 25 and = 3. λ }. }

18 18 VI. CONCLUSIONS In this pape we cay out an in-depth study of the two poblems: what is the lagest possible minimum distance fo an [n k] LRC? How to constuct an [n k] LRC with the lagest possible minimum distance? Fo the fist poblem we deive an intege pogamming based uppe bound on the minimum distance fo LRCs and then give an eplicit bound by solving the intege pogamming poblem. The eplicit bound applies all LRCs satisfying n 1 > n 2. Fo the second poblem we pesent a constuction of linea LRCs that attains the eplicit bound fo n 1 > n 2. Theefoe we have completely solved the two poblems unde the condition n 1 > n 2. Howeve fo n 1 n 2 the two poblems emain unsolved in many cases. REFERENCES [1] V. Cadambe and A. Mazumda An uppe bound on the size of locally ecoveable codes IEEE Int. Symp. Netw. Coding (NetCod) Calgay 2013 pp [2] M. Fobes and S. Yekhanin On the locality of codewod symbols in non-linea codes axiv pepint axiv: [3] P. Gopalan C. Huang H. Simitci and S. Yekhanin On the locality of codewod symbols IEEE Tans. on Infom. Theoy vol. 58 pp Nov [4] C. Huang M. Chen and J. Li Pyamid codes: Fleible schemes to tade space fo access efficiency in eliable data stoage systems in Poc. 6th IEEE Int. Symp. Netw. Comput. Appl. Cambidge 2007 pp. 79C86. [5] C. Huang H. Simitci Y. Xu A. Ogus B. Calde P. Gopalan J. Li and S. Yekhanin Easue coding in Windows Azue Stoage pesented at the USENIX Annu. Tech. Conf. Boston MA [6] R. Lidl Finite fields Cambidge Univesity Pess [7] F. Oggie and A. Datta Self-epaiing homomophic codes fo distibuted stoage systems in Poc. IEEE Infocom Shanghai 2011 pp [8] L. Pamies-Juaez H. D. L. Hollmann and F. Oggie Locally epaiable codes with multiple epai altenatives in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Istanbul 2013 pp [9] D. S. Papailiopoulos and A. G. Dimakis Locally epaiable codes in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Cambidge 2012 pp [10] D. S. Papailiopoulos J. Luo A. G. Dimakis C. Huang and J. Li Simple egeneating codes: netwok coding fo cloud stoage in Poc. IEEE Infocom Olando 2012 pp [11] N. Pakash G. M. Kamath V. Lalitha and P. V. Kuma Optimal linea codes with a local-eo-coection popety in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Cambidge 2012 pp [12] N. Pakash V. Lalitha and P. Kuma. Codes with locality fo two easues in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Honolulu 2014 pp [13] A. S. Rawat D. S. Papailiopoulos A. G. Dimakis and S. Vishwanath Locality and availability in distibuted stoage in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Honolulu 2014 pp [14] M. Sathiamoothy M. Asteis D. Papailiopoulos A. G. Dimakis R. Vadali S. Chen and D. Bothaku Xoing elephants: Novel easue codes fo big data Poceedings of the VLDB Endowment (to appea) [15] N. Silbestein A. S. Rawat O. O. Koyluoglu and S. Vishwanath Optimal locally epaiable codes via ank-metic codes in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Istanbul 2013 pp [16] W. Song S. Dau C. Yuen and T. Li Optimal locally epaiable linea codes IEEE J. Sel. Aeas Commun. vol. 32 pp May [17] I. Tamo and A. Bag Bounds on locally ecoveable codes with multiple ecoveing sets in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Honolulu 2014 pp [18] I. Tamo D. S. Papailiopoulos and A. G. Dimakis Optimal locally epaiable codes and connections to matoid theoy in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Istanbul 2013 pp [19] Itzhak Tamo and Aleande Bag A family of optimal locally ecoveable codes IEEE Tans. on Infom. Theoy vol. 60 pp Aug [20] A. Wang and Z. Zhang Repai locality with multiple easue toleance axiv pepint axiv: [21] A. Wang and Z. Zhang Repai locality fom a combinatoial pespective in Poc. IEEE Int. Symp. Inf. Theoy (ISIT) Honolulu 2014 pp Lemma 19. Fo 1 n 1 APPENDIX A PROOF OF PROPOSITION 13 Ψ() + ma{ λ + 1 ν λ }.

19 19 Poof: Set s = µ t 1 = = t ν = λ + 1 t ν+1 = = t µ = λ a 1 = = a ν = λ a ν+1 = = a µ = λ 1. It is clea that s and {t i a i } i [s] satisfy (7) and then we have Ψ() Min ( + 1 lh 1...h l = Min lh 1...h l ( + l) l 1 (a hi t hi )) whee the minimum is subject to (8). On the othe hand fo any integes l h 1... h l satisfying (8) { (λ + 1)l if l ν t h1 + + t hl (λ + 1)ν + (l ν)λ if l > ν which induces min{λl + l λl + ν}. Theefoe l ma{ λ+1 ν λ } and then ν Ψ() Min ( + l) + ma{ }. lh 1...h l λ + 1 λ Lemma 20. Fo 1 n 1 Ψ() + ma{ λ + 1 ν Poof: We pove the lemma by contadiction. Assume that fo some 1 n 1 ν Ψ() ma{ }. λ + 1 λ Then thee eist integes s and t i a i 1 i s satisfying the constaints (7) and l 1 ν Min ( + 1 (a hi t hi )) ma{ } lh 1...h l λ + 1 λ whee the minimum is subject to the constaint (8). Theefoe fo all integes l and h 1... h l [s] satisfying the constaint (8) it has l 1 ν (a hi t hi ) ma{ }. (15) λ + 1 λ Conside the following two cases. Case 1. 1 (λ + 1)ν. Then ma{ λ+1 ν λ } = λ+1. Fo 1 i s define b i = (λ + 1)a i λt i. Then without loss of geneality we can assume that b 1 b 2 b s. Let h be the smallest intege such that 1 h s and h (a i t i ) λ+1. Note that h eists because s (a i t i ) = n 2 n 1 = µ λ+1. Net we conside the value of t1 + + t h. λ }.

20 20 If t t h thee eists a positive intege h h such that h 1 j=1 t j < h j=1 t j. Then h 1 < h and by (15) h 1 (a i t i ) λ + 1 which contadicts to the minimality of h. If t t h < we compute s b i in two diffeent ways. On the one hand s s b i = ((λ + 1)a i λt i ) On the othe hand we claim that (i) h b i 1 and b i 1 fo h + 1 i s (ii) h s ν µ and then s h b i = ( b i ) + ( which contadicts to (16). In fact the claim (i) holds because h b i = = (λ + 1)n 2 λn 1 = ν µ. (16) s i=h+1 b i ) 1 + ( 1) (s h) = 1 + h s ν µ 1 h ((λ + 1)a i λt i ) = (λ + 1) (a) h (a i t i ) + h (λ + 1) + 1 λ whee (a) follows fom h (a i t i ) λ+1 and h t i <. Then b j 1 h h b i < 0 fo h + 1 j s. To show the claim (ii) obseve that a i t i 1 and h 1 (a i t i ) λ fom the minimality of h. Then we have s µ = n 2 n 1 = (a i t i ) h 1 = (a i t i ) + s (a i t i ) i=h ( 1) (s h + 1). λ + 1 t i

21 21 Because (λ + 1)ν it holds and the claim (ii) follows diectly. Case 2. (λ + 1)ν + 1 n 1. Then ma{ λ+1 µ ν ( 1) (s h + 1) = ν + h s ν λ c i = λa i (λ 1)t i } = ν λ. Simila to Case 1 define 1 i s and assume c 1 c 2 c s. Let g [s] be the smallest positive intege such that g (a i t i ) ν λ. Note that g eists because s (a i t i ) = n 2 n 1 = µ ν λ. Net we conside the value of t t g. Simila to Case 1 t t g contadicts to the minimality of g. Then it follows t t g <. We compute the value of s c i in two diffeent ways. On the one hand s s s c i = λ a i (λ 1) On the othe hand we claim that (i) g c i ν 1 (ii) c i 0 fo g + 1 i s. Then t i = λn 2 (λ 1)n 1 = ν. (17) s c i = g c i + s i=g+1 c i g c i ν 1 which contadicts to (17). Note that g (a i t i ) ν λ and g t i < then the claim (i) follows fom g g g c i = λ (a i t i ) + ν λ + 1 λ ν 1. To show the claim (ii) obseve that c j 1 g g c i ν 1 fo g + 1 j s and g g g (a i t i ) ν λ > ν whee the fist inequality is fom ai t i 1 fo 1 i s and the last inequality is fom (λ + 1)ν + 1. Then it has c j < ν < 1 fo g + 1 j s and the claim (ii) then follows. g APPENDIX B PROOF OF LEMMA 17 (i) Because X = ( ) geneates an [ + 1 ] MDS code thee eist nonzeo elements e 0 e 1... e F q such that e e e = 0. Moeove since c = (1 c 1... c ) is a codewod of the MDS code it has e 0 + e 1 c e c = 0. Theefoe e 0 α 0 + e 1 α i1 + + e α i = 0 fo 1 i λ + 1. Thus (i) follows diectly. (ii) We pove the statement by contadiction. Assume that the vectos in F ae linealy dependent i.e. thee eists e α F q fo each α F such that α F e αα = 0 whee {e α } α F ae not all zeos. In fact t i

22 22 at least two out of {e α } α F ae nonzeo because the vectos in F ae not zeo vectos. We conside the following two cases. Case 1. (F {α 0 }) A i 1 fo 1 i λ + 1. Because at least two out of {e α } α F ae nonzeo thee eists i 0 [λ + 1] such that the coefficients {e α } α Ai0 \{α 0 } ae not all zeo. Then without loss of geneality assume (F {α 0 }) A i0 = {α i0 1 α i α i0 h} whee h 1. Conside the estiction of the linea combination α F e αα to its i 0 th thick ow (i.e. the ((i 0 1) +1)-th ow to the i 0 -th ow) we have h j=1 e α i0 j j = e 0 fo some e F q. It follows that h ae F q -linealy dependent whee h 1 which contadicts the fact that ( 0... ) geneates an [ + 1 ] MDS code. Case 2. Fo some (i 0 j 0 ) [λ + 1] [] (F {α i0 j 0 }) A i 1 fo 1 i λ + 1. Without loss of geneality assume i 0 = j 0 = 1 i.e. (F {α 11 }) A i 1 fo 1 i λ + 1. If thee eists l 2 l λ + 1 such that {e α } α Al \{α 0 } ae not all zeo then simila to Case 1 esticting the linea combination α F e αα to its lth thick ow will lead a contadiction. Theefoe we have e α = 0 fo all α λ+1 i=2 A i {α 0 }. Thus it suffice to check the vectos in F A 1. Similaly a contadiction aises when esticting α F e αα to the fist thick ow. APPENDIX C PROOF OF THE CLAIM Lemma 21. Fo any V Ω with V = k + η thee eist subsets V 1... V µ V such that the following two conditions ae satisfied: (1) µ l=1 V l k; (2) Fo 1 l µ V l λ+1 W (l) i and thee eists ω l V l such that (V l {ω l }) W (l) i 1 fo all i [λ + 1]. Poof: Denote U l = V ( λ+1 i ) fo 1 l µ. Then the poof is completed by two steps. Fist we show that fo all nonempty set U l 1 l µ thee eists a subset V l U l satisfying ; and V l U l Ul 1 W (l) Thee eists ω l V l such that (V l {ω l }) W (l) i 1 fo all i [λ + 1]. Second by setting V l = fo all l [µ] with U l = 0 we pove that V 1 V 2 V µ k. The details ae given below. Step 1. Suppose U l is nonempty. Conside the following two cases. (a) ω (l) 0 U l. Then thee ae at most U l say W (l) 1... W (l) h U l whee h Ul 1 Ul 1 sets out of W (l) 1 W (l) 2... W (l) λ+1. Define V l by deleting ω (l) 11 ω (l) Ul 1 we have (V l {ω (l) 0 }) W (l) i 1 fo all i [λ + 1] and V l U l (b) ω (l) 0 / U l. Similaly thee ae at most sets out of W (l) 1 W (l) 2... W (l) Ul U l {ω (l) 0 } say W (l) 1... W (l) h U l {ω (l) 0 } whee h Ul which ae contained in ω (l) h1 fom U l then. λ+1. Define V l by deleting ω (l) which ae contained in 21 ω (l) Ul fom U l then we have (V l {ω 11}) (l) W (l) i 1 fo all i [λ + 1] and V l U l ( U l. Ul ω (l) h 1 1)

23 23 Step 2. Obseve that µ l=1 V l = l [µ]u l V l Ul 1 ( U l ) l [µ]u l = k + η Ul 1. l [µ]u l k + η U l 1 l [µ]u l k + η ɛ = k + η whee ɛ = {l [l] : U l }. Then it suffices to show k+ η ɛ η. Denote ɛ 1 = {l : 1 l ν U l } and ɛ 2 = {l : ν + 1 l µ U l } then ɛ = ɛ 1 + ɛ 2. Because U 1 + U U µ = k + η and { λ+1 U l W (l) i = (λ + 1) + 1 fo 1 l ν λ W (l) i = λ + 1 fo ν + 1 l µ we have 0 ɛ 1 ν; 0 ɛ 2 µ ν; k + η ɛ 1 ((λ + 1) + 1) + ɛ 2 (λ + 1). Since ɛ 1 ((λ + 1) + 1) + ɛ 2 (λ + 1) ((λ + 1) + 1)(ɛ 1 + ɛ 2 ) and also ɛ 1 ((λ + 1) + 1) + ɛ 2 (λ + 1) = (λ + 1)(ɛ 1 + ɛ 2 ) + ɛ 1 (λ + 1)(ɛ 1 + ɛ 2 ) + ν it follows that ɛ ma{ k+ η k+ η ν }. Thus (λ+1)+1 λ+1 k + η ɛ 1 k + η k + η ν (k + η ma{ }) (λ + 1) + 1 λ η)(λ + 1) (k + η)λ + ν = min{(k } (λ + 1) + 1 λ + 1 (k + η)(λ + 1) (k + η)λ + ν = min{ }. (λ + 1) + 1 λ + 1 Note that η = min{ (λ+1)(k 1)+1 } 1. Then if η = 1 it has and theefoe (k + η)(λ + 1) (λ + 1) + 1 (k+ η)(λ+1) (λ+1)+1 (λ+1)( 1)+1 Thus we conclude that k+ η ɛ λ(k 1)+ν+1 λ( 1)+1 η. Similaly if η = η. (λ+1)(k 1)+1 (λ+1)( 1)+1 (λ + 1)(k 1) ((λ + 1)( 1) + 1)( η + 1) ( η + 1) = (λ + 1) + 1 (λ + 1)( 1) + 1 (λ + 1)(k 1) = ( ( η + 1)) (λ + 1) + 1 (λ + 1)( 1) + 1 < 0 λ(k 1)+ν+1 1 it can be poved that λ( 1)+1 (k+ η)λ+ν η. λ+1