Binary Codes with Locality for Multiple Erasures Having Short Block Length

Transcription

1 Binay Codes with Locality fo Multiple Easues Having Shot Bloc Length S. B. Balaji, K. P. Pasanth and P. Vijay Kuma, Fellow, IEEE Depatment of Electical Communication Engineeing, Indian Institute of Science, Bangaloe. Abstact axiv: v3 [cs.it] Feb 016 This pape consides linea, binay codes having locality paamete, that ae capable of ecoveing fom t easues and which additionally, possess shot bloc length. Both paallel (though othogonal paity checs) and sequential ecovey ae consideed hee. In the case of paallel epai, minimum-bloc-length constuctions ae chaacteized wheneve t ( + ) and examples examined. In the case of sequential epai, the esults include (a) extending and chaacteizing minimum-bloc-length constuctions fo t =, (b) poviding impoved bounds on bloc length fo t = 3 as well as a geneal constuction fo t = 3 having shot bloc length, (c) poviding high-ate constuctions fo ( =, t {4, 5, 6, 7}) and (d) poviding shot-bloc-length constuctions fo geneal (, t). Most of the codes constucted hee ae binay codes. Index Tems Distibuted stoage, codes with locality, sequential epai, codes with availability, othogonal-paity codes. I. INTRODUCTION All codes discussed ae linea and ove a finite field F q. Thoughout, [n,, d min ] will denote the bloc length, dimension and minimum distance of the linea code. Two classes of codes with locality ae consideed hee, namely those that offe paallel and sequential ecovey espectively. While much of the discussion holds fo geneal q, most of the codes constucted in the pape ae binay, coesponding to q =. The focus in the pape is on the constuction of codes having shot o minimum bloc length. This is motivated in pat by pactical implementation consideations, and in pat, with a view to getting some insight into what is to be gained by inceasing the bloc length. A. Paallel and Sequential Recovey a) Codes with paallel ecovey: Codes in this class can be defined as the nullspace of an (m n) paity-chec matix H, whee each ow has weight (+1) and each column has weight t, with nt = m(+1). Additionally, if the suppot sets of the ows in H having a non-zeo enty in the ith column ae given espectively by S (i) j, j = 1,, t, then we must have that S (i) j S (i) l = {i}, 1 j l t. Thus each code symbol c i is potected by a collection of t othogonal paity checs (opc) each of weight ( + 1). The paamete is called the locality paamete and we will fomally efe to this class of codes as (, t) pa codes. When the paametes, t ae nown fom the context, we will simply tem the code as a code with paallel ecovey. This is because ecovey fom a set of t easues can be caied out locally and in paallel. b) Codes with sequential ecovey: The equiement of this class of codes is that given any set of s t eased symbols, {x 1,..., x s }, thee is an aangement of these s symbols (say) {x i1,..., x is } such that thee ae s codewods {h 1,..., h s } in the dual of the code, each of weight +1, with i j suppot(h j ) and suppot(h j ) {i j+1,..., i s } =, 1 j s. The paamete is again the locality paamete and we will fomally efe to this class of codes as (, t) seq codes as it can ecove fom any s t easues sequentially and locally, using a set of s t paity checs, each of weight + 1, as mentioned in the definition above. Again when (, t) ae clea fom the context, we will efe to a code in this class as a code with sequential ecovey. P. Vijay Kuma is also an Adjunct Reseach Pofesso at the Univesity of Southen Califonia. This wo is suppoted in pat by the National Science Foundation unde Gant No and in pat by the joint UGC-ISF eseach pogam.

2 B. Bacgound The notion of codes with locality was intoduced in [1], see also [], [3]. The initial focus was on ecovey fom a single easue and constuctions fo codes that accomplish this can be found in [1], [4], [5], [6]. Thee ae seveal appoaches in the liteatue to local ecovey fom multiple easues. The appoach adopted by the authos of [5], [7], is to use a stonge local code with d min > to potect the code symbols against multiple easues. Thee is a second class of codes, temed as t-availability codes in which each code symbol is coveed by t othogonal paity checs, but these ae only equied to have suppot of size ( + 1) as opposed to the stict equiement of = ( + 1) discussed hee. Codes with t-availability can be found discussed in [8], [9], [10], [11], [1], [13], [6], [14], [15], [16]. The sequential appoach of ecoveing fom multiple easues, intoduced in [17], can also be found discussed in [18], [19], [0]. C. Ou Contibutions The focus hee is on codes with paallel o sequential ecovey, that have shot bloc length. c) Paallel Recovey: In the case of paallel ecovey, we deive a simple lowe bound on the bloc length of this class of codes fo given (, t). It is then shown that a necessay and sufficient condition fo the existence of a minimum-bloc length (n min ), paallel-ecovey code is the existence of a balanced incomplete bloc design (BIBD) with paametes govened by {n min,, t}. Some examples ae noted, including cases in which the codes possess in addition, the highest possible ate fo the given {n min,, t}. d) Sequential Recovey: Ou esults hee can be boen down accoding to the value of the paamete t: t = : hee we genealize the optimal constuction by Paash et al [17] to cove a lage paamete set. Each code constucted hee has minimum bloc length fo a given, and fo the case when, being the dimension of the code, a chaacteization of the class of optimal codes with maximum ate which can sequentially ecove fom easues is povided. t = 3: we deive a lowe bound on bloc length fo a given, that fo 1.8 impoves upon an ealie bound by [18] fo binay codes. A geneal constuction of codes is pesented with ate and shot bloc length (O( 1.5 )) that diffes by at most fom the lowe bound on bloc length deived hee, ove a lage paamete ange. = and t {4, 5, 6, 7}: seveal constuction of codes having shot bloc length and high ate ae povided. Geneal t: some geneal constuctions having shot bloc length ae pesented hee. Most codes constucted hee ae binay codes. The esults on paallel ecovey ae pesented in Section II. Results on sequential ecovey appea in Sections III to VI. +3 II. MINIMUM BLOCK-LENGTH CODES WITH PARALLEL RECOVERY Let H be the coesponding (m n) paity chec matix of an (, t) pa paallel-ecovey code C ove the field F q which includes all the othogonal paities of all symbols. Then each column of H has Hamming weight t and each ow has weight ( + 1). The code C may also be viewed as (d v = t, d c = ( + 1))-egula LDPC code. The coesponding bipatite gaph of the code must necessaily have no cycles of length 4. Let A be the {0, 1} matix ove the eals R given by { 1 hij 0, a ij = 0 else. Given the othogonal natue of the paity checs and ou assumption on the paities involving a code symbol c i, the sum of the inne poducts between distinct ows of A must satisfy: ( ) m ( n ) n a i,l a j,l = ( ) a il a jl t = n. j>i j>i l=1 Using the elation nt = m( + 1), we obtain l=1 m (t 1)( + 1) + 1, n ( + 1) ( + 1). (1) t

3 Ou inteest is in the minimum-bloc-length case, whee (1) holds with equality and fo which a necessay condition is that t ( + 1). We set n min = ( + 1) (+1) t and define a code having length n min to be a minimum-length code. When equality holds, it follows that the inne poduct of evey pai of distinct ows of A is exactly equal to 1. Let us define the column suppot sets B j [m] by i B j iff a i,j = 1 o equivalently, h ij 0. It follows then that the sets {B j } n j=1 fom a (b, v, ˆ, ˆ, λ) balanced incomplete bloc design (BIBD) having paametes b = n, v = m, ˆ = ( + 1), ˆ = t, λ = 1. Convesely a BIBD with these paamete values will yield an (, t) pa bloc code C having minimum possible bloc length. The ate R of C clealy satisfies R 1 t (+1). Example 1. Let Q = s, s and let P G(, Q) denote the pojective plane ove F Q. Thee ae Q +Q+1 points and Q + Q + 1 lines in P G(, Q). Each line contains Q + 1 points and thee ae Q + 1 lines though a point. Set n = Q + Q + 1. Let H be the (n n) paity chec matix of a binay code C i.e., a code ove F, given by h ij = 1 the ith point lies on the jth line. Then it is nown that H has an 3 s + 1 ove F, and that C has d min = Q +, thus C is a binay (Q, Q + 1) pa code having paametes [Q + Q + 1, Q + Q 3 s, Q + ]. A plot of the ate of this code vesus the bound by Tamo et al in [10] as a function of the paamete s is shown in Fig. 1. While this code is well-nown in the liteatue on LDPC codes, ou aim is to daw attention to the fact that this code is a (Q, Q + 1) pa code having minimum bloc length. The paametes of a minimum-bloc-length code obtained by a simila constuction involving lines in the affine plane and (, t) = (Q, Q) ae given by [n,, d min ] = [Q + Q, Q + Q 3 s, Q + 1], whee Q = s, s. Conjectue 1 (Hamada-Sacha Conjectue (Conjectue of [1]). Evey Pojective plane of ode p s, p a pime, has p an at least ( ) p+1 s + 1 with equality if and only if its desaguesian. The above conjectue is as yet unpoven, but if tue, would show that the pojective-plane code descibed in Example 1 would have minimum possible bloc length and the maximum possible ate ove the binay field F among all binay (Q, Q + 1) pa codes with n = Q + Q + 1 and Q = s. Example. Fo t = 3, one can obtain a code by maing use of the Steine Tiple System (STS) associated to the point-line incidence matix of (s 1) dimensional pojective space P G(s 1, ) ove F. Once again, the ows of H coespond to points in the pojective space and the columns to lines. It follows that t = 3 and ( + 1) = s = s 1 1. Let m = s 1. It tuns out that this yields a binay ( s 1, 3) pa code having paametes [ m(m 1) 6, m(m 1) 6 m + s, 4]. A plot compaing the ate of this code and the bound by Tamo et al [10] is shown in Fig. 1. Conjectue (Hamada s Conjectue (1.91 in [])). The p- an of any design D with paametes of a geometic design G in PG(n,q) o AG(n,q) (q = p m ) is at least the p-an of G with equality if and only if D is isomophic to G. This conjectue has been shown to hold tue fo the Steine Tiple system appeaing in Example. Thus the code in Example has, as a binay code, the minimum possible bloc length and maximum possible ate among all binay ( m 1 1, 3) pa codes with n = m(m 1) 6 and m = s 1. III. CODES WITH SEQUENTIAL RECOVERY FROM TWO ERASURES In [17], it is shown that fo eithe sequential o paallel ecovey fom t = easues with locality, the ate of the code is uppe bounded by n + ()

4 Fig. 1: Compaing the ates of the pojective plane and Steine-tiple-system-based codes with the bound in [10]. Fo given dimension and locality paamete, this leads to the lowe bound n +. The authos in [17] povide a constuction of optimal codes whee equality holds in (). The constuction is based on Tuan gaphs and holds wheneve, and in addition, = ( + β), fo some β. The constuction given in [18] fo t = sequential easue coection equies. In the pesent pape, we pesent a simple constuction that has minimum possible bloc length fo given {, } and that holds fo a lage set of paametes, as it only equies that +1 fo. and + fo. Constuction 1 (Sequential ecovey fom t = easues). Let = a + b, 0 b 1. Let G be a gaph on a set of m = nodes with a nodes having degee and an additional node having degee b, fo the case when b > 0. Let each edge in the gaph epesent an infomation symbol in the code and each node epesent a paity chec symbol which coesponds to the sum of the infomation symbols coesponding to the edges connected to that node. The code is systematic and is defined by the infomation symbols coesponding to edges and the paity symbols coesponding to the nodes of G. The dimension of this code, i.e., the numbe of infomation symbols, is clealy equal to and bloc length n = +. Since each paity chec symbol epesented by a node is the sum of at most infomation symbols, the coesponding paity chec involves at most + 1 code symbols. Thus the code has locality. It is staightfowad to see that the code can sequentially ecove fom easues. Noting that the gaph G is egula in the case, b = 0, we will efe to the code descibed in Constuction 1 with b = 0 as the Regula Gaph Code. The paamete sets (, ) fo which the gaph G of the fom descibed in Constuction 1 exists can be detemined fom the Edös-Gallai theoem [3] and the paametes sets tuns out to be {(, ) : = m + 1} and {(, ) : = m + }, when b = 0 and b > 0 espectively. A. Uniqueness of Rate Optimal Codes fo Easues In [4], Paash et. al. intoduced the class of (, δ) codes which have MDS codes having d min > as the local codes. We will efe to these codes fo bette claity as (, δ) MDS codes. In this section, we pove that a ate-optimal sequential code- with-locality fo easues must have a specific fom. Paash et. al. [17] deived bound () on the ate of a (, t = ) seq code. Fom the deivation of this ate bound given in [17], it is staightfowad to see that an [n,, d] code with locality and sequential easue coection, which achieves the bound will have a paity chec matix (afte possible pemutation of code symbols) of the fom [I H ], whee I denote the identity matix of ode n and H is a (n ) matix with all columns having a hamming weight of and all ows having a hamming weight of. Theoem 1. A code with locality capable of sequential ecovey fom easues and achieving the ate uppe bound in () must fall (upto coodinate pemutation) into one of the following classes: 1) A egula gaph code (possibly defined ove a lage field with coefficients fom the lage field in place of 1 s in the paity chec matix), ) A (, 3) MDS code in Paash et. al. [4]. 3) A code that is the diect poduct C = C 1 C whee C 1 is a egula gaph code (possibly defined ove a lage field with coefficients fom the lage field in place of 1 s in the paity chec matix) and C is a (, 3) MDS code in [4].

5 Poof: Assume that H is the paity chec matix of an [n,, d] code with locality and sequential epai capability of t =, with ate. As mentioned befoe, we can wite (afte possible pemutation of code symbols) + H = [I H ] (3) whee I denote the identity matix of ode n and H is a (n ) matix with all columns having a hamming weight of and all ows having a hamming weight of. Conside ows of H, R 1 and R. Assume that suppot(r 1 ) and suppot(r ) intesect at columns C 1, C...C s. Since all columns in H has weight exactly, the columns C 1, C...C s will have non zeo enties in R 1 and R only. Let A denote the s sub matix obtained by consideing the ows R 1 and R and the columns C 1, C...C s only. In ode to ecove fom any instance of easues in the symbols coesponding to the columns of A, any two columns of A must be linealy independent. Thus the s sub matix A foms the geneato matix of an MDS code of bloc length s and dimension. This also says that any vecto, obtained by a linea combination of the two ows of A will have a hamming weight at least s 1. Let us conside two exteme cases: Case 1: s = : This case coesponds to having (, 3) MDS locality intoduced by Paash et.al.[4] fo the set of symbols in suppot(r 1 ) suppot(r ) Case : s 1: In this case two paity checs, epesented by R 1 and R have at most one symbol in common. If these ae the only two cases that can occu fo any pai of ows R i and R j i.e., suppot(r i ) suppot(r j ) {0, 1, } then the set of code symbols can be patitioned into two sets, one set of symbols foming a egula gaph code (with possibly highe field coefficients in place of 1 s in the paity chec matix) and the othe set of symbols foming (, 3) MDS code, with no paities acoss these sets i.e., the code will be a diect poduct of egula gaph code (with possibly highe field coefficients in place of 1 s in the paity chec matix) and (, 3) MDS code (afte possible pemutation of code symbols). Now, we will pove that 1 < s < is not possible fo any pai of ows, whee s denote the size of the intesection of suppot, of the pai of ows. Wlog assume that 1 < s < fo the pai of ows R 1 and R i.e., suppot(r 1 ) suppot(r ) = s. Let C i, C j be two columns belonging to the set of s columns whee suppot(r 1 ) and suppot(r ) intesect. Assume that the symbols coesponding to C i and C j ae eased. In ode to sequentially epai these symbols locally, R 1 and R must linealy combine with some of the emaining ows of H to get a vecto v with the following popeties. 1) Hamming weight of v is less than o equal to ( + 1). ) v has a zeo in the coodinate coesponding to C i and a non zeo value in the coodinate coesponding to C j, o vice vesa. Assume that a linea combination of l ows {R 1, R, R 3...R l } esults in v. Let s ij denote suppot(r i ) suppot(r j ). Clealy, s 1 = s. If s ij > 0, we have shown that the s ij sub matix fomed by the ows R i and R j and the columns in suppot(r i ) suppot(r j ) fom a geneato matix of an MDS code of bloc length s ij and dimension and they linealy combine to fom a vecto of hamming weight at least s ij 1. Thus the hamming weight of v is at least l + 1 i<j l,s (s ij>0 ij 1) + f, whee the facto of l comes fom the identity pat of H (i.e., columns 1 to n of H) and f comes fom the single weight columns in the sub matix L fomed

6 by the ows {R 1, R, R 3...R l } and columns n + 1 to n of H. l + (s ij 1) + f + 1 l + l + 1 i<j l,s ij>0 1 i<j l,s ij>0 1 i<j l,s ij>0 ( ) l s ij + f + 1 ( ) l (s ij ) + f i<j l,s ij>0 (s ij ) (4) Also, by counting the non zeo enties in L ow wise and column wise f + s ij = l (5) substituting (5) and (6) in (4) gives: simplifying and assuming l > we get l + l ( ) l l l 1 1 i<j l,s ij>0 1 i<j l,s ij>0 s ij l Hence we get l + 1, when l >. But when l + 1 the coodinates in the identity pat (columns 1 to n ) will add a Hamming weight of + 1 to v, maing the Hamming weight of v geate than + 1 as it must also have a non zeo Cj th o Ci th coodinate. Hence, if 1 < s <, l > is not possible. Now, assume l = i.e., a linea combination of R 1 and R should give v. The Hamming weight of a linea combination of R 1 and R is at least (s 1) + ( + 1 s) (weight s 1 comes fom the coodinates in suppot(r 1 ) suppot(r ), and weight ( + 1 s) comes fom the emaining coodinates in suppot(r 1 ) suppot(r )). We need, (s 1) + ( + 1 s) + 1 which is not possible as 1 < s <. Hence l = is also not possible. Hence putting togethe we have l not possible but fo 1 < s <, but we need to linealy combine l ows to get v. Hence 1 < s < is not possible. s (6) IV. CODES WITH SEQUENTIAL RECOVERY FROM THREE ERASURES We fist pesent an impoved lowe bound on the bloc length of binay codes that can sequentially ecove fom 3 easues fo This is followed by the constuction of a shot bloc length code that genealizes an instance of the Tuan gaph constuction in [17]. A. Bound on Minimum Bloc Length In [18], W. Song et. al. deived the following lowe bound on the bloc length of codes that can sequentially ecove fom thee easues. + n + (7) Constuctions wee also povided of codes meeting the above bound fo. Hee, we pesent a new bound on bloc length. Simulation shows that the new bound is tighte than (7) fo fo We

7 also povide a few spoadic examples whee this bound tuns out to be tight. Fo binay codes, ou bound taes on the fom: n + min s 1 max{f 1 (s 1 ), f (s 1 ), s 1 }, (8) ( 5) + ( 5) whee f 1 (s 1 ) = + 4(6 + s 1 5s 1) (4 4 + s 1 ) + (4 4 + s 1 ) f (s 1 ) = + 4(1 + 3s 1 4s 1 7). Poof: Let H be the paity chec matix of an (, 3) seq code C (possibly having global paities), with m linealy independent ows, bloc length n and dimension. Let B 0 = span(h : weight(h) + 1, h C ) Let {c 1,..., c m } be a basis of B 0 with weight(c i ) + 1. Let c 1 H =.. Let s extend the basis {c 1,..., c m } of B 0 to a basis of C and fom the paity chec matix H of C with this basis of C. Hence [ ] H H =. whee H 1 contains the exta vectos coming fom extending the basis {c 1,..., c m } of B 0 to a basis of C. The numbe of ow vectos in the above matix must be m. Since is the dimension of the code C with paity chec matix H then n = + m + m. Now H is a paity chec matix of a (, 3) seq code with same bloc length n, as its ow span has all the paities of weight + 1 of C defined by H. Now we conside the code defined by the paity chec matix H with m linealy independent ows which is a (, 3) seq code and deive a lowe bound on m as a function of and. Using n = + m + m and the deived lowe bound on m, we get a lowe bound on n. Let s 1, s be the numbe of columns of H with weights 1 and espectively. Then by simple counting of non zeo enties of H ow wise and column wise: c m H 1 s 1 + s + 3(n s 1 s ) m( + 1) Pemute the columns and ows of H matix such that : 3n m( + 1) s 1 s (9) H = [ Ds1 A 0 B whee D s1 is a diagonal matix of ode s 1 with non zeo diagonal enties. Now the s, two-weight columns ae to the ight of D s1. In these s columns, we cannot have a column with non zeo enties in the fist s 1 ows, as this would imply d min <= 3 (whee d min is the minimum distance of the code defined by the paity chec matix H) as the code defined by paity chec matix H is also a (, 3) seq code and hence d min 4. Hence : Let f 1 = numbe of columns of weight with exactly one non zeo enty in the fist s 1 ows. f = numbe of columns of weight with both non zeo enties in the last m s 1 ows. ].

8 s = f 1 + f f 1 s 1 (m s 1 ) f N(m s 1,, 4) whee N(m s 1,, 4) is the maximum numbe of columns in a paity chec matix with m s 1 ows and column weight, of a code with d min >= 4. Resticting to binay codes, it is staightfowad to see that, N(m s 1,, 4) ( m s 1 ). With a little bit of thought this can be tightened to: N(m s 1,, 4) (m s 1 + 3)(m s 1 + 1) + 1 4( ) m s1 Hence, s s 1 (m s 1 ) + s s 1 (m s 1 ) + (m s 1 + 3)(m s 1 + 1) + 1 (11) 4 Hence substituting both the above bounds (10),(11) on s in (9): ( ) m s1 3n m( + 1) s 1 s 1 (m s 1 ) + 3n m( + 1) s 1 s 1 (m s 1 ) + (m s 1 + 3)(m s 1 + 1) + 1 (13) 4 (1) (On using n + m) leads to: which gives: m + m( 5) (6 + s 1 5s 1 ) 0 (10) (1) m ( 5) + ( 5) + 4(6 + s 1 5s 1) = f 1 (s 1 ) ( 5) + ( 5) + 4(6 6) which when added with gives a bette lowe bound on n than(7), ove some paamete ange in. (13) (On using n + m) leads to: which gives: m m + m(4 4 + s 1 ) (1 + 3s 1 4s 1 7) 0 (4 4 + s 1 ) + (4 4 + s 1 ) + 4(1 + 3s 1 4s 1 7) = f (s 1 ) Taing m min s1 max(f 1 (s 1 ), f (s 1 ), s 1 ) and using n + m, we get n + min s1 max(f 1 (s 1 ), f (s 1 ), s 1 ) Simulation shows that this bound is tighte than Song et. al. bound (7) fo fo We povide some examples which achieve the bound (8). 1) When = 4, = 8, t = 3, ou bound (8) gives n 14 wheeas the bound in (7) gives n 13. The binay code associated to the paity-chec matix shown below achieves ou tighte bound and hence epesents a

9 code of minimum possible bloc length fo = 8, t = H = This code is an example of a moe geneal constuction pesented below in the next subsection (see Fig. ). ) n = 8, = 7, = 0, t = 3 : ((7) gives n 7 fo = 0, = 7. Ou bound (8) gives n 8 fo = 0, = 7. Hence the binay code associated with the paity chec matix given below has the least bloc length fo a binay code fo = 0, = 7.) H = ) n = 10, = 3, = 5, t = 3 : ((7) gives n 9 fo = 5, = 3. Ou bound (8) gives n 10 fo = 5, = 3. Hence the binay code associated with the paity chec matix given below has the least bloc length fo a binay code fo = 3, = 5.) H = B. A Hypegaph-Based Constuction fo t = The constuction of a binay code pesented below may be viewed as a genealization of an instance of the Tuan-gaph-based constuction pesented in [17] fo the sequential ecovey of failed nodes. Constuction. Set b = 3β fo some paamete β 1. Let G be a hypegaph on b nodes constucted by fist patitioning the b nodes into thee subsets of nodes, labelled as A 1, A, A 3, each having β nodes. Next, fo evey tiple of nodes n 1, n, n 3, n i A i, a hypeedge is placed that connects these thee nodes. Thus thee ae a total of β 3 hypeedges. Each hypeedge is then associated to a unique message symbol and each of the nodes in the node subsets A i is associated with a paity-chec symbol. The paity-chec symbol associated to a node n i is the sum of all the message symbols associated with the hypedges connected to n i. The code is defined by the set of message symbols assosiated with hypeedges and the paity chec symbols assosiated with the nodes of G. Thus the code has dimension = β 3 and bloc length n = 3β + β 3. It can be shown that this code has minimum distance 4, hence the code C has paametes [n = β 3 + 3β, = β 3, d = 4]. It follows that the code has ate Claim 1. Constuction gives a ( = β, 3) seq code with [n = β 3 + 3β, = β 3, d = 4].. β 3 β 3 +3β = Poof: The poof poceeds by showing that fo evey instance of 3 code symbol easues, thee is at least one paity chec of weight +1 whose suppot contains exactly one of the eased symbols, and hence the coesponding symbol can be ecoveed and subsequently the emaining two symbols can also be ecoveed using local paities. Assume that x numbe of infomation symbols ae eased and 3 x paity symbols ae eased. We conside the cases x = 0, x = 1, x =, x = 3 and pove each case. case 1 : x = 0 (14). +3.

10 Patition 1 P 1 P I 1 I I 3 Patition P 3 P 4 I 4 I 5 Patition 3 P 5 P 6 I 6 I 7 I 8 Fig. : On the left, an example of the hypeedge-based constuction given in Constuction. In the bipatite gaph shown hee, each node on the ight epesents a hypeedge and hence, a distinct message symbol. Each node P i on the left, epesents a paity chec symbol. Thus this code has bloc length n = 14 and = 8. The plot on the ight shows that thee is at most a diffeence of between the bloc length of Constuction and the lowe bound on bloc length given by (8) fo 1 β 000. Each paity symbol is calculated fom = β infomation symbols associated with it. Hence any numbe of easues among the paity symbols can be epaied locally if none of the infomation symbols ae eased. Case : x = 1 Each hype edge is connected to 3 paity nodes. At most two paity nodes ae allowed to fail in this case. So the eased infomation symbol can be ecoveed using the thid paity symbol which is not eased. Case 3 : x = Let S i denote the set of nodes {n (i) 1, n(i), n(i) 3 } which ae connected by the ith hypeedge. It is easily checed that S i S j and theefoe S i S j 4 fo all 1 i j β 3. Assume that i and j ae the hypeedges epesenting the eased infomation symbols I i and I j espectively. Let the set T ij be defined as: T ij = {S i S j }\{S i S j } T ij The suppot of each of the paity checs associated with the paity symbols epesented by the nodes in T ij will contain eithe I i o I j, but not both. Since T ij, thee ae atleast two paity checs whose suppot contains only one of the eased symbols I i and I j. The thid easue in this case (which is a paity symbol) can affect at most one of these paity checs. Theefoe at least one of the infomation symbols can be ecoveed, and subsequently the emaining symbols can also be ecoveed. Case 4 : x = 3 Assume that the infomation symbols I i, I j and I ae eased, and i, j, ae the coesponding hypeedges. Let S i and T ij be as defined above. Conside I i and I j. If S i S j < then T ij 4. i.e. thee ae at least 4 paity checs whose suppot contains only one of the eased symbols I i and I j, and hence can be used to ecove the coesponding symbol. The thid easue in this case can affect at most thee of these paity checs, since the hypeedge coesponding to the infomation symbol I is connected to exactly 3 nodes. Theefoe at least one of the infomation symbol can be ecoveed, and subsequently the emaining symbols can also be ecoveed. Conside the case when S i S j =. Let S i S j = {n 1, n }. Clealy, n 1 and n belong to two diffeent patitions, say A 1 and A. Hypeedges i and j will be connected to two distinct nodes in the thid patition A 3. Exactly one infomation symbol belonging to the suppot of the paity checs associated to the paity symbols epesented by these two nodes in A 3 has been eased. The thid easue in this case can affect at most one paity chec among those paity checs associated with the nodes in A 3 (Since each hypeedge is connected to exactly one node, in one patition). Hence at least one of the infomation symbols I i and I j can be ecoveed using the paity chec associated to the coesponding node in A 3. Theefoe sequential ecovey is possible in this case. Fig. shows an example constuction fo the case when β = and hence with paametes [n = 14, = 8, d min = 4] with locality paamete = 4 and pemits sequential ecovey fom t = 3 easues. The paity-chec matix of this code was pesented ealie in Section IV-A as an example of a code that achieves bound (8) fo t = 3 and appeas in (14). Rema 1. The ate-optimal constuction given in [18] fo 3 easues equies. The hypegaph

11 constuction in Constuction descibed above on the othe hand, has a much smalle value of this atio, namely =. Futhemoe, the diffeence between the bloc length of Constuction and the bound (8) is fo 1 β 000. Thus, Constuction yields high-ate binay codes fo paamete sets outside the ange of the constuctions appeaing in [18]. V. HIGH RATE CONSTRUCTIONS WITH = FOR SEQUENTIAL RECOVERY In [10], Tamo et al. deived the uppe bound 15 n 1 t j=1 (1 + 1 j ) (15) on the ate of codes with availability. Howeve, to the autho s nowledge, pio to this pape, thee wee no geneal constuctions, eithe sequential o paallel, that achieved this bound fo =, t = 4, 5, 6. In this context, the constuctions fo (, t) seq codes pesented below fo = and t = 5, 6, 7 achieve a lage ate than what appeas on the ight side of (15). Fo t = 4, we povide a constuction having ate close to the bound (15). The ate of the bipatite gaph based constuctions by Rawat et.al.(section VI-A of [19]) is 1 +1 = 0.33, fo =. Ou codes achieve impoved ates compaed to 1 +1 fo t = 4, 5, 6 and an equal ate fo t = 7. Howeve, fo t = 7, the constuction pesented hee has a smalle bloc length. The constuctions will be pesented in gaphical fom. In all of the gaphs, each node epesents a code symbol and a paent node stoes the paity of its childen. Thoughout this section we will use the tems nodes and code symbols synonymously and efe to the code symbols using the same labels as the nodes epesenting them. A. Constuction fo t = 4, = The constuction below yields a systematic ( =, t = 4) seq code with dimension. Let = 4l, l > 1. Aange nodes I 1...I, epesenting infomation symbols, as shown in Fig 3. Constuct paity nodes P 1...P as shown in Fig 3 whee P i is the paity of {I i, I i+1 }, fo i = and P is the paity of {I 1, I }. Add a second laye of / paities Q 1...Q (/). Q i is the paity of {P i, P i+ }, i = This code has ate ++(/) = 0.4. Q 1 Q 3 Q (/)-1 P 1 P 3 P -3 P -1 I 1 I I 3 I 4 I 5 I -3 I - I -1 I P P 4 P - P Q Q 4 Fig. 3: The Fou Easue Coecting Code Q / Claim. The constuction defined in section V-A geneates a (, 4) seq code. Poof: Assume that x numbe of infomation symbols ae eased and 4 x paity symbols ae eased. We conside the cases x = 0, x = 1... x = 4 and pove each case. Hee, we conside only epesentative wost case scenaios. Remaining cases can be analyzed similaly. case 1 : x = 0 All paities ae deived fom infomation symbols. Hence each eased paity symbol can be ecoveed locally using existing paity/infomation symbols. Case : x = 1 WLOG assume that node I 1 was eased. If eithe P 1 o P is not eased, then I 1 can be ecoveed. Theefoe assume that both P 1 and P have been eased. P 1 can be ecoveed using Q 1 & P 1+. Similaly, P can be ecoveed

12 using Q and P. But we can ease only one moe paity symbol. Hence, eithe P 1 o P can be ecoveed and subsequently all the emaining 3 nodes can be ecoveed. Case : x = WLOG assume that node I 1 was eased. It has epai sets {P 1, I } and {P, I } which can be used to ecove I 1. Case.1 : Assume that both P 1 and P ae eased. Both of them can be ecoveed using the second laye of paity symbols (Q i s). I 1 can be ecoveed since at most one of the symbols I and I is allowed to fail. Case. : Assume that P 1 and Q 1 ae eased. I 1 can be ecoveed using the set {P, I }. Hence assume that I is eased. But I can be ecoveed using P 1 and I 1. Subsequently all the emaining eased nodes can be ecoveed. Case.3 : Assume that P 1 ae eased. I 1 can be ecoveed using the set {P, I }. Since easue of P is aleady handled in Case.1, assume that I is eased. But I can be ecoveed using P 1 and I 1. Hence assume that P 1 is eased. Now P 1,P 1 can be ecoveed fom highe laye paities Q is since > 4. Subsequently all the emaining eased nodes can be ecoveed. Case 3 : x = 3 Case 3.1 : Assume that P 1 was eased. But P 1 can be ecoveed using Q 1 and P 1+ easoning as in case 4 below. Case 3. : Assume that Q 1 was eased.. The thee infomation symbols can be ecoveed due to simila But Q 1 can be ecoveed using P 1 and P 1+. The thee infomation symbols can be ecoveed due to simila easoning as in case 4 below. Case 4 : x = 4 Assume that I 1 is eased. Thee ae sets {P 1, I } and {P, I } which can be used to ecove I 1. Assume that both nodes I and I have been eased so that both these sets cannot be used to ecove I 1. Fo ecoveing I the set of code symbols {P, I 3 } can be used and fo ecoveing I the set {P 1, I 1 } can be used. Since we ae allowed to ease at most one moe infomation symbol, eithe I o I can be ecoveed. Rest of the 3 infomation symbols can be ecoveed due to simila easoning. The following constuctions use a simila appoach to geneate codes fo t = 5, 6 and 7, =. The poof of sequential ecovey fo t = 5, 6, 7, = is simila to the t = 4, = case given above. B. Constuction fo t = 5, = A (, 5) seq code can be constucted fom the (, 4) seq code constucted above, by adding additional paity symbols R 1... R (/8). R i is the paity of {Q i 1, Q i 1+ }. Hee we need the additional equiement that 8. 4 R 1 R R /8 Q 1 Q 3 Q (/)-1 P 1 P 3 P -3 P -1 I 1 I I 3 I 4 I 5 I -3 I - I -1 I P P 4 P - P Q Q 4 Q / Fig. 4: The Five Easue Code

13 C. Constuction fo t = 6, = Rate of the code = + + (/) + (/8) = A (, 6) seq code can be constucted fom the (, 5) seq code constucted as descibed above, by adding additional paities S 1... S (/8) and T 1...T /8. S i is the paity of Q i and Q i+(/4). T i is the paity of P (4i ) and P 4i. Hee we need the additional equiement of 8 and 16. R 1 R R /8 Q 1 Q 3 Q /-1 P 1 P 3 P 5 P 7 P -3 P -1 I 1 I I 3 I 4 I 5 I 6 I 7 I 8 I -3 I - I -1 I P P 4 P 6 P - P T 1 T Q Q 4 Q 6 Q /- Q / S 1 S S /8 Fig. 5: The Six Easue Code : Note that the new paities T coves only half of P paities in the bottom (i.e. P, P 4... P ). Hence they ae /8 in numbe. D. Constuction fo t = 7, = Rate of the code = + + (/) + (/8) + (/8) + (/8) = A (, 7) seq code can be constucted fom the (, 6) seq code constucted as descibed above, by adding additional paities U 1... U (/16) and V 1... V (/16). U i is the paity of T i and T i+. V i is the paity of S i and S 16 i+ Hee 16 we need the additional equiement of 16. Rate of the code is : ++(/)+(/8)+(/8)+(/8)+(/16)+(/16) = The above ate is same as the ate 1 +1 = achieved by constuction in [19] based on bipatite gaphs (Section VI-A of [19]) but ou constuction equies a minimum bloc length of 48 (fo = 16) wheeas the t t 4 constuction mentioned in [19] equies bloc length of ( + 1) + 4 = 3 4 = 81 [6].

14 I 1 I I 3 I 4 I 5 I 6 I 7 I 8 P P 4 P 6 T 1 T U 1 U Q Q 4 S 1 S V 1 V Fig. 6: The seven Easue Code (patial diagam) t Bound (15) Rate of codes in Section V TABLE I: Compaison of code ate of constuctions fo = vesus bound in [10]. A. Constuction using Othogonal Latin Squaes VI. CONSTRUCTIONS FOR GENERAL t Let {L 1 L t } be a set of (t ) paiwise othogonal Latin squaes of size ( ). Necessaily, (t ) ( 1). Let L t 1, L t be an additional two ( ) matices with (i, j) th enties given by L t 1,(i,j) = i, L t,(i,j) = j. While L t 1, L t ae not Latin squaes, any two squaes in the enlaged set L = {L 1,, L t+1 } continue to be paiwise othogonal. Next, let A be the (t ) matix constucted fom L as follows. The columns of A ae indexed by a pai (a, b) of coodinates, 1 a, b. Then { 1 L i,(a,b) = i (mod ) + 1 A i,(a,b) = 0 else. Let H be the paity-chec matix given by H = I A I (16) whee 1 denotes the (1 ) vecto of all ones, I denotes the identity matix of ode, epeated t times along the diagonal as shown. Claim 3. The code with paity chec matix as defined in (16) is a (, t + 1) seq code, when t is even. Poof: Thoughout this poof fo the matix H, we identify the Suppot(R) fo some ow R of H with indices of columns in which R has non zeo enties and the code symbols coesponding to the indices of columns in which R has non zeo enties synonymously. We can divide the ows of A into t sets, S i, 1 i t, S i = {R (i 1)+1,, R i } (whee R i indicates the i th ow of A) and the suppot sets of the ows in S i i.e., Suppot(R (i 1)+j ), 1 j depends on L i and fom a patition of columns of A. Each ow of A contain exactly, non zeo enties. Clealy, the suppot of distinct ows R i, R j S fo some, ae disjoint. Since the elements in L ae mutually othogonal, the suppot of two ows R i, R j, i j such that R i S 1, R j S, 1 will have Suppot(R i ) Suppot(R j ) 1. These two obsevations pove that each symbol coesponding to the columns of A is potected by t othogonal paities in A of weight each. Hence, A denote the paity chec matix

15 of a ( 1, t) pa code. This also implies that each symbol coesponding to the columns of A is potected by t othogonal paities in H of weight + 1. Let the set of fist symbols of the code (coesponding to the columns of A) be denoted by B 1 and the set of emaining t + 1 code symbols be denoted by B. Assume that thee ae s eased symbols T 1 = {x 1,..., x s } in B 1 and t s + 1 eased symbols T = {x s+1,..., x t+1 } in B. Conside the case when s = 1. Let T 1 = {x 1 }. x 1 is coveed by t othogonal paities coesponding to ows R a i, 1 i t whee 1 a i t (R j denotes the jth ow of H) with suppots D ai, 1 i t. If D ai T = fo some i, then x 1 can be ecoveed. Hence, assume that wlog D ai T = {x 1+i }, 1 i t. Since R ai S 1 fo some i and D ai T = {x 1+i } fo that i, x 1+i can be ecoveed using the last ow paity of H. This ecoveed symbol can be used to ecove x 1, and subsequently all symbols can be ecoveed. Now conside the case when 1 < s < t + 1. Let x 1 T 1 be potected by t othogonal paities coesponding to ows R a i, 1 i t whee 1 a i t with suppots D ai, 1 i t. If D ai (T 1 {x 1 } T ) = fo some i, then x 1 can be ecoveed. Hence, assume that wlog D ai (T 1 {x 1 } T ) = {x 1+i }, 1 i t. Since s > 1, x T 1. x is also potected by t othogonal paities coesponding to ows R b i, 1 i t whee 1 b i t with suppots D bi, 1 i t. If D bi (T 1 {x } T ) = fo some i, then x can be ecoveed. Hence, assume that wlog D bi (T 1 {x }) = 1, D bi T = 0, 1 i s 1 and D bi T = 1, D bi (T 1 {x }) = 0, s i t. Hence x s+j1 D bj fo some s j t and 1 j 1 t s + 1. Since x s+j1 Suppot(R ) fo exactly one in 1 t and x s+j1 D as+j1 1,x s+j1 D bj, this implies D as+j1 = D 1 bj as x s+j1 B. But this is not possible as x 1 D as+j1 and x 1 1 / D bj as s j t. Hence the symbol x can be ecoveed. Similaly all eased symbols can be ecoveed. Now, conside the case when s = t + 1. Let the t othogonal paities potecting x i be coesponding to Rj i (ows of H), 1 i t + 1, 1 j t. If one of the symbol can be ecoveed then the est of the symbols can be ecoveed by thei t othogonal paities. Hence fo none of the symbols to be ecoveable, the only possibility is that Suppot(Rj i ) T 1 {x i } = 1, 1 i t + 1, 1 j t. Hence Suppot(R) T 1 {0, } fo any ow R of H. If we conside ows R 1,.., R (fist ows) of H with suppots D 1,.., D, D i D j = and [ ] i=1 D i. Now since i D i T 1 = t + 1 which is odd but i D i T 1 is even as D i T 1 {0, }, this leads to a contadiction. Hence one of the eased symbols can be ecoveed and subsequently the emaining eased symbols can be ecoveed. This code has bloc length + t + 1 and ate +t+1. In compaison, the ate of the constuction given in [5] is +t+1 with a bloc length of ( ) +t+1 t+1. Thus ou constuction achieves both a bette ate and a smalle bloc length, made possible by adopting a sequential appoach to ecovey as opposed to using othogonal paities. The constuction in [19] based on bipatite gaphs (Section VI-A of [19]) has a highe ate of 1 +1 but equies bloc t t 4 length of size ( + 1) + 4 [6]. B. Constuction using poduct of sequential codes Let C 1 be a [n 1, 1 ] (, t 1 ) seq code and C, be a [n, ] (, t ) seq code. Let C be the code obtained by taing the poduct of C 1 and C which will be a [n 1 n, 1 ] code with locality. The sequential easue coecting capability of C is given by: Claim 4. The code C constucted as descibed above can ecove fom (t 1 + 1)(t + 1) 1 easues sequentially. t 1 easues x x... x x x x x Fig. 7: Poduct code : X indicates node failues.

16 Poof: Conside a codewod in C as shown in Figue 7 consisting of a (, t 1 ) seq code in the ows and (, t ) seq code in the columns. Assume that thee ae t 1 easues in a ow of the codewod. If t 1 t 1, then the nodes can be ecoveed using C 1. Hence assume that t 1 > t 1. C can ecove one of these symbols if the numbe of easues in the column coesponding to that symbol is less than o equal to t. Hence, the code C fails to epai the eased symbols only if the numbe of easues is geate than o equal to (t 1 + 1)(t + 1), which completes the poof. 1) Suboptimality of the poduct of δ numbe of [3,] single paity chec codes fo δ 9 ([0]): Tae the poduct of the code given in Section V-D fo = 16 with the poduct of thee [7,3] Simplex codes i.e., Let C = Poduct of (Code given in Section V-D fo = 16, [7,3] Simplex code, [7,3] Simplex code, [7,3] Simplex code) The ate of the esulting code C will be : with locality = and t = 9 1 and bloc length n = If we tae the poduct of nine [3, ] single paity chec codes, the esulting code will have n = 3 9 = 19683, ate = and =, t = 9 1. Thus C achieves a bette ate with a smalle bloc length than the code given by poduct of nine [3,] single paity chec codes. Beyond this fo t = δ 1 fo δ > 9, we can simply tae the code given by poduct of C and the code obtained fom poduct of δ 9 numbe of [3, ] single paity chec codes, and achieve a bette ate with a smalle bloc length than the the poduct of δ numbe of [3, ] single paity chec codes with the same locality OF = and the same easue coecting capability of t = δ 1. Hence poduct code is not optimal fo sequential ecovey, fo =,t = δ 1 and δ 9. ) An Example Constuction: Let H t denote the paity chec matix of a (, t)-sequential easue coecting code with paametes [n,, d ]. Constuct a new matix H as shown below. H t H t 0 0 H =....., (17) 0 0 H t 0 I n I n I n I n H t is epeated times along the diagonal. I n denote the n n identity matix. Claim 5. A code with paity chec matix H as defined by (17) has all symbol locality and can coect t + 1 easues using sequential appoach. Poof: Poof follows by obseving that the given paity chec matix is the paity chec matix of the code obtained by taing the poduct of single paity chec code and the [n,, d ] code. The esulting code will have the following paametes Bloc Length = ( + 1)n Dimension = Minimum Distance = t + REFERENCES [1] P. Gopalan, C. Huang, H. Simitci, and S. Yehanin, On the Locality of Codewod Symbols, IEEE Tans. Inf. Theoy, vol. 58, no. 11, pp , Nov. 01. [] D. Papailiopoulos and A. Dimais, Locally epaiable codes, in Infomation Theoy Poceedings (ISIT), 01 IEEE Intenational Symposium on, July 01, pp [3] F. Oggie and A. Datta, Self-epaiing homomophic codes fo distibuted stoage systems, in INFOCOM, 011 Poceedings IEEE, Apil 011, pp [4] C. Huang, M. Chen, and J. Li, Pyamid codes: Flexible schemes to tade space fo access efficiency in eliable data stoage systems, in Netwo Computing and Applications, 007. NCA 007. Sixth IEEE Intenational Symposium on, July 007, pp [5] G. Kamath, N. Paash, V. Lalitha, and P. Kuma, Codes with local egeneation, in Infomation Theoy and Applications Woshop (ITA), 013, Feb 013, pp [6] I. Tamo and A. Bag, A family of optimal locally ecoveable codes, IEEE Tans. Inf. Theoy, vol. 60, no. 8, pp , 014. [7] W. Song, S. H. Dau, C. Yuen, and T. Li, Optimal locally epaiable linea codes, Selected Aeas in Communications, IEEE Jounal on, vol. 3, no. 5, pp , May 014.

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