On the Combinatorics of Locally Repairable Codes via Matroid Theory

Transcription

1 On the Combinatoics of Locally Repaiable Codes via Matoid Theoy Thomas Westebäc, Ragna Feij-Hollanti, Toni Envall, and Camilla Hollanti 1 Abstact axiv: v3 [cs.it] 21 Jan 2016 This pape povides a lin between matoid theoy and locally epaiable codes (LRCs) that ae eithe linea o moe geneally almost affine. Using this lin, new esults on both LRCs and matoid theoy ae deived. The paametes (n,, d,, δ) of LRCs ae genealized to matoids, and the matoid analogue of the genealized Singleton bound in [P. Gopalan et al., On the locality of codewod symbols, IEEE Tans. Inf. Theoy] fo linea LRCs is given fo matoids. It is shown that the given bound is not tight fo cetain classes of paametes, implying a nonexistence esult fo the coesponding locally epaiable almost affine codes, that ae coined pefect in this pape. Constuctions of classes of matoids with a lage span of the paametes (n,, d,, δ) and the coesponding local epai sets ae given. Using these matoid constuctions, new LRCs ae constucted with pescibed paametes. The existence esults on linea LRCs and the nonexistence esults on almost affine LRCs given in this pape stengthen the nonexistence and existence esults on pefect linea LRCs given in [W. Song et al., Optimal locally epaiable codes, IEEE J. Sel. Aeas Comm.]. T. Westebäc, T. Envall, and C. Hollanti ae with the Depatment of Mathematics and Systems Analysis, Aalto Univesity, Finland. R. Feij-Hollanti is with the Depatment of Communications and Netwoing, Aalto Univesity, Finland. s: {thomas.westebac, agna.feij, toni.envall, camilla.hollanti}@aalto.fi. The eseach of R. Feij-Hollanti is patially suppoted by the Finnish Academy of Science and Lettes. The eseach of C. Hollanti is suppoted by the Academy of Finland gants #276031, #282938, and #283262, and by Magnus Ehnooth Foundation, Finland. The suppot fom the Euopean Science Foundation unde the ESF COST Action IC1104 is also gatefully acnowledged. Peliminay and patial esults of this pape wee pesented at the 2014 IEEE Infomation Theoy Woshop (ITW) in Hobat, Tasmania [1].

2 I. INTRODUCTION Due to the eve-gowing need fo moe efficient and scalable systems fo cloud stoage and data stoage in geneal, distibuted stoage has become an inceasingly impotant ingedient in many data systems. In thei seminal pape [2], Dimais et al. intoduced netwo coding techniques fo lagescale distibuted stoage systems such as data centes, cloud stoage, pee-to-pee stoage systems and stoage in wieless netwos. These techniques can, fo example, consideably impove the stoage efficiency compaed to taditional stoage techniques such as eplication and easue coding. Failing devices ae not uncommon in lage-scale distibuted stoage systems [3]. A cental poblem fo this type of stoage is theefoe to design codes that have good distibuted epai popeties. Seveal cost metics and elated tadeoffs [2], [4], [5], [6], [7], [8] ae studied in the liteatue, fo example epai bandwidth [2], [4], dis-i/o [9], and epai locality [10], [11], [12]. In this pape epai locality is the subject of inteest. The notion of a locally epaiable code (LRC) was intoduced in [13], and such epai-efficient codes ae aleady used in existing distibuted stoage systems, e.g., in the Hadoop Distibuted File System RAID used by Faceboo and Windows Azue Stoage [14]. Thee ae two notions of symbol locality consideed in the liteatue: infomation locality only equies infomation symbols to be locally epaiable, while all-symbol locality equies this to be tue fo all code symbols. The subject of inteest in this pape is the all-symbol locality. It is well-nown that nonlinea codes often achieve bette pefomance than linea ones, e.g., in the context of coding ates fo eo-coecting codes and maximal thoughput fo netwo codes. Almost affine codes wee intoduced in [15] as a genealization of linea codes. This class of codes contains codes ove abitay alphabet size, not necessaily pime powe. In this pape, we ae studying LRCs in the geneality of almost affine codes. We will conside five ey invaiants (n,, d,, δ) of locally epaiable codes. The technical definitions ae given in Section II-A, but in shot, a good code should have lage ate /n as well as high global and local failue toleance d and δ, espectively. In addition, it is desiable to have small, which will detemine the maximum numbe of nodes that have to be contacted fo epai within a local epai set. In this pape, ou main tools fo analyzing LRCs come fom matoid theoy. This is a banch of algebaic combinatoics with natual lins to a geat numbe of diffeent topics, e.g., to coding theoy, gaph theoy, matching theoy and combinatoial optimization. Matoids wee intoduced in 2

3 [16] in ode to abstactly captue popeties analogous to linea independence in vecto spaces and independence in gaphs. Since its intoduction, matoid theoy has been successfully used to solve poblems in many aeas of mathematics and compute science. Matoid theoy and the theoy of linea codes ae closely elated since evey matix ove a field defines a matoid. Despite this fact, until athe ecently matoid theoy has only played a mino pat in the development of coding theoy. One pioneeing wo in this aea is the pape by Geene fom 1976 [17]. In this pape he descibes how the weight enumeato of a linea code C is detemined by the Tutte polynomial of the associated matoid of C. Using this esult, Geene gives an elegant poof of the MacWilliams identity [18]. Genealizations of these esults have then been pesented in seveal papes, fo example in [19], [20]. Anothe impotant instance of matoidal methods in coding theoy is the development of a decomposition theoy of binay linea codes [21]. Today, matoid theoy also plays an impotant ole in infomation theoy and coding theoy, fo example in the aeas of netwo coding, secet shaing, index coding, and infomation inequalities [22], [23], [24]. In this pape, while ou main goal is investigating almost affine LRCs with the aid of matoid theoy, ideas fom the theoy of LRCs will also be utilized to acquie new esults in matoid theoy. A. Related wo One of the most classical theoems in coding theoy is the Singleton bound, discussed in Section II-B [25]. Its classical vesion bounds the minimum distance d of a code fom above in tems of the length n and dimension. Recent wo shapens the bound in tems of the local paametes (, δ) [10], [26], [27], [28], as well as in tems of othe paametes [13], [29], [30]. Thee ae diffeent constuctions of LRCs that ae optimal in the sense that they achieve a genealized Singleton bound, e.g. [14], [26], [31], [32], [33]. Song et al. [32] investigate fo which paametes (n,,, δ) thee exists a linea LRC with all-symbol locality and minimum distance d achieving the genealized Singleton bound fom [26]. The paamete set (n,,, δ) is divided into eight diffeent classes. In fou of these classes it is poven that thee ae linea LRCs achieving the bound, in two of these classes it is poven that thee ae no linea LRCs achieving the bound, and the existence of linea LRCs achieving the bound in the emaining two cases is an open question. Independently to the eseach in this pape, Wang and Zhang used linea pogamming appoaches to stengthen these esults when δ = 2 [28]. It was shown in [14], that the -locality of a linea LRC is a matoid invaiant. This was used in 3

4 [14] to pove that the minimum distance of a class of linea LRCs achieves a genealized Singleton bound. Moeove, thee ae seveal instances of esults in the theoy of linea codes that have been genealized to all matoids. Examples on how these esults can be intepeted fo othe objects that can epesent a matoid, such as gaphs, tansvesals and cetain designs can be found in [34]. Recently, the pesent authos have studied locally epaiable codes with all-symbol locality [35]. Methods to modify aleady existing codes wee pesented and it was shown that with high pobability, a cetain andom matix will be a geneato matix fo a locally epaiable code with a good minimum distance. Constuctions wee given fo thee infinite classes of optimal vecto-linea locally epaiable codes ove an alphabet of small size. The pesent pape extends and deviates fom this wo by studying the combinatoics of LRCs in geneal and elating LRCs to matoid theoy. This allows fo the deivation of fundamental bounds fo matoids and linea and almost affine LRCs, as well as fo the chaacteization of the matoids achieving this bound. In this pape, we have chosen to call the codes and matoids achieving the genealized Singleton bound pefect instead of optimal, eseving the tem optimal fo the best existing solution, i.e., fo codes achieving a tight bound instead of the (in some cases loose) Singleton bound. See Definition III.2 and the follow-up footnote fo moe details. B. Contibutions and oganization The fist contibution of this pape is to extend the definitions of the paametes (n,, d,, δ) in [26] fom linea codes to the much lage class of almost affine codes, and to show that these paametes ae matoid invaiant fo all almost affine LRCs. We then poceed to pove the main esults of this pape, which can be summaized as follows: (i) A matoid analogue of the genealized Singleton bound in [26] is given fo (n,, d,, δ)-matoids, and in paticula to all almost affine codes in Theoem III.3. (ii) In Theoem III.4, some necessay stuctual popeties ae given fo an (n,, d,, δ)-matoid meeting the genealized Singleton bound. (iii) In Theoem IV.1, a class of matoids is given with diffeent values of the paametes (n,, d,, δ). Simple and explicit constuctions of matoids in this class ae given in Theoem IV.1, Theoem IV.2, and Coollay IV.2, and in Examples IV.1, IV.2 and IV.3. (iv) In Section V-B, we pove that the matoids fom Theoem IV.1 ae epesentable ove finite fields of lage enough size. Hence we obtain fou explicit constuctions of linea LRCs with given 4

5 paametes. The epesentability is deived by constucting a gaph suppoting a gammoid isomophic to the matoid in Theoem IV.1, and using esults on epesentability of gammoids [36]. (v) Theoem IV.4 chaacteizes values of (n,,, δ) fo which thee exist (n,, d,, δ)-matoids meeting the bound (i). In paticula, the nonexistence esults fo linea LRCs in [32] ae extended to the nonexistence of almost affine codes and matoids. Moeove, in Theoem V.4 and Theoem V.5, we settle the existence in one of the egimes left open in [32], leaving open only a mino subegime of b > a 1, whee a = and b = ( + δ 1) n +δ 1 n. This complements ecent and independent eseach by Wang and Zhang [28], whee they settle the existence in the subegime n +1 > b and δ = 2 using intege pogamming techniques. The poofs of some of the longe theoems and the explicit constuctions of matoids with pescibed paametes ae given in the Appendix. II. PRELIMINARIES A. Paametes (n,, d,, δ) of locally epaiable codes In this subsection, we intoduce the paametes (n,, d,, δ) defined in [26] fo linea locally epaiable codes. We extend this definition to the much wide class of almost affine codes, to be intoduced in II-F. Figue 1 seves as a visual aid fo the technical definitions. The infomation symbols (a, b, c, d, e, f) ae stoed on twelve nodes as in the figue. Equivalently, we thin of the content of the twelve nodes as a codewod, and of the content of an individual node as a code symbol. Within each of the local clouds (o locality sets), thee symbols ae enough to detemine the othe two. Thus, Figue 1 depicts a (12, 6, 3, 3, 3)-LRC, accoding to the following definitions. Let C A n be a code such that C = A, whee A is a finite set, also efeed to as the alphabet. Fo any subset X = {i 1,..., i m } [n] = {1, 2,..., n}, let C X denote the pojection of the code into A X, that is C X = {(c i1,..., c im ) : c = (c 1,..., c n ) C}. (1) The code C X is also called a punctued code in the coding theoy liteatue. The minimum (Hamming) distance d of C can be defined in tems of pojections as d = min{ X : X [n] and C [n]\x < C }. (2) Fo 1 and δ 2, an (, δ)-locality set of C is a subset S [n] such that (i) S + δ 1 5

6 Fig. 1. A stoage system fom a (12, 6, 3, 3, 3)-LRC. (ii) Fo evey l S, L = {i 1,..., i L } S \ {l} and L = S (δ 1), c l is a function of (c i1,..., c i L )), whee c = (c 1,..., c n ) C. We say that C is a locally epaiable code (LRC) with all-symbol locality (, δ) if all the n symbols of the code ae contained in an (, δ)-locality set. The locality sets can be also efeed to as the local epai sets. We ema that the symbols in a locality set S can be used to ecove up to δ 1 lost symbols in the same locality set. Futhe, we note that each of the following statements ae equivalent to statement (ii) above: (ii ) Fo any l S, L = {i 1,..., i L } S \ {l}, and L = S (δ 1), we have C L {l} = C L, (ii ) Fo any L S with L S (δ 1), we have C L = C S, (ii ) d(c S ) δ, whee d(c S ) is the minimum distance of C S. An LRC with paametes (n, ), minimum distance d, and all-symbol locality (, δ) is an (n,, d,, δ)- LRC. Since we focus only on all-symbol locality in this pape, we will hencefoth use the tem LRC to mean a locally epaiable code with all-symbol locality. B. The Singleton bound Fo any [n, ]-linea code with minimum distance d, the Singleton bound is given by d n + 1. (3) 6

7 This bound was genealized fo locally epaiable codes in [10] (the case δ = 2) and [26] (geneal δ) as follows. A linea LRC with paametes (n,, d,, δ) satisfies ( ) d n (δ 1). (4) While the bounds in [10] and [26] ae stated assuming only infomation locality, so ae of couse in paticula still valid unde the stonge assumption of all-symbol locality. Othe genealizations of the Singleton bound fo linea and nonlinea LRCs can be found in [13], [29], [30]. C. Gaphs, G = (V, E) Let us fix some standad gaph-theoetic notation that will be used at two stages in the constuctions. A (finite) diected gaph G = (V, E) is a pai of a finite vetex set V, whose elements ae called nodes o vetices, and an edge set E V V of pais called acs o edges. Gaphs ae often dawn with the vetices as points and acs (v, u) as aows v u. We call v the tail of (v, u), and u the head. A path fom S V to T V is a sequence v 0, v 1,..., v n, whee v 0 S, v n T, and (v i, v i+1 ) E fo each i = 0,..., n 1. If v 0 = v n, then the path is called a (diected) cycle. An impotant case of gaphs is when E is symmetic, i.e., (u, v) E if and only if (v, u) E. In such case, it is customay to identify the two pais (u, v) and (v, u) with the set {u, v}, and ease all the heads of the aows in the dawing. When taling about a gaph without specifying that it is diected, the symmetic situation is assumed. Obseve that this definition allows fo loops edges (whee the tail and the head is the same), but not multiple edges. In this pape, we will assume that all gaphs, both symmetic and diected, ae without multiple edges and loops. D. Posets and lattices, (P, ) Befoe studying matoids, we need a minimum of bacgound on poset and lattice theoy. We efe the eade to [37] fo moe infomation on posets and lattices. The mateial in this section is used only in the technical wo with the lattice of cyclic flats of matoids. A collection of sets P 2 E odeed by inclusion defines a (finite) poset (P, ). A chain C of (P, ) is a set of elements X 0,..., X m P such that X 0 X 1... X m. The length of a chain C is defined as the intege l(c) = C 1 = m. Fo X, Y P, let L X,Y = {Z P : Z X and Z Y }, U X,Y = {Z P : X Z and Y Z}. 7

8 An element Z L X,Y is the meet of X and Y, denoted by X Y, if it contains evey V L X,Y. Dually, Z U X,Y is the join of X and Y, denoted by X Y, if it is contained in evey V U X,Y. A poset (P, ) is a lattice if evey pai of elements of P has a meet and a join. If (P, ) is a (finite) lattice, then thee ae two elements 0 P, 1 P P such that 0 P X and X 1 P fo all X P. The atoms and coatoms of a lattice (L, ) ae defined as espectively. 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 }, E. Matoids, M = (ρ, E) Matoids can be defined in many equivalent ways, fo example by thei an function, nullity function, independent sets, cicuits and moe [38]. Fo ou pupose, the following definition will be the most useful. Let 2 E denote the set of all subsets of E. A matoid M on a finite set E is defined by a an function ρ : 2 E Z satisfying the following axioms: (R1) 0 ρ(x) X fo X E, (R2) X Y E ρ(x) ρ(y ), (R3) X, Y E ρ(x) + ρ(y ) ρ(x Y ) + ρ(x Y ). (5) The nullity function η : 2 E Z of the matoid M = (E, ρ) is defined by η(x) = X ρ(x), fo X E. Let X be any subset of E. The subset X is independent if ρ(x) = X, othewise it is dependent. A dependent set X is a cicuit if all pope subsets of X ae independent, i.e., ρ(x) = X 1 and ρ(y ) = Y fo all subsets Y X. The closue of X is defined as cl(x) = {x E : ρ(x x) = ρ(x)}. The subset X is a flat if cl(x) = X. It is cyclic if it is a (possible empty) union of cicuits. The sets of cicuits, independent sets, cyclic sets and cyclic flats of a matoid M is denoted by C(M), I(M), U(M) and Z(M), espectively. We omit the subscipt M when the matoid is clea and wite C, I, U and Z, espectively. The set of cyclic flats togethe with inclusion defines the lattice of cyclic flats (Z, ) of the matoid. The estiction of M to X is the matoid M X = (ρ X, X) whee ρ X (Y ) = ρ(y ), fo all subsets Y X. (6) 8

9 F. Almost affine codes and thei associated matoids A code C A n, whee A is a finite set of size s 2, is almost affine if log s ( C X ) Z fo each X [n]. Note that if C is an almost affine code, then all pojections C X of C ae also almost affine. In [15] it is poven that evey almost affine code C A n induces a matoid M C = (ρ C, [n]), whee ρ C (X) = log s ( C X ). (7) Examples of matoids which cannot be epesented by any almost affine code ae given in [39]. Moeove, an example of a matoid which can be epesented by an almost affine code ove a thee lette alphabet, but not by any linea code is given in [15]. This example is the so-called non-pappus matoid. Example II.1. An example of a matoid M G = (ρ, E) is defined by the matix a b c G =, (8) d e f which we thin of as a geneato matix of a linea code C ove the field F 5. The code C is the ow span of G, E = {1,..., 12} is the set of columns, and the an of a subset of E is the an of the coesponding submatix, i.e., ρ(i) = an(g I ) fo I E, whee G I is the submatix of G whose columns ae the columns indexed by I. Below ae some 9

10 independent sets, cicuits, cyclic flats and an functions of some subsets of E fo the matoid M. I = {, {2, 3, 7}, {3, 4, 5}, {7, 8, 9}, [6],...}, C = {{1, 2, 3, 7}, {4, 5, 8, 11},...}, Z = {{1, 2, 3, 7, 10}, {3, 4, 5, 8, 11}, {1, 2, 3, 4, 5, 7, 8, 10, 11}, [12],...}, ρ( ) = 0, ρ({3, 4, 5}) = ρ({4, 5, 8, 11}) = ρ({3, 4, 5, 8, 11}) = 3, ρ([6]) = ρ([12]) = 6. The eade can veify that the code geneated by this matix coesponds to the stoage system in Figue 1, when the ows ae the infomation symbols. G. Basic popeties of matoids and the lattice of cyclic flats Fo the applications in this pape, the most impotant matoid attibute is its lattice of cyclic flats. This is because the minimal cyclic flats of matoids will coespond to local epai sets of the LRC. In this subsection, we pesent basic popeties of the lattice of cyclic flats, that will be needed in late pats of the pape. Poposition II.1 (see [40]). Let M = (ρ, E) be a matoid. Then (i) ρ(x) = min{ρ(f ) + X \ F : F Z}, fo X E, (ii) Define D = {X : thee is F Z with X F and X = ρ(f ) + 1}. Then C is the set of minimal elements in D, odeed by inclusion. (iii) (Z, ) is a lattice with the following meet and join fo X, Y Z, X Y = {C C:C X Y } C and X Y = cl(x Y ). The assetion (i) in Poposition II.1 shows that a matoid is detemined by its cyclic flats and thei ans. Convesely, the following theoem gives an axiomatic scheme fo a collection of subsets on E and a function on these sets to define the cyclic flats of a matoid and thei ans. This will allow us to constuct matoids with pescibed paametes in Section III. Theoem II.1 (see [40] Th. 3.2). Let Z 2 E and let ρ be a function ρ : Z Z. Thee is a matoid M on E fo which Z is the set of cyclic flats and ρ is the an function esticted to the sets in Z if 10

11 and only if (Z0) Z is a lattice unde inclusion, (Z1) ρ(0 Z ) = 0, (Z2) X, Y Z and X Y 0 < ρ(y ) ρ(x) < Y X, (Z3) X, Y Z ρ(x) + ρ(y ) ρ(x Y ) + ρ(x Y ) + (X Y ) \ (X Y ). The esults in the poposition below ae basic matoid esults that will be needed seveal times in the poofs of othe esults given late in this pape. We give a poof fo the esults in Poposition II.2 that we have not been able to find in the liteatue. Fo the othe esults we only give a efeence. Poposition II.2. Let M = (ρ, E) be a matoid and let X, Y be subsets of E, then (i) If X Y, then η(x) η(y ), (ii) η(x Y ) η(x) + η(y ) η(x Y ), (iii) If ρ(x) < ρ(e) and 1 Z = E, then η(x) max{η(z) : Z coa Z }, (iv) cl(u) Z(M) fo U U(M), (v) U(M X) = {U X : U U(M)}, (vi) C(M X) = {C X : C C(M)}, (vii) Z(M X) = {Z Z(M) : Z X} if X F(M), (viii) X / U(M) if and only if x X such that ρ(x x) < ρ(x), (ix) ρ(cl(x)) = ρ(x), (x) If X Y, then cl(x) cl(y ). Poof: Popeties (i), (ii), (v), (vii) and (viii) can be found in [41, Lemma 2.2.4, Lemma 2.3.1, the paagaph unde Lemma 2.4.5]. Popety (iv) is a consequence of [38, Poposition (ii)]. Fo (iii), assume that ρ(x) < ρ(e) and 1 Z = E. Thus, cl(x) E and η(x) η(cl(x)). Let U be the lagest cyclic set such that U cl(x). Fom [41, Lemma 2.4.8, Lemma 2.5.2], we have that η(cl(x)) = η(u) and that U is a cyclic flat. Popety (iv) now follows fom the fact that ρ(u) ρ(cl(x)) < ρ(e) = ρ(1 Z ). Popety (vi) is a diect consequence of (v). Popety (ix) is a consequence of popety (x) which can be found in [38, Lemma 1.4.2] 11

12 Example II.2. Continuing with Example II.1, and emembeing that the elements of M G ae the columns of G, we see that the cyclic flats of M G ae the submatices in Figue 2. The atomic cyclic flats ae thus the submatices coesponding to column sets {1, 2, 3, 7, 10}, {3, 4, 5, 8, 11} and {1, 5, 6, 9, 12}. Remembeing fom (8) that the ows ae indexed by the infomation symbols (a, b, c, d, e, f), these atomic cyclic flats agee exactly with the local clouds in Figue 1. Fig. 2. The lattice Z(M G) of cyclic flats of the matoid M(G) in Example II.2. III. LOCALLY REPAIRABLE MATROIDS A. The paametes (n,, d,, δ) fo matoids In this subsection we show that the paametes (n,, d,, δ) ae matoid invaiants fo an almost affine LRC. This will allow us to extend the definition of these paametes to matoids in geneal. Let C be an almost affine (n,, d,, δ)-lrc ove some finite alphabet A. By the definition given in Eq. (7), we now that C X = A ρc(x), which specializes to = ρ C ([n]) when X = [n]. In [15] it is poven that M CX = M C X fo X [n]. Consequently, since the pojection C X is also almost affine, (2) implies that d(c X ) = min{ Y : Y X and ρ C (X \ Y ) < ρ C (X)}, 12

13 whee d(c X ) denotes the minimum distance of C X. Using the obsevations above and the definition of an (n,, d,, δ)-lrc given in Section II-A, we conclude the following theoem. Theoem III.1. Let C be an almost affine LRC with the associated matoid M C = (ρ C, [n]). Then, the paametes (n,, d,, δ) of C ae matoid invaiants, whee (i) = ρ C ([n]), (ii) d = min{ X : X [n] and ρ C ([n] \ X) < }, (iii) C has all-symbol locality (, δ) if and only if, fo evey j [n] thee exists a subset S j [n] such that a) j S j, b) S j + δ 1, c) d(c Sj ) = min{ X : X S j and ρ C (X) < ρ C (S j )} δ. These esults can now be taen as the definition of the paametes (n,, d,, δ) fo an abitay matoid. Definition III.1. Let M = (ρ, E) be a matoid. Then we call M an (n,, d,, δ)-matoid, whee (i) n = E, (ii) = ρ(e), (iii) d = min{ X : X E and ρ(e \ X) < }, (iv) The paametes 0 < ρ(e) and δ 2 ae such that fo all x E, thee exists a subset S x E with a) x S x, b) S x + δ 1, c) d(m S x ) = min{ X : X S x and ρ(s x \ X) < ρ(s x )} δ. A subset S E is called a (, δ)-locality set of the elements x S if the statements b) c) above ae satisfied by S. The paametes n and ae obviously defined fo all matoids. We note that the paamete d is finite if and only > 0. Futhemoe, we notice that evey element x E is contained in some cyclic set S x if and only if 1 Z = E. If this is the case, and = max{ S x 1 : x X}, then M has (, 2)-locality. As a consequence of the obsevations above, we get the following poposition. Poposition III.1. A matoid M = (ρ, E) is an (n,, d,, δ)-matoid with finite values of (n,, d,, δ) if and only if 0 < ρ(e) and 1 Z = E. 13

14 Obseve that if M has (, δ)-locality, then by Definition III.1 (iv), M has (, δ )-locality fo and 2 δ δ with + δ + δ. So neithe the values of (, δ) no the locality sets S x ae in geneal uniquely detemined fo a matoid M. B. A genealized Singleton bound fo (n,, d,, δ)-matoids The main esult of this subsection is Theoem III.3 which gives a Singleton-type bound on the paametes (n,, d,, δ) fo matoids. In the case of linea LRCs with infomation locality and tivial failue toleance δ = 2, i.e., only toleating one failue, the bound was given in [10]. The coe ingedients of the poof of Theoem III.3 ae the same as in [10], intepeted fo matoids. Fist, we elate the paametes (n,, d,, δ) of a matoid to its lattice of cyclic flats in Theoem III.2. Then in Lemma III.1, we obtain a lage cyclic flat Y m 1 of an less than. In Theoem III.3 we elate Y m 1 to d, theeby poving the theoem. Theoem III.2. Let M = (ρ, E) be an (n,, d,, δ)-matoid with 0 < ρ(e) and 1 Z = E. Then (i) d = n + 1 max{η(z) : Z coa Z }, (ii) Fo each x E, thee is a cyclic set S x U(M) such that a) x S x, b) S x + δ 1, c) d(m S x ) = η(s x ) + 1 max{η(z) : Z coa Z(M Sx)} δ. Poof: The poof is given in the Appendix. As η(z) is non-negative fo evey Z, Theoem III.2 (ii) c) gives δ + ρ(s x ) 1 S x, which togethe with Theoem III.2 (ii) b) shows that ρ(s x ) (9) fo any (, δ)-locality S x. Moeove, we obseve that fo any atom S in a lattice of cyclic flats with 0 Z =, we can use any subset S S as a locality set when S > ρ(s). Howeve, diffeent choices of locality sets may give diffeent values on the paametes (, δ). Example III.1. Repesenting the cyclic flats associated to the matoid M G fom Example II.2 just by thei coesponding sets and ans in Figue 3, we use Theoem III.2 to get the paametes (n,, d,, δ) of the linea LRC that is geneated by the matix G given in Example II.1. 14

15 Fig. 3. The lattice Z(M G) of cyclic flats of the matoid M(G) in Example II.2, without efeence to the matix G. The values fo (n,, d) ae n = 12, = 6, d = = 3. Using S 1 = {1, 2, 3, 7, 10}, S 2 = {3, 4, 5, 8, 11} and S 3 = {1, 5, 6, 9, 12} as the locality sets, we get the paametes (, δ) = (3, 3). Fom Theoem III.2, we deive a chain of cyclic flats, fom which we will extact a lage cyclic flat, to be used in the poof of Theoem III.3. Lemma III.1. Let M = (ρ, E) be an (n,, d,, δ)-matoid. Then thee is a chain 0 Z = Y 0 Y 1... Y m = E in (Z(M), ) such that fo j = 1,..., m we have (i) ρ(y j ) ρ(y j 1 ) +, (ii) η(y j ) η(y j 1 ) + (δ 1). Poof: The poof is given in the Appendix. We ae now eady to pove the genealized Singleton bound fo matoids. Theoem III.3. Let M = (ρ, E) be an (n,, d,, δ)-matoid. Then ( ) d n (δ 1). Poof: Let C : 0 Z = Y 0 Y 1... Y m = E 15

16 be a chain of (Z, ) given in Lemma III.1. Then η(y m 1 ) (m 1)(δ 1), by Lemma III.1 (ii). On the othe hand, by Lemma III.1 (i) we have that = ρ(y m ) m, so m. Combining these esults we get η(y m 1 ) (m 1)(δ 1) ( 1)(δ 1). Since Y m 1 Z \ {1 Z }, we have so Theoem III.2 (i) yields max{η(z) : η(z) coa Z } η(y m 1 ), d n + 1 η(y m 1 ) n + 1 ( ) 1 (δ 1). We also give thee additional bounds on the paametes of a matoid. Poposition III.2. Let M = (ρ, E) be an (n,, d,, δ)-matoid. Then (i) δ d, (ii) n (δ 1), (iii) n +δ 1. Poof: The poof is given in the appendix. In the case of codes, Poposition III.2 (i) and (iii) have natual intepetations. Indeed, (i) says that the local minimum distance is bounded fom above by the global minimum distance, and (iii) says that the global code ate is bounded fom above by the local code ate. C. A stuctue theoem fo matoids achieving the genealized Singleton bound Definition III.2. We will call an (n,, d,, δ)-matoid pefect if it meets the genealized Singleton bound of Theoem III.3 with equality, i.e. if d = n + 1 In analogy, we will call a LRC satisfying (10) a pefect LRC 1. ( ) 1 (δ 1). (10) 1 We point out that, typically, codes achieving these ind of bounds have been called optimal in the liteatue. Howeve, we feel that the notion optimal should be saved fo the code that is the best we can do. Thus, saying that an optimal code does not exist when the bound cannot be eached with equality feels wong, since we can still find a code with minimum distance only slightly smalle than the bound, and this code is the best possible solution in this case and thus deseves to be called optimal. Theefoe, we have opted to call the codes achieving the bound pefect. This is say that, even though pefect codes do not exist fo all paametes, optimal solutions can still be found. 16

17 These notions should not be confused with those of a pefect matoid design o a pefect code in classical coding theoy liteatue. Theoem III.4 gives some necessay stuctual popeties fo pefect (n,,, δ)-matoids with <. We will use this stuctue theoem to pove that fo cetain values of (n,,, δ), thee ae no pefect (n,,, δ)-matoids, and consequently no pefect LRCs. The degeneate case when = is easie, and is consideed in Section IV-B1. A collection of sets X 1..., X j is said to have a non tivial union if X l i [j]\{l} X i, fo l = 1,..., j. Theoem III.4. Let M = (ρ, E) be an (n,, d,, δ)-matoid with < and ( ) d = n (δ 1). Let then {S x : x E} U(M) be a collection of cyclic sets fo which the statements a) c) in Theoem III.2 (iv) ae satisfied. Then (i) 0 Z =, (ii) fo each x E, a) η(s x ) = (δ 1), b) S x is an atom in Z(M), and in paticula a cyclic flat. (iii) Fo each collection F 1,... F j of cyclic flats in {S x : x E} that has a non tivial union, c) η( j i=1 F j(δ 1) if j < i) =, n (δ 1) if j, d) j j i=1 F i=1 i = F i if j <, E if j, e) ρ( j i=1 F i) = j i=1 F i j(δ 1) if j <, if j. f) F j ( j 1 i=1 F i) F j δ if j. Poof: The poof is given in the Appendix. By the stuctue theoem III.4 above we get the following coollay. Coollay III.1. Let M = (ρ, E) be an (n,, d,, δ)-matoid with < and ( ) d = n (δ 1). Then M has a collection of cyclic flats F 1,..., F m such that 17

18 (i) {F i } i [m] has a non tivial union, (ii) F i + δ 1 fo i = 1,..., m, (iii) η(f i ) = δ 1, (iv) {X Z(M) : X F i } = {, F i } fo i = 1,..., m, (v) i [m] F i = E, (vi) statements c) f) in Theoem III.4 holds fo evey collection of flats {F i } i I with I [m] and I, (vii) i I\{j} F i + F j (δ 1) ( i I\{j} F i) F j fo I [m], I = and j I. Poof: The statements (i) (v) follows diectly fom Theoem III.4 (i) (ii) and Theoem III.2 (iv). Statement (vi) is a consequence of (i) and Theoem III.4 (iii), since (i) implies that {F i } i I has a non tivial union. Fo statement (vii) we fist obseve by (iv), (vi) and Poposition II.1 (iii) that ( i I\{j} F i ) F j = ( Hence, by (vi) and axiom (Z3) in Theoem II.1, = ρ( i I F i) i I\{j} F i ) F j =. ρ( i I\{j} F i) + ρ(f j ) ρ( ) ( i I\{j} F i) F j = i I\{j} F i ( 1)(δ 1) + Fj (δ 1) ( i I\{j} F i) F j. We ema that stuctue theoems simila in spiit to the above have been given fo linea (n,, d,, δ)- LRCs in [10] and [42]. Namely, Theoem 2.2 in [42] coves the case when, showing that local epai sets coespond to linea [ + δ 1,, δ]-mds codes and ae mutually disjoint. Theoem 7 in [10] poves the same in the special case δ = 2. Coollay III.1 (iv) means that the local matoid M F i is unifom of an F i (δ 1), fo i = 1,..., m. When the matoid comes fom a linea code, the code in question is thus an [ F i, F i (δ 1), δ]-mds code. By (vi) and (vii) in Coollay III.1, we obtain conditions on how lage the intesections of union of subsets of the cyclic flats {F i } i [m] can be. These esults imply the coesponding esults on linea LRCs. IV. CONSTRUCTIONS AND CLASSES OF (n,, d,, δ)-matroids The genealized Singleton bound theoem fo matoids gives an uppe bound fo the value of d in tems of the paametes (n,,, δ) fo a matoid. In subsection IV-A we will give some constuctions 18

19 on (n,, d,, δ)-matoids. These constuctions will then be used in Subsection IV-B, whee we will investigate, given diffeent classes of the paametes (n,,, δ), whethe o not pefect (n,,, δ)- matoids exist. A. Combinatoial constuctions of (n,, d,, δ)-matoids In this section we will give fou inceasingly specialized constuctions of (n,, d,, δ)-matoids. The constuctions ae puely combinatoial, and poceed by assigning the atomic cyclic flats, togethe with the an function on the lattice of cyclic flats. In Section V, we pove that the matoids we have constucted can be epesented by linea codes. 1) Geneal constuction of (n,, d,, δ)-matoids: Let F 1,..., F m be a collection of subsets of a finite set E and define F I = i I F i fo I [m]. Futhe, let be a nonnegative intege and ρ a function ρ : {F i } i [m] Z satisfying (i) 0 < ρ(f i ) < F i fo i [m], (ii) F [m] = E, (iii) F [m] + i [m] (ρ(f i) F i ), (iv) I [m], j [m] \ I F I F j < ρ(f j ). Define Z < = {F J : F J + (ρ(f i ) F i ) < } i J and Z = Z < {E}. Now, we extend the function ρ to a function Z Z, by ρ(f J ) = F J + i J (ρ(f i) F i ) fo F J Z <, ρ(e) =. Note that the extension of ρ given in (11) is well defined, as by (iii), E is not in Z <. Also note that F = and ρ(f ) = 0. Finally, we define I = {X E : F I X ρ(f I ) fo all I [m]}. Theoem IV.1. Let F 1,..., F m be a collection of subsets of a finite set E, a nonnegative intege and ρ : {F i } i [m] Z a function satisfying (i) (iv). Then Z and ρ : Z Z, defined in (11), define an (n,, d,, δ)-matoid M(F 1,..., F m ; ; ρ) on E fo which Z is the collection of cyclic flats, ρ is the an function esticted to the cyclic flats, I is the set of independent sets, and (i) n = E, (ii) = ρ(e), (11) 19

20 (iii) d = n + 1 max{ i I η(f i) : F I Z < }, (iv) δ = 1 + min i [m] { F i ρ(f i )}, (v) = max i [m] {ρ(f i )}. Poof: The poof is given in the Appendix. Example IV.1. Let F 1, F 2, F 3 be disjoint sets of cadinality 4, with ρ(f 1 ) = ρ(f 2 ) = 3 and ρ(f 3 ) = 2. Moeove, let (,, δ) = (7, 3, 3). By Theoem IV.1, this coesponds to a matoid of size 14 and minimum distance 4. 2) Specialized constuction of (n,, d,, δ)-matoids: To constuct (n,, d,, δ)-matoids with lage d in Section IV-B, we will use a special case of the constuction in IV.1. We epesent the atomic cyclic flats F i by nodes in a gaph, with labelled edges epesenting the intesections between the flats. The constuction of a lattice of cyclic flats fom a weighted gaph can be made much moe geneal by assigning weights to the nodes, epesenting the size and an of the coesponding flats. Howeve, in this section we specialize all paametes to obtain matoids that achieve the Singleton bound. Let G be a gaph with vetices [m] and edges W, and let γ : W Z 1 be a positive intege-valued function on the edge set. Moeove, let (,, δ) be thee integes with 0 < < and δ 2, such that (i) G has no tiangles, (ii) m w W γ(w), (iii) > j γ({i, j}) fo evey i [m]. Fom the gaph G we constuct the sets F 1,..., F m and the an function ρ by fist assigning the following: (iv) ρ(f i ) = fo i [m], (v) F i = + δ 1 fo i [m], (vi) F i F j = γ({i, j}) fo {i, j} W. Note that (v) (vii) uniquely defines the sets F 1,..., F m and thei ans, up to isomophism. This can be seen by induction ove m, obseving that (iv) guaantees that the intesections F i F j can be chosen to be disjoint fo diffeent j. This is equied, as thee is no 3-cycle in the gaph G, so F h F i F j = 0 fo all thee distinct elements h, i, j [m]. Also note that, while n is not a paamete of the gaph constuction, it is a function of the paametes, 20

21 as we have n = i F i = m( + δ 1) w W γ(w). Theoem IV.2. Let F 1,..., F m and ρ : {F i } Z be constucted fom a weighted gaph (G, γ) with paametes (,, δ) accoding to (i) (vi). Then ({F i }, ρ) satisfies (i) (iv) in IV-A1. In paticula, {F i } ae the atomic cyclic flats of an (n,, d,, δ)-matoid with (i) n = ( + δ 1)m w W γ(w), (ii) d = n + 1 (δ 1) max{ I : I w W I I γ(w) < }. Poof: The poof is given in the Appendix. Now, in addition, we assume that G has gith at least max{4, + 1}, and that the weight function γ does not tae too lage values. Then we get the following theoem, on the existence of pefect (n,,, δ)-matoids. Coollay IV.1. Let (G, γ) be a weighted gaph, and let (,, δ) be integes such that (i) (iii) is satisfied. Let b = w W γ(w), and a =. Assume moeove that G has no l-cycles, fo l, and that w W I I γ(w) a fo evey I [m] with I =. Then thee exists a (n,, d,, δ)-matoid with (i) n = ( + δ 1)m b, (ii) d = n + 1 ( 1)(δ 1). Poof: We need to pove that { 1 max I : I If I = 1, then I If, on the othe hand, I =, then I w W I I w W I I γ(w) = w W I I γ(w) < = γ(w) ( 1) <. w W I I by assumption. Thus, the coollay follows fom Theoem IV.2. γ(w) } a. a =, Coollay IV.2. Let (G, γ) be a weighted gaph, and let (,, δ) be integes such that (i) (iii) is satisfied. Let b = w W γ(w), and a =. Assume moeove that G has no l-cycles, fo 21

22 l, and that 1 γ(w) with a 1 fo evey w W. Then thee is an (n,, d,, δ)-matoid (i) n = ( + δ 1)m b, (ii) d = n + 1 ( 1)(δ 1). Poof: Since G has no l-cycles fo l, we have fo evey I [m] with I = that 1. Since γ(w), we then get W I I so Theoem IV.1 applies. w W I I γ(w) a 1 ( ) a 1 a, 1 We ema that in ode to find as small n as possible fo a chosen (,, δ, a, b) in Coollay IV.2, we want to find a good gaph with as few nodes as possible. To find such a gaph, pefeable popeties fo the gaph ae: many small cycles of length max{4, + 1}, lage values of γ on evey edge, a i.e. γ(w) = fo w W, and that the sum of γ-values incident to each node is lage, i.e. j 1 γ({i, j}) = 1 fo all nodes i [m]). Example IV.2. Let G denote the gaph below on the vetex set [6], whee the values of γ ae witten above the edges in the gaph, and (,, δ) = (14, 4, 2). We get b = γ(w) = 3 and a = = 2 Fig. 4. The gaph G(γ; 14, 4, 2, 2, 3) By Coollay IV.2, this gaph coesponds to a (27, 14, 11, 4, 2)-matoid on the gound set [27], with six atomic cyclic flats F 1,... F 6, whee F 1 = {1,..., 5}, F 2 = {1, 6,..., 9}, F 3 = {6, 10,..., 13}, F 4 = {14,..., 18}, F 4 = {14, 19,..., 22} and F 5 = {23,..., 27}. Example IV.3. Let G = G(γ;,, δ, a, b) denote the gaph below on the vetex set [11]. The γ-values fo the edges ae witten in the gaph and (,, δ, a, b) = (19, 9, 5, 8, 21). By Coollay IV.2, this gaph coesponds to a (122, 19, 96, 9, 5)-matoid, whose lattice of cyclic flats has 11 atoms. 22

23 Fig. 5. The gaph G(γ; 19, 9, 5, 8, 21) B. The maximal d fo (n,,, δ)-matoids We now by Theoem III.3, that the inequality d n ( 1)(δ 1) holds fo any (n,, d,, δ)-matoid. It is then vey natual to as what is the maximal value of d, fo which thee exists an (n,,, δ)-matoid, fo given (n,,, δ) with 0 < n (δ 1) and δ 2. We will denote this maximal value d max = d max (n,,, δ). The case = is degeneate, and we will conside this fist. The case when < will be futhe divided into fou subcases in Theoem IV.4. Theoem IV.4 will late tanslate into esults fo linea LRCs in Theoem V.4 and Theoem V.5. 1) The maximal value of d when = : A well nown class of matoids is the class of unifom matoids [38], defined as U n = (ρ, E), whee This implies that the cyclic sets of U n is and that the cyclic flats ae E = n and ρ(x) = min{ X, }. (12) U(U n) = { } {X E : X + 1}, Z = {0 Z, 1 Z }, with 0 Z =, 1 Z = E, ρ(0 Z ) = 0 and ρ(1 Z ) =. If =, the genealized Singleton bound given in Theoem III.3 educes to the classical Singleton bound, d = n + 1. Then using Theoem III.2 (iii), we get that Z = {, E}, so M is the unifom matoid U n. Fo (, δ)-locality, let S x = U n fo each x E and δ = d = n + 1. Then S x = + (δ 1) and d(s x ) = δ. Consequently, U n is a matoid with paametes (n,, d,, δ) = (n,, n + 1,, n + 1). 2) The maximal value of d when < : As the fist esult of this section, we pove that n (δ 1) d max (n,,, δ) n ( 1)(δ 1), 23

24 whee the second inequality is Theoem III.3 evisited. We will then use the gaph constuctions given in Theoem IV.2 and Theoem IV.1, in ode to constuct matoids with lage d. In the cases when d max < n ( 1)(δ 1), we will use Theoem III.4 to pove this. Theoem IV.3. Fo any (n,,, δ) satisfying 1 < n (δ 1) and 2 δ < n, thee exists a (n,, d,, δ)-matoid, whee Poof: Let m = n +δ 1 d = n (δ 1)., and let F 1,... F m 1 be disjoint sets with an and size + δ 1. Let F m be disjoint fom all of F 1,... F m 1, with size F m = n (m 1)( + δ 1) and an ρ(f m ) = F m δ + 1. Finally, let M be defined by Z(M) = {F I }, whee F I = i I F i, and ρ(f I ) = min{ i I It is eadily seen that M has minimum distance ρ(f i ), }. d = n + 1 (δ 1) max{ I : ρ(f I ) < } n + 1 (δ 1). In paticula, when δ = 2, this means that the optimal minimum distance is one of n + 1 and n. The emainde of this section aims at deciding which of these two possibilities is the case fo fixed Befoe stating the technical theoem on d max, we need the following qualitative esult. Poposition IV.1. Let M be an (n,, d,, δ)-matoid and let a = and b = δ 1) n. Then the following hold, n if b a, + δ if b > a, Poof: Let n +δ 1 n +δ 1 = + t. Note that n (δ 1) by Poposition III.2. Hence, (δ 1) n This implies that t 0 if b a and t 1 if b > a. = ( + t)( + δ 1) b ( a) = (δ 1) + t( + δ 1) (b a). ( + 24

25 Theoem IV.4. Let (n,,, δ) be integes such that 0 < < n (δ 1), = a and n = ( + δ 1) b. Let d max = d max (n,,, δ) be the lagest d such that thee exists an n +δ 1 (n,, d,, δ)-matoid. Then the following hold. (i) If a b, then d max = n + 1 ( 1 ) (δ 1); (ii) If b > a and b, then d max n + 1 (δ 1) + (b ). (iii) If b > a and a < 1, then d max = n + 1 ( ) 1 (δ 1) if and only if /2 a and ( n 1 + (b a) ), + δ 1 t whee t = a/ ( 1 a ) ; (iv) If b > a 1, 3 and n b + δ 1 stu whee s =, t = 1, u = a 1 s +1 2 (t(u 1) + 2) + y,, x = b b stu stu s, and 0 if stu b, y = x x u min{ x u, 1} if stu b, then d max = n + 1 ( 1 ) (δ 1); (v) If b > a 1, = 2, and n + δ 1 then d max = n + 1 ( 1 ) (δ 1). Poof: The poof is given in the Appendix. b a + 1 if 2a 1, b if 2a > 1, In the poof of Theoem IV.4(iv), we will notice that a simple bound, but in geneal not as good, is n +δ 1 b stu (t(u 1) + 2). Example IV.4. Examples of constuctions of matoids in Theoem IV.4(i), (iii) and (iv) given by the poofs of the theoem ae given in Example IV.1, IV.2 and IV.3 espectively. 25

26 V. APPLICATIONS OF (n,, d,, δ)-matroids TO (n,, d,, δ)-lrcs In this section we will use the pevious esults on (n,, d,, δ)-matoids to get new esults on linea and almost affine (n,, d,, δ)-lrcs. All the poofs of the non-existence of matoids immediately give coesponding bounds fo codes. To veify the othe diection, obtaining codes with pescibed paamete values fom matoids with the same paametes, we will show that the class of matoids given in Theoem IV.1 is a subclass of a class of matoids called gammoids. Gammoids have the popety that they ae epesentable ove any finite field of sufficiently lage size. The main esult in this section is Theoem V.1. Theoem V.1. Let M(F 1,..., F m ; ; ρ) be an (n,, d,, δ)-matoid that we get in Theoem IV.1. Then fo evey lage enough finite field thee is a linea LRC ove the field with paametes (n,, d,, δ). A. Tansvesal matoids and gammoids We stat by giving a shot intoduction to gammoids. Fo moe infomation on this fascinating class of matoids we efe the eade to [38], [43]. A gammoid is associated to a diected gaph G as follows. Definition V.1. Let G = (V, D) be a diected gaph, with S V and T V.The gammoid M(G) is a matoid M(G) on S whee the independent sets of M(G) equals I(M(G)) = {X S : a set of X vetex-disjoint paths fom X to T }. Ou inteest in gammoids in this pape stems fom the following esult. Theoem V.2 ([36]). Evey gammoid ove a finite set E is epesentable ove evey finite field of size geate than o equal to 2 E. Many natual classes of gammoids, can be epesented ove fields of much smalle size than 2 n. Fo example, a unifom matoid Un (12) is a gammoid associated to a complete bipatite gaph with V = S T, S = n, T = and D = S T. Howeve, unifom matoids ae epesented by linea [n,, d = n + 1]-MDS codes, which exist ove F q when q n. B. Constuctions of linea (n,, d,, δ)-lrcs C(F 1,..., F m ; ; ρ) Theoem V.1 follows immediately fom Lemma V.1 and Theoem V.2. The ey element is the constuction of a diected gaph whose associated gammoid is the matoid fom Theoem IV.1. This constuction is detailed in Algoithm 1. 26

27 Algoithm 1 Input: (F 1,..., F m ; ; ρ). Output: G = (V, D, S, T ) 1: S = E, H =, D =, T = [] 2: Label e S with s(e) = {i : e F i }. 3: h is a function H 2 [m] 4: fo all e E do 5: if s(e) 2 then 6: H H {u e } 7: h(u e ) = s(e) 8: fo all i [m] do 9: l i = ρ(f i ) {u H : i h(u)} 10: H H {v i 1,..., vi l i } 11: h(v i 1 ) =... = h(vi l i ) = {i} 12: fo all (e, u) S H do 13: if s(e) h(u) then 14: D D ( e, u) 15: D D H T 16: V = S (H T ) 17: Output (V, D, S, T ) Lemma V.1. Let F 1,..., F m be a collection of subsets of a finite set E whose union is all of E, and wite F I = i I F i. Let ρ : {F i } i [m] Z satisfy (i) 0 ρ(f i ) F i, (ii) F [m] + i [m] (ρ(f i) F i ), (iii) F I F j < ρ(f j ) wheneve j I. Then the gammoid M(G), that we get fom Algoithm 1 is equal to the matoid M(F 1,..., F m ; ; ρ) that we get in TheoemIV.1. Poof: The poof is given in the Appendix. Example V.1. Conside the matoid M G, associated to the stoage system in Figue 1 and the code in Example II.1, and whose lattice of cyclic flats is witten out in Example II.2. By Lemma V.1, this is the gammoid associated to the following gaph, with T = = 6 and S = n = 12. Note that in 27

28 this paticula setting, Line 15 in Algoithm 1 is supefluous, could be eplaced by assigning H = T, since H aleady has only 6 nodes. Indeed, the inclusion of the bipatite gaph (H, T ) coesponds to tuncating the gammoid at an. Fig. 6. A downwad diected gaph suppoting the matoid associated to a (12, 6, 3, 3, 3)-LRC. C. Bounds on the paametes (n,, d,, δ) fo LRCs In this section we will give esults on the paametes (n,, d,, δ) fo linea, and moe geneally almost affine LRCs. The esults ae diect consequences of the coesponding esults fo matoids, thans to the epesentability esults in Theoem V.1 and the matoid invaiance of the paametes (n,, d,, δ), fom Theoem III.1. We will theefoe not give any futhe poofs in this section. Obseve that this means that the same bounds ae valid fo matoids, almost affine codes, and linea codes. Theoem V.3. If C is an almost affine LRCs with the paametes (n,, d,, δ), then ( ) d n (δ 1). Poposition V.1. Let C be an almost affine LRC with paametes (n,, d,, δ). Then (i) δ d, (ii) n (δ 1), (iii) n +δ 1. Theoem V.4. Let C be an almost affine LRC with paametes (n,,, d, δ), and let a = and b = ( + δ 1) n. Then the following hold. n +δ 1 (i) If b > a and a < /2, then d < n + 1 ( 1 ) (δ 1); (ii) If b > a and /2 a 1, then d < n + 1 ( 1 ) (δ 1). 28

29 Theoem V.5. Let (n,,, δ) be integes such that 0 < < n (δ 1), a = and b = ( + δ 1) n. Let d max = d max (n,,, δ) be the lagest d such that thee exists a n +δ 1 linea LRC with paametes (n,, d,, δ). Then the following hold. (i) If a b, then d max = n + 1 ( 1 ) (δ 1); (ii) If b > a, then n + 1 (δ 1) if b 1, d max n + 1 (δ 1) + (b ) if b ; (iii) If b > a, /2 a < 1 and n +δ 1 ( ) 1 + (b a) t, whee t = ( a/ ) 1 a, then ( ) d max = n (δ 1); (iv) If b > a 1, 3 and n +δ 1 b stu (t(u 1) + 2) + y, whee s =, t = 1 b b, u =, x = and then a 1 s +1 2 stu stu s 0 if stu b, y = x x u min{ x u, 1} if stu b, d max = n + 1 (v) If b > a 1, = 2 and n + δ 1 then ( ) 1 (δ 1); b a + 1 if 2a 1, b 1 2 d max = n if 2a > 1, ( ) 1 (δ 1). Just lie in the ema below Theoem IV.4, a simple bound, but in geneal not as good, in Theoem V.5(iv) is n +δ 1 b stu (t(u 1) + 2). It was poven in [42] Coollay 2.3 that linea LRCs with all-symbol locality in the case when and + δ 1 n cannot achieve the Singleton-type bound given in Theoem V.3. This coesponds to the case a = 0 and b > 0 in Theoem V.4. Hence, by Theoem V.5 (ii), we obtain that ( ) d max = n (δ 1) 1, fo linea (n,, d,, δ)-lrcs when and b = + δ 2. 29

30 VI. CONCLUSIONS Recent pogess in coding theoy has poven matoid theoy to be a valuable tool in many diffeent contexts. This tend caies ove to locally epaiable codes. Especially the lattice of cyclic flats is a useful object to study, as its elements coespond to the local epai sets. We have thooughly studied linea and moe geneally almost affine LRCs with all-symbol locality, as well as the connections of these codes to matoid theoy. We deived a genealized Singleton bound fo matoids and nonexistence esults fo cetain classes of (n,, d,, δ)-matoids. These esults can then be diectly tanslated to nonexistence esults fo almost affine LRCs. Futhe, we have given seveal constuctions of matoids with pescibed values of the paametes (n,, d,, δ). Using these matoid constuctions, novel constuctions of linea LRCs ae given, using the epesentability of gammoids. Seveal classes of optimal linea LRCs then aise fom these constuctions. As futue wo, (non)existence esults fo matoids and linea and almost affine LRCs achieving the genealized Singleton bound emain open fo cetain classes of paametes (n,,, δ), when 1 a < b. In addition, the size of the undelying finite field that ou linea (n,, d,, δ)-lrcs can be constucted ove is left fo futue eseach. We expect that the uppe bound 2 n aising fom the elated bound fo all gammoids is loose fo ou class of matoids. We conjectue that all ou matoids fom Section IV-A ae epesentable ove fields of size polynomial in n. REFERENCES [1] T. Westebäc, T. Envall, and C. Hollanti, Almost affine locally epaiable codes and matoid theoy, 2014 IEEE Inf. Theoy Woshop (ITW), [2] A. G. Dimais, P. B. Godfey, Y. Wu, M. J. Wainwight, and K. Ramchandan, Netwo coding fo distibuted stoage systems, IEEE Tans. Inf. Theoy, 56(9), pp , Septembe [3] S. Ghemawat, H. Gobioff, and S. T. Leung, The Google file system, In SOSP03, Poceedings of the nineteenth ACM symposium on Opeating systems pinciples, [4] T. Envall, S. El Rouayheb, C. Hollanti, and H. V. Poo, Capacity and secuity of heteogeneous distibuted stoage systems, IEEE J. Sel. Aeas Comm., 31(12), pp , Dec [5] B. Sasidhaan, and P. V. Kuma, High-ate egeneating codes though layeing, 2013 IEEE Int. Symp. Inf. Theoy (ISIT), pp , [6] C. Tian, V. Aggawal, and V. A. Vaishampayan, Exact-epai egeneating codes via layeed easue coection and bloc designs, 2013 IEEE Int. Symp. Inf. Theoy (ISIT), pp , [7] T. Envall, Codes between MBR and MSR points with exact epai popety, IEEE Tans. Inf. Theoy, 60(11), pp , Nov