Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures

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1 Bounds on Binary Locay Repairabe Codes Toerating Mutipe Erasures Matthias Grezet, Ragnar Freij-Hoanti, Thomas Westerbäc, Otay Omez and Camia Hoanti Department of Mathematics and Systems Anaysis, Aato University, Finand, Emai: Department of Eectrica and Computer Engineering, Technica University of Munich, Germany, Emai: Department of Mathematics, Anara University, Turey, Emai: arxiv: v2 [cs.it] 19 Feb 2018 Abstract Recenty, ocay repairabe codes has gained significant interest for their potentia appications in distributed storage systems. However, most constructions in existence are over fieds with size that grows with the number of servers, which maes the systems computationay expensive and difficut to maintain. Here, we study inear ocay repairabe codes over the binary fied, toerating mutipe oca erasures. We derive bounds on the minimum distance on such codes, and give exampes of LRCs achieving these bounds. Our main technica toos come from matroid theory, and as a byproduct of our proofs, we show that the attice of cycic fats of a simpe binary matroid is atomic. I. INTRODUCTION In modern distributed storage systems (DSSs) faiures happen frequenty, whence decreasing the number of connections required for node repair is crucia. Removing even one connection ocay can easiy impy huge gains in the overa system functionaity, thans to shortened queues and improved data avaiabiity. Consequenty, ocay repairabe codes (LRCs) have gained a ot of interest in the past few years [1] [4]. Namey, LRCs aow to repair a sma number of faiures ocay, i.e., by ony contacting few cose-by nodes and hence avoiding congesting the system. Reated Singetontype bounds have been derived for various cases, see [5] [7]. The first bound on the minimum distance for fixed fied size was obtained by Cadambe and Mazumdar in [8]. Recenty, this bound was improved and generaized, via the observation that any og-convex bound on the oca ran of a code can be bown up to obtain a bound on the goba ran [9]. Interestingy, the bounds in [9] do not depend on the inearity of the code. However, a the bounds in [8], [9] are impicit, except for specia casses of codes. In this paper, we consider binary codes motivated by the fact that the computationa compexity when retrieving a fie or repairing a node grows with the fied size. We derive new, improved Singeton-type bounds for this specia case aongside with sporadic exampes, in particuar when the oca repair sets can toerate mutipe faiures. In contrast to the bounds in [8], [9], our bounds are expicit, and do not depend on any prior bounds on binary codes of shorter ength. As our main contribution, in Theorem 6, we obtain a cosedform bound on the minimum distance d of a binary (n, )- code of ength n and dimension and with a-symbo (r, δ)- ocaity, where the oca distance δ > 2. Such bounds were previousy ony nown when δ = 2. The bound is in terms of the ran of the repair sets, but can easiy be transformed to bounds in terms of the size r + δ 1. Interestingy, whie the two parameters r and can be assumed to agree when δ = 2, as we as when the fied size is unbounded, this is no onger the case over the binary fied with δ > 2. Whie both parameters are of independent interest in appications, we have chosen to focus on the number of nodes that need to be contacted for oca repair, rather than on the size of the oca couds 1. In addition, in Section III we prove that every eement of a non-degenerate binary ocay repairabe code without repication is contained in an atomic cycic fat, and hence that the attice of cycic fats is atomic. From a practica point of view, this impies a hierarchy of faiure toerance, as expained in the end of Section III. In particuar, whenever a symbo e fais, we can start by downoading nodes in an atomic cycic fat in order to repair e. If it turns out that some other nodes in this oca set have faied as we, we can repair them whie sti eeping the part that we aready downoaded, and simpy contact some more nodes in the corresponding repair set to repair a the faied symbos. Thus, we do not have to restart from the beginning if we find out during the repair process that a sma amount of other nodes have faied as we. Severa constructions are nown for optima LRCs over the binary fied, for specified ranges of parameters, and amost excusivey in the case δ = 2. The first such construction, for codes with exponentiay ow rate and ocaity r = 2, 3, was obtained by deeting carefuy chosen coumns from the simpex code [10]. These constructions are aso optima when taing the avaiabiity t, i.e., the number of disjoint sets that can recover a given symbo, into consideration. A sighty more fexibe famiy of codes, aowing for higher rate, was given in [11], [12], where aso a sight improvement over the Cadambe-Mazumdar bound was given for inear codes. In the ream of mutipe erasures, i.e., when δ > 2, rateoptima codes were studied in [13]. There, rate-optima codes were characterized when δ = 3, and anaogous constructions without optimaity proof were given for δ > 3. However, to the best of our nowedge, no previous wor has studied bounds on the goba minimum distance in the regime δ > 2. 1 Note that here, whie r sti essentiay bounds the size of the oca sets, we never need to contact that many nodes, since r 1 as shown in Prop. 5. From another point of view, r can be seen as a parameter describing how far one needs to go in order to repair.

2 II. PRELIMINARIES OF LRCS AND MATROIDS As is common practice, we say that C is an (n,, d)-code if it has ength n, dimension, and minimum Hamming distance d. A inear (n,, d)-code C over a finite fied F is a nondegenerate storage code if d 2 and there is no zero coumn in a generator matrix of C. To study LRCs in more detai, we consider punctured codes C Y, where Y [n] is a set of coordinates of the code C. For a fixed code C, we denote by d Y the minimum Hamming distance of the punctured code C Y. Definition 1. Let C be an (n,, d)-code over F n. A symbo x [n] has ocaity (r, δ) if there exists a subset R of [n], caed repair set of x, such that x R, R r + δ 1, and d R δ. Definition 2. A inear (n,, d, r, δ)-lrc over a finite fied F n is a non-degenerate inear (n,, d)-code C over F n such that every coordinate x [n] has ocaity (r, δ). In the iterature, this is specificay caed a-symbo ocaity. The parameters (n,, d, r, δ) can immediatey be defined and studied for matroids in genera, as in [2], [14], [15]. A. Matroid fundamentas Matroids have many equivaent definitions in the iterature. Here, we choose to present matroids via their ran functions. Much of the contents in this section can be found in more detai in [16]. Definition 3. A (finite) matroid M = (E, ρ) is a finite set E together with a ran function ρ : 2 E Z such that for a subsets X, Y E (R.1) 0 ρ(x) X, (R.2) X Y ρ(x) ρ(y ), (R.3) ρ(x) + ρ(y ) ρ(x Y ) + ρ(x Y ). A subset X E is caed independent if ρ(x) = X. If X is independent and ρ(x) = ρ(e), then X is caed a basis. A subset that is not independent is caed dependent. A circuit is a minima dependent subset of E, that is, a dependent set whose proper subsets are a independent. Strongy reated to the ran function is the nuity function η : 2 E Z, defined by η(x) = X ρ(x) for X E. Any matrix G over a fied F generates a matroid M G = (E, ρ), where E is the set of coumns of G, and ρ(x) is the ran of G(X) over F, where G(X) denotes the submatrix of G formed by the coumns indexed by X. As eementary row operations preserve the row space of G(X) for a X E, it foows that row-equivaent matrices generate the same matroid. Thus, there is a straightforward connection between inear codes and matroids. Let C be a inear code over a fied F. Then any two different generator matrices of C wi have the same row space by definition, so they wi generate the same matroid. Therefore, without any inconsistency, we can denote the matroid associated to these generator matrices by M C = (E, ρ C ). The ran function ρ C can be defined directy from the code without referring to a generator matrix, via ρ C (X) = dim(c X) for X E. One way of defining a new matroid from an existing one is obtained by restricting the matroid to one of its subsets. For a given set X E, we define the restriction of M to X to be the matroid M X = (X, ρ X ) by ρ X (Y ) = ρ(y ) for a subsets Y X. Two matroids M 1 = (E 1, ρ 1 ) and M 2 = (E 2, ρ 2 ) are isomorphic if there exists a bijection ψ : E 1 E 2 such that ρ 2 (ψ(x)) = ρ 1 (X) for a subsets X E 1. Definition 4. A matroid that is isomorphic to M G for some matrix G over F is said to be representabe over F. We aso say that such a matroid is F-representabe. A binary matroid is a matroid that is F 2 -representabe. By the Critica Theorem [17], the matroid M C determines the supports of a inear code C. Consequenty, since binary codes are determined uniquey by the support of the codewords, binary matroids are in one-to-one correspondence with binary codes. This is in sharp contrast to inear codes over arger fieds, where many interesting properties are not determined by the associated matroid. An important exampe of such a property is the covering radius [18]. Definition 5. A matroid is caed simpe if if has no circuits consisting of 1 or 2 eements. A eement e E is caed a co-oop if ρ(e e) < ρ(e). B. Fundamentas on cycic fats The main too from matroid theory in this paper are the cycic fats. We wi define them using the cosure and cycic operators. Let M = (E, ρ) be a matroid. The cosure operator c : 2 E 2 E and cycic operator cyc : 2 E 2 E are defined by (i) (ii) c(x) = X {e E X : ρ(x e) = ρ(x)}, cyc(x) = {e X : ρ(x e) = ρ(x)}. A subset X E is a fat if c(x) = X and a cycic set if cyc(x) = X. Therefore, X is a cycic fat if ρ(x y) > ρ(x) and ρ(x x) = ρ(x) for a y E X and x X. The coection of fats, cycic sets, and cycic fats of M are denoted by F(M), U(M), and Z(M), respectivey. Some more fundamenta properties of fats, cycic sets, and cycic fats are given in [19]. The cycic fats of a inear code C of F E can be described as sets X E such that C (X y) > C X and C (X x) = C X for a y E X and x X. Before going deeper in the study of Z(M), we need a minimum bacground on poset and attice theory. We wi use the standard notation of and for the meet and join operator, we wi denote by 0 L and 1 L the bottom and top eement of a attice L, and we wi denote by the covering

3 reation, i.e., for X, Y (L, ), we say that X Y if X < Y and there is no Z L with X < Z < Y. The atoms and coatoms of a attice (L, ) are defined as A L = {X L \ 0 L : Y L such that 0 L Y X}, coa L = {X L \ 1 L : Y L such that X Y 1 L }, respectivey. A attice L is said to be atomic if every eement of L is the join of atoms. For further information about posets and attices, we refer the reader to [20]. We can now give a crucia property of the set of cycic fats. Proposition 1 (See [19]). Let M = (E, ρ) be a matroid. Then 1) (Z(M), ) is a attice with X Y = c(x Y ) and X Y = cyc(x Y ). 2) 1 Z = cyc(e) and 0 Z = c( ). C. Reation between LRCs and the attice of cycic fats Recenty, some wor has been done to emphasise the reation between cycic fats and inear codes over finite fieds. In [14], the authors proved that the minimum distance can be expressed in terms of the nuity of certain cycic fats: Proposition 2. Let C be a non-degenerate (n,, d)-code and M = (E, ρ) the matroid associated to C. Then, d = η(e) + 1 max{η(z) : Z Z(M) {E}}. Moreover, [21] gives us necessary conditions on the structure of the attice of cycic fats when the code and hence the matroid are binary. The ey resuts from [21] are the foowing proposition and theorem that constrain the edges of the associated Hasse diagram. Proposition 3 ( [21]). Let M = (E, ρ) be a binary matroid. Then, every X, Y Z(M) with X Y satisfy exacty one of the foowing: ρ(y ) ρ(x) = > 1 and η(y ) η(x) = 1. We ca such an edge in the Hasse diagram of Z(M) a ran edge and abe it ρ =. ρ(y ) ρ(x) = 1 and η(y ) η(x) = > 1. We ca such an edge a nuity edge and abe it η =. ρ(y ) ρ(x) = 1 and η(y ) η(x) = 1. We ca such an edge a eementary edge. Theorem 1 (Announced in [21]). Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with d > 2 and without repication. Let M = (E, ρ) be the associated matroid. Then Z = Z(M) satisfies the foowing: 1) and E are cycic fats. 2) Every covering reation Z E is a nuity edge abeed with a number d 1. 3) If δ = 2, then for every i E, there is X Z with i X such that ρ(x) r. 4) If δ > 2, then for every i E, there is X Z with i X such that a) Every covering reation Y X is a nuity edge abeed with a number δ 1. b) Every cycic fat Y with Y X has ran r 2. Proof. From non-degeneracy, it foows that and E are cycic fats. If Z E, then by Proposition 2, d η(e) + 1 η(z). As d > 2, this means that 1 < d 1 η(e) η(z), so Z E is a nuity edge. This proves 1) and 2). Now for every i E, there is R E with i R, d R δ and R δ + 1 r. In particuar, when δ > 1, R must be a cycic set, and its cosure X is a cycic fat with ρ(x) = ρ(r) and d X d R δ. Now, (3) foows as ρ(x) = ρ(r) R d R + 1 R δ + 1 r. Finay, assume δ > 2 et Y X. Since d X = d R δ, it foows from 2) that Y X is a nuity edge abeed with a number δ 1. Consider now the attice Z(M R). We have 0 Z(M R) = and 1 Z(M R) = R since R is cycic. As there is no repication, ρ(r) 2 and η(r) δ 1 2 by Proposition 2. It foows from Proposition 3 that there exists a cycic fat Z c Z(M R) different from and R. Then it foows that max{η(z) : Z Z(M R) {R}} 1, η(r) = d R + max{η(z) : Z Z(M R) {R}} 1 δ, and ρ(r) = R η(r) r 1. Hence, ρ(x) r 1 and a cycic fats Y X have ran r 2. III. LATTICE STRUCTURE AND REPAIR PROPERTIES The first part of this section is devoted to understanding how restricting to binary inear codes affects the structure of the attice of cycic fats. We wi see that Proposition 3 has a strong impact on the structure of the atoms of Z, which in return infuences the whoe attice. Indeed the main resut of this section consists of proving that the attice of cycic fats has the property of being atomic. In the second part, we wi discuss the meaning of these resuts for binary inear codes and LRCs. In particuar, we wi see that every non-degenerate binary inear (n,, d)-code without repication is aready a binary inear (n,, d, r, 2)- LRC for a certain r. Furthermore, for LRCs with δ > 2, we wi see that these codes have a hierarchy in faiure toerance. A. Structura properties of the attice of cycic fats We wi first begin by the reation between binary inear codes and the associated matroid. The foowing proposition is a reformuation of Proposition 8 in [14] together with the easy observation that, in a binary inear code, two symbos are dependent if and ony if they are equa. Proposition 4. Let C be a binary inear (n,, d)-code. Then C is non-degenerate with no repication if and ony if the associated matroid M = (E, ρ) is simpe and contains no co-oops. Now that we have estabish the type of matroids that is reveant to our case, we can study the impications of

4 Proposition 3 for the attice of cycic fats. We begin by an immediate consequence on the atoms of Z(M). Lemma 1. Let M = (E, ρ) be a simpe binary matroid. Then, Z is an atom of Z(M) if and ony if η(z) = 1. Proof. Since M is simpe, it guaranties that = 0 Z. Furthermore, it aso means that, for a cycic fats Z, we have ρ(z) > 1. Hence, by Proposition 3, every atom Z at wi have a ran edge, i.e., η(z at ) = 1. The next two emmas in atoms in Z(M) to certain short circuits in M. Lemma 2. Let M = (E, ρ) be a simpe binary matroid. Let C be a circuit of M. Then c(c) is an atom of Z(M) if and ony if c(c) = C. Proof. By emma 1, c(c) is an atom of Z(M) if and ony if η(c(c)) = 1. But since C is a circuit, we have η(c) = 1. Now η(c(c)) = 1 + c(c) C. Hence c(c) is an atom if and ony if c(c) = C. Lemma 3. Let M = (E, ρ) be a simpe binary matroid that contains no co-oops. Let e E and C be a circuit containing e of minima ength. Then c(c) = C. Proof. First, the existence of a circuit containing e is guaranteed by the fact that M contains no co-oops. Let C be a minimum ength circuit containing e. Now, consider a binary representation {x f } f M of M. We can express x e by a inear combination of eements {x f : f C \{e}}. Since C is a binary circuit, we wi need a eements in C \ {e} with coefficient equa to 1. Hence x e = x f. f C\{e} Assume for a contradiction there exists e c(c) C. Then x e = x f. Since M is binary and simpe, we have f D C 2 D < C. If e D, then we have found a circuit smaer than C containing e which is a contradiction to the minimaity of C. If e / D, then x e = x f + x f = x e + x f. f D f (C\{e})\D f (C\{e})\D Thus, e is in the circuit {e} {e } ((C \ {e}) \ D) with cardinaity {e } (C \ D) < C by the fact that D 2. Again, this is a contradiction to the minimaity of C. Hence c(c) = C. By combining Lemma 2 and 3, we obtain the foowing resut. Lemma 4. Let M = (E, ρ) be a simpe binary matroid that contains no co-oops. Then every eement e E is contained in an atom. Theorem 2. Let M = (E, ρ) be a simpe binary matroid that contains no co-oops. Then the attice of cycic fats Z(M) is atomic. Proof. By Lemma 4, for every e E, there exist an atom Zat e Z(M) with e Zat. e Thus, Zat e Zat e = E. e M e M For Y Z(M), we can restrict the matroid to M Y = (Y, ρ). Since Z(M Y ) = {Z : Z Y, Z Z(M)} and M Y contains no co-oops, we are bac to the previous case. Hence Y = e Y Z e at and this proves that Z(M) is atomic. Indeed, we have proven a sighty stronger property than atomicity. Namey, any eement in Z(M) is equa not ony to the join, but aso to the union of a the atoms that it contains. B. Hierarchy of faiure toerance By Proposition 4, non-degenerate binary inear (n,, d)- codes with no repication are associated with simpe binary matroids that contain no co-oops. Thus, we can reinterpret these resuts for this type of binary codes. Probaby the most significant one is Lemma 4 stating that every eement of a given matroid M is contained in some atoms, meaning that every symbo of the associated code is automaticay contained in a sma dependent set. Furthermore, by Proposition 2, the minimum distance of an atom is 2. Thus, every symbo is directy contained in a sma repair set with δ = 2, i.e., in a repair set that can correct exacty one erasure. Hence we obtain the foowing theorem. Theorem 3. For every non-degenerate binary inear (n,, d)- code C with no repication, C is aso an (n,, d, r, 2)-LRC for some r {2,..., }. Proof. Let M = (E, ρ) be the associated matroid to C. By Proposition 4, M is simpe and contains no co-oops. By Lemma 4, for a e E, there exists an atom Zat e Z(M) with e Zat. e Define r = max e E ρ(ze at). Then the coection {Zat} e gives a ocaity (r, δ = 2) to C. Indeed, by Proposition 3 and 2 we have η(zat) e = 1 and d Z e at = 2, which gives us δ = 2. For the size criterion, we have Z e at = ρ(z e at) + η(z e at) = ρ(z e at) + δ 1 max e E {ρ(ze at)} + δ 1 = r + δ 1. Now, if we want to be abe to correct more than one erasure, then the repair sets cannot be atoms of Z(M) as these have d Z = 2. They have to be at east one eve above some atoms. However, the previous theorem sti hods, meaning that for every symbo, there is aso an atom containing it. Thus, we get a natura hierarchy in faiure toerance. If one node fais, then we can contact the cose-by nodes in the atom to repair

5 it. If more nodes fai, but no more than δ 1, we can contact other repair sets to fix them. And if more than δ 1 nodes fai, then we need to use the goba properties of the code. Moreover, by the remar foowing Theorem 2, it foows that repair sets are unions of a the atoms beow them. Since the coection of repair sets contains every symbo, we can choose the coection of atoms that wi give us the (r, 2) ocaity to be inside repair sets. The foowing coroary summaries the previous observations in one statement. Coroary 1. Let C be a non-degenerate binary inear (n,, d, r, δ)-lrc with no repication and with δ > 2. Let {R i } i I be the ist of repair sets. Then, there exists a coection of sets {X j } j J such that for a X j, there exists R i with X j R i such that C is aso an (n,, d, r, 2)-LRC. From a practica point of view, this reinforces the usefuness of the faiure toerance hierarchy. For exampe, suppose that the symbo e E fais. We can start by downoading nodes in the atom Z e at in order to repair e. Now, if we reaize that some other nodes in Z e at have faied as we, we can eep the part that we aready downoaded from Z e at and contact more nodes in the corresponding repair set to repair a the faied symbos in Z e at. Thus, we do not have to restart from the beginning if we find out during the repair process that a sma amount of other nodes have faied as we. IV. IMPROVING THE SINGLETON-TYPE BOUND FOR δ > 2 The goa of this section is to improve the existing bound for non-degenerate inear (n,, d, r, δ)-codes C when the codes are binary, contain no repication and δ > 2. It has been proven in [22] that, for a inear (n,, d, r, δ)-code over F q, we have d n + 1 ( r 1 ) (δ 1) (1) Since the predominant parameter of this study wi be the ran of the repair sets instead of the size, we assume from now on that repair sets are maxima after fixing its ran or, in matroid terms, that repair sets are cycic fats. We wi denote these repair sets by Z i instead of R i to avoid confusion. Remember that Proposition 2 ins the minimum distance of a code to the coatoms among the cycic fats. So we woud ie to construct a cycic fat that is as cose as possibe to the coatoms eve to give an accurate ower bound on max{η(z) : Z Z(M) {E}}. We wi do this by creating a chain in Z(M) made of joins of repair sets and we wi ca these type of chains repair-setschain or, for short, rps-chain. Definition 6. Let C be a non-degenerate (n,, d, r, δ)-code and {Z i } i I the coection of repair sets. Let M = (E, ρ) the associated matroid. An rps-chain = Y 0 Y 1... Y m = E is a chain in Z(M) defined inductivey by 1) Let Y 0 =. 2) Given Y i 1 E, we choose x i E \ Y i 1 and Z i with x i Z i arbitrariy, and assign Y i = Y i 1 Z i. 3) If Y i = E, we set m = i. Notice that this chain is not uniquey defined since we can choose symbos and corresponding repair sets freey. Now that we have created this chain, we are interested in how the ran and the nuity can increase at each step. This wi be usefu when evauating the ran and the nuity at the end of the chain. First, we wi define a new parameter that represents the maximum ran of a repair set. Definition 7. Let C be an (n,, d, r, δ)-lrc. Let {Z i } i I be the ist of repair sets and M = (E, ρ) the associated matroid to C. We define to be := max i I ρ(z i) Lemma 5. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication and M = (E, ρ) the associated matroid. Then, every rps-chain (Y i ) m i=0 has the foowing properties. 1) ρ(y i ) ρ(y i 1 ) for a i = 1,..., m. 2) η(y i ) η(y i 1 ) δ 1 for a i = 2,..., m. Proof. Let {Z i } i I be the coection of repair sets and (Y i ) m i=0 an arbitrary rps-chain. By ran axioms, cosure property and by Proposition 5, we have ρ(y i ) ρ(y i 1 ) + ρ(z i ) ρ(y i 1 Z i ) ρ(y i 1 ) + for a i = 1,..., m. Moreover, by nuity properties coming from ran axioms and by Proposition 5, we have η(y i ) η(y i 1 Z i ) η(y i 1 ) + η(z i ) η(y i 1 Z i ) η(y i 1 ) + η(z i ) max{η(z) : Z Z(Z i ) Z i } η(y i 1 ) + d Zi 1 η(y i 1 ) + δ 1 for a i = 2,..., m. We are now ready to present the first of our main resuts, a Singeton-type bound on the parameters n,, d, and δ. This wi ater be improved in Section V. Theorem 4. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Then, ( ) d n 1 (δ 1). (2) Proof. Let M = M(E, ρ) be the matroid associated to C, {Z i } i I the coection of repair sets and (Y i ) m i=0 an rps-chain. Our goa is to obtain a ower bound on max{η(z) : Z Z(M) {E}} since it wi give us a upper bound for the minimum distance by Proposition 2. To do this, we wi give a ower bound on the nuity of the penutimate set in the rps-chain using Lemma 5. Indeed, we have η(y m 1 ) (m 2)(δ 1) + η(z 1 ).

6 Now, by Theorem 2, there exists an atom Z at beow Z i. Then, max{η(z) : Z Z(Z i ) Z i } η(z at ) = 1. Thus, by Proposition 2, we have η(z i ) δ 1 + max{η(z) : Z Z(Z i ) Z i } δ. (3) Hence, η(y m 1 ) (m 2)(δ 1) + δ = (m 1)(δ 1) + 1. To evauate m, we use the upper bound on the ran. We have = ρ(y m ) m m. Combining the two previous resuts, we get ( ) η(y m 1 ) 1 (δ 1) + 1. Appying this and Proposition 2, we obtain d C = n + 1 max{η(z) : Z Z(M) {E}} n + 1 η(y m 1 ) ( ) n 1 (δ 1). Even if this bound oos graphicay the same as the previous nown bound with an extra 1, the meaning is truy different since the ceiing depends on the maximum ran of the repair sets and not on the parameter r reated to the size. To iustrate this, and to mae the comparison easier, we wi give a bound on the ran with r. This wi aso aow us to give as a coroary a new version of the ast bound that ony depends on n,, d, r and δ. Proposition 5. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Let {Z i } i I be the coection of repair sets. Then, ρ(z i ) r 1 for a i I. In particuar, this impies that r 1. Proof. We aready saw in Equation (3) that η(z i ) δ. Now, using the definition of a repair set and the ower bound for η(z i ), we get ρ(z i ) = Z i η(z i ) r 1. We can now reformuate the bound in Theorem 4: Coroary 2. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Then, ( ) d n 1 (δ 1) r 1 Compared to the previous bound (1), we have improved by dropping the +1 and by repacing r by r 1. V. TOOL FOR ACHIEVING A BETTER BOUND This section is an attempt to be more precise on the evauation of rps-chains which wi ead to a further improvement in the bound. We wi first discuss about the proof of Theorem 4 and Lemma 5. The main ingredient in the previous section is the evauation of the ran and nuity difference at each step on rps-chains in Lemma 5. Indeed, if ρ(y i ) ρ(y i 1 ) =, then we must have Y i 1 Z i = and ρ(z i ) =. In particuar, the intersection condition means that η(y i ) η(y i 1 ) δ 1. Hence, there is no code nor an rps-chain that can simutaneousy achieve both bounds in Lemma 5. The idea is to introduce an indicator function that wi capture when the intersection is a coatom of a repair set. To be more concise, we wi denote by A i the event when Y i 1 Z i coa Z(M Zi). First, this is a necessary condition to have η(y i ) η(y i 1 ) = δ 1. Secondy, this wi aso impy that ρ(y i ) ρ(y i 1 ) = 1 since every covering reation of a repair set has a nuity edge. Thus, this indicator wi be abe to separate the two previous extrema cases impossibe to reach at the same time and aows us to have a more precise view of each of the chain step. Thus, the foowing emma with the new indicator is an improvement of Lemma 5. Lemma 6. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication, and et M = (E, ρ) be the associated matroid. Let {Z i } i I be the coection of repair sets and (Y i ) m i=0 an associated rps-chain. Then (Y i ) m i=0 has the foowing properties: 1) ρ(y i ) ρ(y i 1 ) ( 1)1 Ai for a i = 2,..., m. 2) η(y i ) η(y i 1 ) δ 1 Ai for a i = 2,..., m. Proof. Let i {2,..., m}. We wi begin by proving the upper bound for the ran. By ran axioms, cosure property and by Proposition 5, we have ρ(y i ) ρ(y i 1 ) ρ(z i ) ρ(y i 1 Z i ) ρ(y i 1 Z i ). When the intersection is not an coatom, we obtain the same bound as in Lemma 5. However, if Y i 1 Z i coa Z(M Zi), then, by Theorem 1.4, we have ρ(y i 1 Z i ) = ρ(z i ) 1. Thus ρ(y i ) ρ(y i 1 ) = 1. Hence, we have ρ(y i ) ρ(y i 1 ) ( 1)1 Ai. We wi now prove the ower bound for the nuity. By nuity properties coming from ran axioms, we have η(y i ) η(y i 1 ) η(z i ) η(y i 1 Z i ). If Y i 1 Z i coa Z(M Zi), then η(z i ) η(y i 1 Z i ) d Zi 1 δ 1. Thus, we obtain the same bound as in Lemma 5. However, if Y i 1 Z i / coa Z(M Zi), then it impies that η(y i 1 Z i ) max{η(z) : Z coa Z(M Zi)} 1 Therefore, = max{η(z) : Z Z(M Z i )} 1. η(z i ) η(y i 1 Z i ) d Zi δ,

7 and η(y i ) η(y i 1 ) δ 1 Ai. In order to use Lemma 6 to get a Singeton-type bound with these assumptions, we need to define a new parameter that wi count the number of times the intersection Y i 1 Z i is a coatom of Z i. Since we aso want to compare different codes, we wi focus on the proportion of atom intersections instead of the actua number of atom intersections. Definition 8. Let (Y i ) m i=0 be an rps-chain. Define 0 α 1 by αm = #{i : Y i 1 Z i coa Z(M Zi)}. We say that (Y i ) m i=0 is an rps α -chain. We can now derive a new Singeton-type bound with the extra parameter α. Theorem 5. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Let α [0, 1] be such that C has an rps α -chain. Then, d n δ ( 1)α (δ α) Proof. Let M = M(E, ρ) be the matroid associated to C, {Z i } i I the coection of repair sets and (Y i ) m i=0 an rps α- chain. We can use Lemma 6 to get a ower bound on m and η(y m 1 ). Now, the indicator 1 Ai = 1 exacty αm times in the rps α -chain. We first need to get a ower bound on m. We have, Thus, = ρ(y m ) (m αm) + αm = m( ( 1)α) m ( 1)α Next, we can obtain a ower bound on η(y m 1 ). We have, η(y m 1 ) δ(m 1) αm = m(δ α) δ (δ α) δ ( 1)α Finay, by Proposition 2, we obtain d n δ ( 1)α (δ α) Since the bound is vaid for a α, we can optimize α to get a bound for a type of rps-chain. Theorem 6. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Then, ( ) d n δ + 1 ( 1). Proof. Seect any rps-chain in C, and observe that this is an rpsᾱ-chain for some ᾱ [0, 1]. By Theorem 5, we then have d n δ (δ ᾱ) ( 1)ᾱ n δ min (δ α). α [0,1] ( 1)α For fixed vaue of h α := ( 1)α, the expression h α (δ α) is ceary minimized when α taes its argest possibe vaue, i.e.when ( 1)α = h α, so α = 1 h( 1). Now, when 0 α 1, h α can tae any integer vaue between and. So we can rewrite the bound as ( d n δ min h δ ) h Z [,] 1 + h( 1) ( = n δ min h δ ) +. h Z [,] 1 1 The coefficient of h is ceary positive, so the minimum is achieved when h =. Hence, we get the bound ( d n δ δ ) = n δ δ ( ) 1. 1 ( 1 It is straightforward to verify that the term 1 ) evauates to 1 if = 1 mod, and 0 otherwise. Thus, we can write the bound as ( ) d n δ + 1 ( 1). Now, since we have two bounds with the same assumptions, we can compare them and find the minimum of the two, to get the best bound. It turns out that the atter of the two bounds is aways at east as strong as the first one, expect when r =. In that case ony, the first bound is stronger, and the difference between the two bounds is one. Hence, we obtain the foowing goba bound. Theorem 7. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Then, ( ) d n δ + 1 ( 1) and (+1). (4) We can appy the bound from Proposition 5 to obtain a bound on the parameters n,, d, r and δ ony. Coroary 3. Let C be a non-degenerate, binary inear (n,, d, r, δ)-lrc with δ > 2 and without repication. Then, ( ) d n δ + 1 (r 1) ( 1) and r r 1

8 The foowing graph is a comparison of the previous nown bound (1) and the new one from Coroary 3 for two different vaues of r. We can see that the new bound is aways better than (or equivaenty, smaer) or equa to the previous bound. Fig. 1: Comparison of the previous bound and the new bound for n = 50 and δ = 3. Finay, we provide one sma exampe that achieves the bound from Coroary 3. Exampe 1. Let C be the binary inear (10, 4, 4)-code given by the foowing generator matrix: G = We can define three repair sets by their corresponding coumns in G. Z 1 = {1, 2, 3, 5, 6, 8}, Z 2 = {2, 3, 6, 7, 9, 10}, and For a i {1, 2, 3}, we have Z 3 = {1, 4, 6, 7, 8, 10}. Z i = 6, ρ(z i ) = 3, and d Zi = 3. Hence we obtain a binary inear (10, 4, 4, 4, 3)-LRC that achieves the bound from Coroary 3. First, even if this code has not the most interesting parameters, mainy because the rate of the code is under onehaf, it proves the achievabiity of the bound. Secondy, it iustrates perfecty the notion of faiure toerance hierarchy mentioned in Section III. To avoid confusion, we denote now by {c 1, c 2,..., c 10 } the set of coumns of G. Suppose that the coumn c 1 fais. Then, we can repair it by summing coumns c 2 and c 4 in the corresponding atom. If c 1 and c 2 fai, then we form a basis in the repair set Z 1 and we repair both coumns via the basis. For exampe the set {c 3, c 6, c 8 } is a basis of the first repair set Z 1, and c 1 = c 6 c 8 and c 2 = c 3 c 6. Finay, if the three coumns c 1, c 2 and c 5 fai, we use the goba correcting properties of the code. Here, we can choose the basis {c 3, c 4, c 6, c 7 } and we have c 1 = c 4 c 7, c 2 = c 3 c 6, and c 5 = c 3 c 4 c 6 c 7. The most commony used fied-dependent distance bound for LRCs is the Cadambe Mazumdar bound [8], which is ony stated without reference to the oca minimum distance (or equivaenty, for δ = 2). It can straightforwardy be generaized to codes toerating more than one oca erasure. To the best of our nowedge, this generaization has not previousy appeared in the iterature. An unfortunate confict of notation in the iterature caused an erroneous version of Proposition 6 to occur in [23, Proposition 6]. In [24], a simiar bound was presented when more than one erasure is toerated, but the system assumptions used there are quite different from those in the present paper, and hence the bounds are not directy comparabe, Proposition 6. Let C be a non-degenerate inear (n,, d, r, δ)-lrc over F q. Then, [ ] min tr + opt(n (q) t(r + δ 1), d), (5) t Z + where (q) opt(n, d) is the maximum ran of a inear code of ength n and minimum distance d over F q. However, the determination of (q) opt is a cassica open probem in coding theory. Moreover, even given a formua for opt, (q) evauating (5) may be a tedious tas. In that sense, the bound (4) is more expicit than this one. In order to compare the bounds (4) and (5) for binary codes, we choose to estimate opt in (5) via the Potin bound for binary inear codes with specific vaues of n, d, r, δ and, and denote by max the maximum dimension given by (5). Then we have, on one hand + (5) max + δ 1 ( 1) and 1 max δ 1 (max 1) and max 1 (6) and on the other hand Thm.6 + δ 1 ( 1) and 1 n d δ, (7) so the eft hand side in (6) and (7) is bounded by the minimum of the right hand sides of these inequaities. The best bound wi be the one that gives the minimum of the two aternative right hand sides. This aows for a partia but computabe comparison of (4) and (5). Using this method, we found parameters vaues for which (6) is tighter than (7) and vice-versa. In particuar, the parameters from exampe 1 achieve (7) with equaity whie it is one off in (6). This shows that for certain parameters vaues, Theorem 7 is stronger than Proposition 6. In [23], it was erroneousy caimed that Proposition 6 is aways stronger than Theorem 7, due to an error in the statement of Proposition 6. However, this wor is ony a first step toward an expicit bound for binary LRCs and further improvements can be done by expoiting the techniques deveoped in this paper, via more detaied study of the cycic fats. Such improvements can be made to give stronger versions both of Theorem 7 and of

9 Proposition 6. Due to interest of space, this is eft to an extended version of this artice. VI. CONCLUSION We have derived new expicit tradeoffs between the oca and goba minimum distances of binary ocay repairabe codes, together with an exampe of a code meeting this bound with equaity. Appications to distributed data storage were discussed, with the motivation to decrease computationa compexity thans to a sma fied size. Moreover, we demonstrated an intrinsic hierarchy of repair sets that naturay occurs for binary inear storage codes. Acnowedgment. This wor is supported in part by the Academy of Finand (grants #276031, # and # to C. Hoanti), and by the TU Munich Institute for Advanced Study, funded by the German Exceence Initiative and the EU 7th Framewor Programme (grant #291763). [18] T. Britz and C. G. Rutherford, Covering radii are not matroid invariants, Discrete Mathematics, vo. 296, pp , [19] J. E. Bonin and A. de Mier, The attice of cycic fats of a matroid, Annas of Combinatorics, vo. 12, pp , [20] R. P. Staney, Enumerative combinatorics, 2nd ed. Cambridge University Press, 2011, vo. 1. [21] M. Grezet, R. Freij-Hoanti, T. Westerbäc, and C. Hoanti, On binary matroid minors and appications to data storage over sma fieds, in Internationa Caste Meeting on Coding Theory and Appications, 2017, pp [22] G. M. Kamath, N. Praash, V. Laitha, and P. V. Kumar, Codes with oca regeneration and erasure correction, IEEE Transactions on Information Theory, vo. 60, no. 8, pp , [23] M. Grezet, R. Freij-Hoanti, T. Westerbäc, O. Omez, and C. Hoanti, Bounds on binary ocay repairabe codes toerating mutipe erasures, in Internationa Zürich Seminar on Information and Communication. IEEE/ETH, [24] A. S. Rawat, A. Mazumdar, and S. Vishwanath, Cooperative oca repair in distributed storage, EURASIP J. Adv. Signa Proc., pp. 1 17, REFERENCES [1] P. Gopaan, C. Huang, H. Simitci, and S. Yehanin, On the ocaity of codeword symbos, IEEE Transactions on Information Theory, vo. 58, no. 11, pp , [2] I. Tamo, D. Papaiiopouos, and A. Dimais, Optima ocay repairabe codes and connections to matroid theory, IEEE Transactions on Information Theory, vo. 62, pp , [3] I. Tamo and A. Barg, A famiy of optima ocay recoverabe codes, IEEE Transactions on Information Theory, vo. 60, no. 8, pp , [4] A. Wang and Z. Zhang, Repair ocaity with mutipe erasure toerance, IEEE Transactions on Information Theory, vo. 60, no. 11, pp , [5] I. Tamo, A. Barg, and A. Froov, Bounds on the parameters of ocay recoverabe codes, IEEE Transactions on Information Theory, vo. 62, no. 6, pp , [6] A. Pöänen, T. Westerbäc, R. Freij-Hoanti, and C. Hoanti, Improved singeton-type bounds for ocay repairabe codes, in Internationa Symposium on Information Theory. IEEE, 2016, pp [7] A. Wang and Z. Zhang, An integer programming-based bound for ocay repairabe codes, IEEE Transactions on Information Theory, vo. 61, no. 10, pp , [8] V. Cadambe and A. Mazumdar, An upper bound on the size of ocay recoverabe codes, in Internationa Symposium on Networ Coding, 2013, pp [9] A. Agarwa, A. Barg, S. Hu, A. Mazumdar, and I. Tamo, Combinatoria aphabet-dependent bounds for ocay repairabe codes, 2017, arxiv: [10] N. Siberstein and A. Zeh, Optima binary ocay repairabe codes via anticodes, in Internationa Symposium on Information Theory. IEEE, 2015, pp [11] P. Huang, E. Yaaobi, H. Uchiawa, and P. Siege, Cycic inear binary ocay repairabe codes, in Information Theory Worshop. IEEE, [12], Binary inear ocay repairabe codes, IEEE Transactions on Information Theory, vo. 62, pp , [13] S. Baaji, K. Prashant, and P. V. Kumar, Binary codes with ocaity for mutipe erasures having short boc ength, in Internationa Symposium on Information Theory, 2016, pp [14] T. Westerbäc, R. Freij-Hoanti, T. Ernva, and C. Hoanti, On the combinatorics of ocay repairabe codes via matroid theory, IEEE Transactions on Information Theory, vo. 62, pp , [15] R. Freij-Hoanti, T. Westerbäc, and C. Hoanti, Locay repairabe codes with avaiabiity and hierarchy: matroid theory via exampes, in Internationa Zürich Seminar on Communications (invited paper). IEEE/ETH, 2016, pp [16] R. Freij-Hoanti, C. Hoanti, and T. Westerbäc, Matroid theory and storage codes: bounds and constructions, 2017, invited boo chapter in Networ Coding and Subspace Designs, Springer, to appear. [17] H. Crapo and G. C. Rota, On the Foundations of Combinatoria Theory: Combinatoria Geometries,, preiminary edition ed. MIT Press, Cambridge, MA, London 1970.