Long MDS Codes for Optimal Repair Bandwidth

Transcription

1 Long MDS Codes for Optmal Repar Bandwdth Zhyng Wang, Itzhak Tamo, and Jehoshua Bruck Electrcal Engneerng Department, Calforna Insttute of Technology, Pasadena, CA 91125, USA Electrcal and Computer Engneerng, Ben-Guron Unversty of the Negev, Beer Sheva 84105, Israel {zhyng, tamo, Abstract MDS codes are erasure-correctng codes that can correct the maxmum number of erasures gven the number of redundancy or party symbols If an MDS code has r partes and no more than r erasures occur, then by transmttng all the remanng data n the code one can recover the orgnal nformaton However, t was shown that n order to recover a sngle symbol erasure, only a fracton of 1/r of the nformaton needs to be transmtted Ths fracton s called the repar bandwdth (fracton) Explct code constructons were gven n prevous works If we vew each symbol n the code as a vector or a column, then the code forms a 2D array and such codes are especally wdely used n storage systems In ths paper, we ask the followng queston: gven the length of the column l, can we construct hgh-rate MDS array codes wth optmal repar bandwdth of 1/r, whose code length s as long as possble? In ths paper, we gve code constructons such that the code length s (r + 1) log r l I INTRODUCTION MDS (maxmum dstance separable) codes are optmal error-correctng codes n the sense that they have the largest mnmum dstance gven the number of party symbols If each symbol s a vector or a column, we call such a code an MDS array code (eg [2], [7], [11], [22], [23]) In (dstrbuted) storage systems, each column s usually stored n a dfferent dsk, and MDS array codes are wdely used to protect data aganst erasures due to ther error correcton ablty and low computatonal complexty In ths paper, we call each symbol a column or a node, and the column length, or the vector sze of a symbol, s denoted by l If an MDS code has r partes, then t can correct up to r erasures of entre columns In ths paper, we not only would lke to recover any e erasures, e r, but also care about the effcency n recovery: what s the fracton of the remanng data transmtted n order to correct e erasures? We call ths fracton the repar bandwdth (fracton) For example, f e = r erasures happen, t s obvous that we have to transmt all of the remanng nformaton, therefore, the fracton s 1 For e = 1 erasure t was shown n [8] (whch also formulated the repar problem) that ths fracton s actually lowered bounded by 1/r If e r symbols are erased and we repar them exactly as they were, ths fracton s lower bounded by e/r [17] If ths bound s acheved for some code, we say t has optmal repar Snce the repar of nformaton s much more crucal than redundancy, and we study manly hgh-rate codes, we wll focus on the optmal repar of nformaton or systematc nodes Moreover, snce sngle erasure s the most common scenaro n practce, we assume e = 1 For example, n Fgure 1, we show an MDS code wth 4 systematc nodes, r = 2 party nodes, and column length l = 2 One can check that ths code can correct any two erasures, therefore t s an MDS code In order to repar any systematc node, only 1/r = 1/2 fracton of the remanng nformaton s transmtted Thus ths code has optmal repar In [12] [14], [20], [21] codes achevng the repar bandwdth lower bound were studed where the number of systematc nodes s less than the number of party nodes (low code rate) For arbtrary code rate, [6], [15] proved that the lower bound s asymptotcally achevable when the column length l goes to nfnty And [3] [5], [9], [10], [16], [17], [19] studed codes wth more systematc nodes than party nodes (hgh code rate) and fnte l, and acheved the lower bound of the repar bandwdth If we are nterested n the code length, e, the number of systematc nodes gven l, low-rate codes have a lnear code length l + 1 [13], [14]; on the other hand, hghrate constructons are relatvely short For example, suppose that we have 2 party nodes, then the number of systematc nodes s only log l n all of the constructons, except for [5] t s 2 log l In [18] t s shown that an upper bound for the code length s k 1 + l( l l/2 ), but the tghtness of ths bound s not known It s obvous that there s a bg gap between ths upper bound and the constructed codes The man contrbuton of ths paper s to construct codes wth 2 party nodes and 3 log l systematc nodes The code uses a fnte feld of sze log l Moreover, we wll gve a general constructon of hgh-rate codes wth (r + 1) log r l systematc nodes for arbtrary number of partes r It turns out that ths constructon s a combnaton of the code n [5] and also [3], [10], [16] The rest of the paper s organzed as follows: n Secton II we wll formally ntroduce the repar bandwdth and the code length problem In Secton III codes wth 2 party nodes are constructed, and we show that the code length s 3 log l Generalzed code constructons for arbtrary number of partes are gven n Secton IV and fnally we conclude n Secton V II PROBLEM SETTINGS An (n, k, l) MDS array code s an (n k)-erasurecorrectng code such that each symbol s a column of length l The number of systematc symbols s k and the number of party symbols s r = n k We call each symbol a column or a node, and k the code length We assume that the code s systematc, hence the frst k nodes of the code

2 N1 N2 N3 N4 P1 P2 a b c d a + b + c + d 2a + w + 2b + 3c + d w x y z w + x + y + z 3w + b + 3x + 2y + z Fgure 1 (n=6,k=4,l=2) MDS code over fnte feld F 4 generated by prmtve polynomal x 2 + x + 1 Here 2 s a prmtve element of the feld The frst 4 nodes are systematc and the last 2 are partes To repar N1 transmt the frst row from every remanng node To repar N2 transmt the second row To repar N3 transmt the sum of both rows And to repar N4 transmt the sum of the frst row and 2 tmes the second row from nodes N1, N2, N3, P1, and the sum of the frst row and 3 tmes the second row from node P2 are nformaton or systematc nodes, and the last r nodes are party or redundancy nodes Suppose the columns of the code are C 1, C 2,, C n, each beng a column vector n F l, for some fnte feld F We assume that for party node k +, nformaton node j, the codng matrx s A,j of sze l l, [r], j [k] And the party columns are computed as C k+ = k A,j C j, j=1 for all [r] For example, n Fgure 1, the codng matrces are A 1,j = I for all j [k] and A 2,j, j = 1, 2, 3, 4 are ( ) ( ) ( ) ( ) ,,, Here the fnte feld s F 4 generated by x 2 + x + 1 In our constructons, we requre that A 1,j = I for all j [k] Hence the frst party s the row sum of the nformaton array Even though ths assumpton s not necessarly true for an arbtrary lnear MDS array code, t can be shown that any lnear code can be equvalently transformed nto one wth such codng matrces [18] Suppose a code has optmal repar for any systematc node, [k], meanng only a fracton of 1/r data s transmtted n order to repar t When a systematc node s erased, we are gong to use sze l/r l matrces S,j, j =, j [n], to repar the node: From a survvng node j, we are gong to compute and transmt S,j C j, whch s only 1/r of the nformaton n ths node Notatons: In order to smplfy the notatons, we wrte S,j and S,k+t A t,j both as matrces of sze l/r l and the subspaces of ther row spans Optmal repar of a systematc node s equvalent to the followng subspace property: There exst matrces S,j, j =, j [n], all wth sze l/r l, such that for all j =, j [k], t [r], S,j = S,k+t A t,j, (1) where the equalty s defned on the row spans nstead of the matrces And r S,k+t A t, = F l (2) t=1 Here the sum of two subspaces A, B of F l s defned as A + B = {a + b : a A, b B} Obvously, the dmenson of each subspace S,k+t A t, s no more than l/r, and the sum of r such subspaces has dmenson no more than l Ths means these subspaces ntersect only on the zero vector Therefore, the sum s actually the drect sum of vector spaces Moreover, we know that each S,k+t has full rank l/r We clam that (1) (2) are necessary and suffcent condtons for optmal repar The sketch of the proof s as follows: suppose the code has optmal repar bandwdth, then we need to transmt l/r elements from each survvng column Suppose we transmt S,j C j from a systematc node j =, j [k], and S,k+t C k+t = k z=1 S,k+tA t,z C z from a party node k + t [k + 1, k + r] Our goal s to recover C and cancel out all C j, j =, j [k] In order to cancel out C j, (1) must be satsfed In order to solve C, all equatons related to C must have full rank l, so (2) s satsfed One the other hand, f (1) (2) are satsfed, one can transmt S,j C j from each node j, j =, j [n] and optmally repar the node Smlar nterference algnment technque was frst ntroduced n [6], [15] for the repar problem Also, [13] was the frst to formally prove smlar condtons It s shown n [18] that we can further smplfy our repar strategy of node and assume S,j = S, for all j =, j [n] by equvalent transformaton of the codng matrces (probably wth an excepton of the strategy of one node) Then the subspace property becomes for any j =, j [k], t [r], S = S A t,j (3) Agan the equalty means equalty of row spans And the sum of subspaces satsfes r S A t, = F l (4) t=1 Notce that f (3) s satsfed, we can say that S s an nvarant subspace of A t,j (multpled on the left) for all party nodes k + t and all nformaton nodes j = If A t,j s dagonalzable and has l lnearly ndependent left egenvectors, an nvarant subspace has a set of bass whch are all egenvectors of A t,j As a result, our goal s to fnd matrces A t,j and ther nvarant subspaces And by usng suffcently large fnte feld and varyng the egenvalues of the codng matrces, we are able to ensure that the codes are MDS Therefore, we wll frst focus on fndng egenvectors of the codng matrces and then dscuss about the egenvalues For example, n Fgure 1, the matrces S, = 1, 2, 3 are (1, 0), (0, 1), (1, 1) One can check that the subspace property (3)(4) s satsfed for [3] For nstance, snce S 3 = (1, 1) s an egenvector for A t,j, t = 1, 2, j = 1, 2, 4, we have S 3 = S 3 A t,j And t s easy to check that S 3 S 3 A 2,3 = span(1, 1) span(3, 2) = F 2 For the node N4, the matrces S 4,j s are not equal In fact S 4,j = (1, 2) for j = 1, 2, 3, 5 and S 4,6 = (1, 3)

3 III CODE CONSTRUCTIONS WITH 2 PARITIES In ths secton, we are gong to construct codes wth column length l = 2 m, k = 3m systematc nodes, and r = 2 party nodes Here m s some nteger As we showed n the prevous secton, we can assume the codng matrces are ( ) I I, (5) A 1 A k where A 1, = I and A 2, = A correspond to party 1 and 2 respectvely Now we only need to fnd codng matrces A s, and subspaces S s For now we only care about egenvectors of A, not ts egenvalues because egenvectors determne the repar bandwdth Later we wll show that usng a large enough fnte feld, we can choose the egenvalues such that the code s ndeed MDS In the followng constructon, for any [k], A has two dfferent egenvalues λ,0, λ,1, each correspondng to l/2 = 2 m 1 egenvectors Denote these egenvectors as for egenvalue λ,0, and V,0 = V,1 = v,1 v,2 v,l/2 v,l/2+1 v,l/2+2 v,l for egenvalue λ,1 Therefore, A can be computed as ( ) ( ) 1 V,0 λ,0 I ( ) l A = 2 2 l V,0 V,1 λ,1 I l2 2 l V,1 By abuse of notatons, we also use V,0, V,1 to represent the egenspace correspondng to λ,0, λ,1, respectvely Namely, V,0 = span{v,1,, v,l/2 } and V,1 = span{v,l/2+1,, v,l } When a systematc node s erased, [k], we are gong to use S to rebuld t The subspace property becomes S = S A j, j =, j [k], (6) S + S A = F l (7) In the followng constructon, e a, a [0, l 1], are some bass of F l, for example, one can thnk of them as the standard bass The subscrpt a s represented by ts bnary expanson, a = (a 1, a 2,, a m ) For example, f l = 16, m = 4, a = 5, then e 5 = e (0,1,0,1) and a 1 = a 3 = 0, a 2 = a 4 = 1 In order to construct the code, we frst defne 3 sets of vectors for [m]: P,0 = {e a : a = 0}, P,1 = {e a : a = 1}, Q = {e a + e b : a + b = 1, a j = b j, j = } For example, f m = 2, = 1, then P 1,0 = {e (0,0), e (0,1) } = {e 0, e 1 }, P 1,1 = {e (1,0), e (1,1) } = {e 2, e 3 }, and Q 1 = {e (0,0) + e (1,0), e (0,1) + e (1,1) } = {e 0 + e 2, e 1 + e 3 } Notaton: The subscrpt for sets P,u, Q and a (the -th dgt of vector a) s wrtten modulo m For example, f [tm + 1, (t + 1)m] for some nteger t, then P,u := P tm,u Constructon 1 The (n = 3m + 2, k = 3m, l = 2 m ) code has codng matrces A, [k], each wth two dstnct egenvalues, and egenvectors V,0, V,1 When node s erased, we are gong to use S to rebuld We construct the code as follows: 1) For [m], V,0 = span(q ), V,1 = span(p,1 ), S = span(p,0 ) 2) For [m + 1, 2m], V,0 = span(p,0 ), V,1 = span(q ), S = span(p,1 ) 3) For [2m + 1, 3m], V,0 = span(p,0 ), V,1 = span(p,1 ), S = span(q ) Example 1 Deletng the node N4, Fgure 1 s a code usng Constructon 1 and l = 2 Another example of l = 4 s shown n Fgure 2 One can check (6) holds For nstance, S 1 = span{e 0, e 1 } = span{e 0 + e 1, e 1 } s an nvarant subspace of A 2 So S 1 = S 1 A 2 If the two egenvalues of A are dstnct, t s easy to show that S S A = F 4, [6] The above example shows that for m = 1, 2, the constructed code has optmal repar It s true n general, as the followng theorem suggests Theorem 2 Constructon 1 s a code wth optmal repar bandwdth 1/2 for rebuldng any systematc node Proof: By symmetry of the frst two cases n the constructon, we are only gong to show that the rebuldng of node, [m] [2m + 1, 3m] s optmal Namely, the subspace property (6)(7) s satsfed Recall that S A j = S s equvalent to S beng an nvarant subspace of A j Case 1: [m] When j [tm + 1, (t + 1)m], j tm =, t {0, 1}, defne B = {e a : a j = 1 t, a = 0} {e a + e b : a j + b j = 1, a = b = 0, a z = b z, z =, j} Then t s easy to see that S = span(p,0 ) = span(b) Moreover, each vector n set B s an egenvector of A j, therefore S s an nvarant subspace of A j When j m =, S = V j,0 = span(p,0 ), so S s an egenspace of A j When j [2m + 1, 3m], we can see that every vector n P,0 s a vector n V j,0 = span(p j,0 ) or n V j,1 = span(p j,1 ), hence t s an egenvector of A j When j =, consder a vector e a P,0, then a = 0 And e a = (e a + e b ) e b where b = 1, b j = a j for all j = Here both e a + e b and e b are egenvectors of A e a A = (e a + e b )A e b A = λ,0 (e a + e b ) λ,1 e b = (λ,0 λ,1 )e b + λ,0 e a Because λ,0 = λ,1, we get span{e a A, e a } = span(e a, e b ) Hence S A + S = span{e a, e b : a = 0, b = 1, a j = b j, j = } = F l

4 N1 N2 N3 N4 N5 N6 1st egenspace e 0 + e 2 e 0 + e 1 e 0 e 0 e 0 e 0 of A e 1 + e 3 e 2 + e 3 e 1 e 2 e 1 e 2 2nd egenspace e 2 e 1 e 0 + e 2 e 0 + e 1 e 2 e 1 of A e 3 e 3 e 1 + e 3 e 2 + e 3 e 3 e 3 e 0 e 0 e 2 e 1 e 0 + e 2 e 0 + e 1 S e 1 e 2 e 3 e 3 e 1 + e 3 e 2 + e 3 Fgure 2 (n=8,k=6,l=4) code The frst party node s assumed to be the row sum, and the second party s computed usng codng matrces A In order to rebuld node, S s multpled to each survvng node The frst 2m = 4 nodes have optmal access, and the last m = 2 nodes have optmal update Case 2: [2m + 1, 3m] When j = m or j = 2m, S = span(q ) s an egenspace of A j When j [tm + 1, (t + 1)m], and j = tm for t {0, 1}, defne D = {e a + e b : a j = b j = 1 t, a + b = 1, a z = b z, z =, j} {e a + e b + e c + e d : a j = b j = 0, c j = d j = 1, a + b = 1, c + d = 1, a z = b z = c z = d z, z =, j} We can see that S = span(q ) = span(d) and every vector n D s an egenvector of A j When j [2m + 1, 3m], j = We can see that Q = {e a + e b : a j = b j = 0, a + b = 1, a z = b z, z =, j} {e a + e b : a j = b j = 1, a + b = 1, a z = b z, z =, j} Apparently, every vector n Q s a sum of two vectors n P j,0 or two vectors n P j,1 So S = span(q ) s an nvarant subspace of A j When j =, consder any e a + e b Q, where a = 1, b = 0, a z = b z, z = We have (e a + e b )A = λ,1 e a + λ,0 e b Because λ,0 = λ,1, we get span{(e a + e b )A, e a + e b } = span{e a, e b } Thus S A + S = span{e a, e b : a = 1, b = 0, a z = b z, z = } = F l It should be noted that f we shorten the code and keep only the frst 2m systematc nodes n the code, then t s actually equvalent to the code n [5] The reparng of the frst 2m nodes does not requre computaton wthn each remanng node, snce only standard bases are multpled to the survvng columns (eg Fgure 2) We call such repar optmal access It s shown n [18] that f a code has optmal access, then the code has no more than 2m nodes On the other hand, the shortened code wth the last m systematc nodes n the above constructon s equvalent to that of [3], [10], [16] Snce the codng matrces A, [2m + 1, 3m] are all dagonal, every nformaton entry s ncluded n only r + 1 entres n the code We say such a code has optmal update In [18] t s proven that an optmal-update code wth dagonal codng matrces has no more than m nodes Therefore, our code s a combnaton of the longest optmal-access code and the longest optmalupdate code, whch provdes tradeoff among access, update, and the code length The shortenng technque was also used n [13] n order to get optmal-repar code wth dfferent code rates In addton, f we try to extend an optmal-access code C wth length 2m to a code D wth length k, so that C s a shortened code of D, then the followng theorem shows that k = 3m s largest code length Therefore, our constructon s longest n the sense of extendng C Theorem 3 Any extended code of an optmal-access code of length 2m wll have no more than 3m systematc nodes Proof: Let C be an optmal-access code of length 2m Let D be an extended code of C By equvalently transformng the codng matrces (see [18]), we can always assume the codng matrces of the partes n D are ( ) I I I I A 1 A 2m A 2m+1 A k Here the frst 2m column blocks corresponds to the codng matrces of C Frst consder the code C, that s, the frst 2m nodes If C has optmal access, then S s the span of l/2 standard bass, for [2m] Snce there are 2m systematc nodes, on average each e z appears 2m 2 l 1 l = m tmes, for z [0, l 1] We clam that each e z appears exactly m tmes Otherwse, there exsts one e z that appears n {S : I}, for some I > m, I [2m] So I S 1 However, by [18] we know when I > m, I S = 0 So every e z, z [0, l 1], must appear n m of the S s, say e z S, J, J = m, J [2m] Agan by [18] when J = m, we have J S = 1, so J S = e z So these m subspaces ntersect only on e z Now consder the extended code D Snce every S, J, s an nvarant subspace of A j, j [2m + 1, k] by the subspace property, we know ther ntersecton, e z s also an nvarant subspace of A j In other words, e z s an egenvector of A j Ths result s true for all z [0, l 1] Hence, we know the standard bass are all the egenvectors of A j, j [2m + 1, k] Equvalently, A j are all dagonal So the last k 2m nodes n D are optmal update By [18], there are only m nodes that are all optmal update So k 3m Next let us dscuss about the fnte feld sze of the code In order to make the code MDS, t s equvalent that we should be able to recover from any two column erasures In other words, any 1 1 or 2 2 submatrces of the matrx (5) should be nvertble Therefore, all egenvalues λ,s should be nonzero, [k], s {0, 1} Moreover, the followng matrx should be nvertble for all = j: [ ] I I A A j Or equvalently, A A j should be nvertble

5 Let us frst look at an example Suppose m = 2, = 1, j = 2 (see Fgure 2), then A 1 A 2 s λ 1,0 λ 2,0 λ 2,1 λ 2,0 λ 1,0 λ 1,1 0 0 λ 1,0 λ 2,1 0 λ 1,0 λ 1,1 0 0 λ 1,1 λ 2,0 λ 2,1 λ 2, λ 1,1 λ 2,1 (8) We can smply compute the determnant by expandng along the frst column and the last row The remanng 2 2 submatrx n the mddle s dagonal: [ ] λ1,0 λ 2,1 0 (9) 0 λ 1,1 λ 2,0 Hence, the determnant det(a 1 A 2 ) s (λ 1,0 λ 2,0 )(λ 1,0 λ 2,1 )(λ 1,1 λ 2,0 )(λ 1,1 λ 2,1 ) For another example, let m = 2, = 1, j = 3, then A 1 A 3 s λ 1,0 λ 3,0 0 λ 1,0 λ 1,1 0 0 λ 1,0 λ 3,0 0 λ 1,0 λ 1,1 λ 3,0 λ 3,1 0 λ 1,1 λ 3,1 0 0 λ 3,0 λ 3,1 0 λ 1,1 λ 3,1 (10) Snce we can permutate rows and columns of a matrx and not change ts rank, the above matrx can be changed nto: λ 1,0 λ 3,0 λ 1,0 λ 1,1 0 0 λ 3,0 λ 3,1 λ 1,1 λ 3, λ 1,0 λ 3,0 λ 1,0 λ 1,1 0 0 λ 3,0 λ 3,1 λ 1,1 λ 3,1 And ts determnant s det(a 1 A 3 ) = (λ 1,0 λ 3,1 ) 2 (λ 3,0 λ 1,1 ) 2 (11) Now let us dscuss n general the fnte feld sze of the code Constructon 2 Let the elements of the code be over F q, wth q 2m + 1 Let c be a prmtve element n F q and wrte < >:= mod m Assgn the egenvalues of the codng matrces to be { c λ,s = <>+sm, [2m] c <>+(1 s)m (12), [2m + 1, 3m] If we have an extra systematc column wth A 3m+1 = I (see column N4 n Fgure 1), we can use a feld of sze 2m + 2 and smply modfy the above constructon by { c λ,s = <>+sm+1, [2m] c <>+(1 s)m+1, [2m + 1, 3m] For example, when m = 1, the coeffcents n Fgure 1 are assgned usng the above formula, where the feld sze s 4 and c = 2 For another example, f m = 2, we can use fnte feld F 5 and c = 2, then assgn the egenvalues to be (λ 1,0,, λ 6,0 ) = (1, 2, 1, 2, 4, 3), (λ 1,1,, λ 6,1 ) = (4, 3, 4, 3, 1, 2) Theorem 4 The above constructon guarantees that the constructed code s MDS and has optmal repar bandwdth The fnte feld sze s q 2m + 1 Proof: We clam that f we check any two ndces = j [3m], then the followng condtons are necessary and suffcent for A A j to be nvertble Assume r, s {0, 1} 1) λ,s = λ j,r, for any = j mod m 2) λ,s = λ j,1 s, for [m], j = + m 3) λ,s = λ j,s, for [2m], j [2m + 1, 3m], = j mod m If we have an extra systematc column wth A 3m+1 = I, then A I s nvertble ff 4) λ,s = 1 By the proof of Theorem 2 we already know that optmal repar bandwdth s equvalent to 5) λ,0 = λ,1 It can be easly checked that the above condtons are satsfed by Constructon 2 Here we only prove condton 1 for, j [m] and condton 2 The rest cases all follow smlar deas Wthout loss of generalty we can assume {e } s standard bass, because the bass wll not change the value of det(a A j ) When, j [m], V,0 = Q, V,1 = P,1, and V j,0 = Q j, V j,1 = P,1 So V,1, V j,1 share the same egenvectors B = {e a : a = a j = 1} If we vew each element n B as an nteger n [0, 2 m 1] (each vector n B s the bnary representaton of an nteger), we can say A, A j both have only one nonzero element n each row n B On the other hand, columns of V 1, V 1 j correspond to the rght egenvectors of A, A j, respectvely And t s easy to show that they share the rght egenvectors C = {e T a : a = a j = 0}, where the superscrpt T means transpose Hence, A, A j both have only one nonzero element n each column n C To compute the determnant of A A j, we can expand along rows B and columns C The remanng submatrx wll be dagonal snce we already elmnated all the non-dagonal elements Then t s easy to verfy condton 1 See (8)(9) for an example When [m], j = + m, V,0 = Q, V,1 = P,1 and V j,0 = P,0, V j,1 = Q Therefore both A, A j have nonzero elements at the dagonal locatons Also A has nonzero elements at row P,0 and column P,1 Smlarly A j has nonzero elements at row P,1 and column P,0 Let a = (0,, 0, 1, 0,, 0) be a bnary vector of length m and the only 1 s at locaton And let us vew e 0, e a as the correspondng ntegers 0, 2 m Then we can see that rows {e 0, e a } and columns {e 0, e a } have only four nonzero elements We can permutate the rows/columns of a matrx and not change ts rank Therefore move these two rows/columns to rows/columns 0, 1, and we get a block dagonal matrx Followng the same procedure, we wll get block dagonal matrx, where each block s of sze 2 2 And the determnant s smple to compute See (10)(11) for an example We can see that the feld sze q s about 2/3 of the number of systematc nodes and s not a constant Also the code has

6 parameters (n = 3m + 2, k = 3m, l = 2 m ) On the other hand, the (n = m + 3, k = m + 1, l = 2 m ) code n [17] has constant feld of sze q = 3 So the proposed code has longer k but longer (actual) column length l log q as well Nonetheless, t may be possble to alter the structure of A s a bt (for example, do not requre A to be dagonalzable) and obtan a constant feld sze And ths wll be one of our future work drectons IV CODES WITH ARBITRARY NUMBER OF PARITIES In ths secton, we wll gve constructons of codes wth arbtrary number of party nodes Our code wll have l = r m rows, k = (r + 1)m systematc nodes, and r party nodes, for any r 2, m 1 Suppose A s, s the codng matrx for party node k + s and nformaton node From Secton II, we assume A 1, = I for all In our constructon, we are gong to add the followng assumptons Every A s, has r dstnct egenvalues, each correspondng to l/r = r m 1 lnearly ndependent egenvectors, for s [2, r] Moreover, gven an nformaton node [k], all matrces A s,, s [2, r], share the same egenspaces V,0, V,1,, V, If these egenspaces correspond to egenvalues λ,0, λ,1,, λ, for A 2,, then we assume they correspond to egenvalues λ s 1,0, λs 1,1,, λs 1, for A s, By abuse of notatons, V,u represents both the egenspace and the l/r l matrx contanng l/r ndependent egenvectors Under these assumptons, t s easy to see that f we wrte A s, as V,0 V, 1 λ s 1,0 I λ s 1, I V,0 V,, where the dentty matrces are of sze r l r l, then A s, = A2, s 1, for all s [r] Hence, we are gong to wrte A = A 2,, thus A s, = A s 1, and our constructon wll only focus on the matrx A As a result, the subspace property becomes S = S A j, j =, j [k] (13) S + S A + S A S A = F l (14) Note that such choce of egenvalues s not the unque way to construct the matrces, but t guarantees that the code has optmal repar bandwdth Also, when the fnte feld sze s large enough, we can fnd approprate values of λ,u s such that the code s MDS At last, snce each V,u has dmenson l/r and corresponds to l/r ndependent egenvectors, we know that any vector n the subspace V,u s an egenvector of A Let {e 0, e 1,, e r m 1} be the standard bass of F l And we are gong to use the r-ary expanson to represent the ndex of a base An ndex a [0, r m 1] s wrtten as a = (a 1, a 2,, a m ), where a s ts -th dgt For example, when r = 3, m = 4, we have e 5 = e (0,0,1,2) Defne for [k], u [0, r 1] the followng sets of vectors: P,u = {e a : a = u}, Q = { e a : a j [0, r 1], j = } a =0 P,0 P,1 P,2 Q e 0 e 3 e 6 e 0 + e 3 + e 6 1 e 1 e 4 e 7 e 1 + e 4 + e 7 e 2 e 5 e 8 e 2 + e 5 + e 8 e 0 e 1 e 2 e 0 + e 1 + e 2 2 e 3 e 4 e 5 e 3 + e 4 + e 5 e 6 e 7 e 8 e 6 + e 7 + e 8 Fgure 3 Sets of vectors used to construct a code wth r = 3 partes and column length l = 3 2 = 9 So P,u s the set of bases whose ndex s u n the -th dgt The sum n Q s over all e a such that the j-th dgt of a s some fxed value for all j =, and the -th dgt vares n [0, r 1] In other words, a vector n Q s the summaton of the correspondng bases n P,u, u For example, when r = 3, m = 2, P 1,0 = {e (0,0), e (0,1), e (0,2) } = {e 0, e 1, e 2 }, P 1,1 = {e 3, e 4, e 5 }, P 1,2 = {e 6, e 7, e 8 }, and Q 1 = {e 0 + e 3 + e 6, e 1 + e 4 + e 7, e 2 + e 5 + e 8 } Notatons: If a = (a 1, a 2,, a m ) s an r-ary vector, denote by a (u) = (a 1,, a 1, u, a +1,, a m ) the vector that s the same as a except dgt, u [0, r 1] In the followng, all of the subscrpt for sets P,u, Q and for dgt a are computed modulo m For example, f [tm + 1, (t + 1)m] for some nteger t, then Q := Q tm Constructon 3 The (n = (r + 1)m + r, k = (r + 1)m, l = r m ) code s constructed as follows For nformaton node [tm + 1, (t + 1)m], t [0, r 1], the u-th egenspace (u [0, r 1]) of codng matrx A and the rebuldng subspace S are defned as V,u = span(p,u ), u = t, V,t = span(q ), S = span(p,t ) For nformaton node [rm + 1, (r + 1)m], the egenspaces and rebuldng subspaces are V,u = span(p,u ), u [0, r 1] S = span(q ) Example 5 Fgure 3 llustrated the subspaces P,u, Q for r = 3 partes and column length l = 9 Fgure 4 s a code constructed from these subspaces and has 8 systematc nodes One can see that f a node s erased, one can transmt only a subspace of dmenson 3 to rebuld, whch corresponds to only 1/3 repar bandwdth fracton The three codng matrces for systematc node are I, A, A 2, for [8] The followng theorem shows that the code ndeed has optmal repar bandwdth 1/r Theorem 6 Constructon 3 has optmal repar bandwdth 1/r when rebuldng one systematc node Proof: By symmetry of the constructon, we are only gong to show that the subspace property (13)(14) s satsfed for [1, m] [rm + 1, (r + 1)m] Also S A j = S mples that S has a bass that are all egenvectors of A j

7 V,0 Q 1 Q 2 P 1,0 P 2,0 P 1,0 P 2,0 P 1,0 P 2,0 V,1 P 1,1 P 2,1 Q 1 Q 2 P 1,1 P 2,1 P 1,1 P 2,1 V,2 P 1,2 P 2,2 P 1,2 P 2,2 Q 1 Q 2 P 1,2 P 2,2 S P 1,0 P 2,0 P 1,1 P 2,1 P 1,2 P 2,2 Q 1 Q 2 Fgure 4 An (n = 11, k = 8, l = 9) code Sets P,u and Q are lsted n Fgure 3 V,u s the u-th egenspace of the codng matrx A S s the subspace used to rebuld systematc node Case 1: [1, m] Before we begn to explore the dfferent cases, let us defne the followng sets of vectors B u = {e a : a = 0, a j = u}, u [0, r 1], C t = { e a : a = 0, a z [0, r 1], z =, j} a j =0 In the defnton of C t, the sum s over all e a such that the -th dgt of a s 0, the the z-th dgt s some fxed value, z =, j, and the j-th dgt vares n [0, r 1] Then one can see that B u P j,u, C t Q j j [tm + 1, (t + 1)m], for some t [0, r 1] and j tm = Then the egenspaces of A j are V j,u = span(p j,u ), u = t, and V j,t = span(q j ) Then t s clear that S = span(p,0 ) = span({b u : u = t} C t ) Also every vector of B u, u = t and C t s an egenvector of A j j [rm + 1, (r + 1)m], j rm = The egenspaces of A j are V j,u = span(p j,u ), u [0, r 1] And S = span(p,0 ) = span{b u : u [0, r 1]} and every vector n B u, u s an egenvector of A j j tm =, t [1, r] Then the frst egenspace of A j s V j,0 = span(p,0 ) = S j = In ths case we want to check (14) n the subspace property Suppose the dstnct egenvalues of A are λ 0, λ 1,, λ Then the egenvalues for A s wll be λ0 s, λs 1,, λs, for s [0, r 1] Notce that S = span(p,0 ) = span{e a (0) : a Zr m } and e a (0) As = ( = λ s 0 u=0 u=0 = λ s 0 e a (0) + e a (u) e a (1) e a ())A e a (u) λ s 1 e a (1) λ s e a () u=1 (λ s 0 λs u)e a (u) Wrtng the equatons for all s [0, r 1] n a matrx, we get wth M = e a (0) e a (0) A e a (0)A 2 e a (0) A = M e a (0) e a(1) e a (), λ 0 λ 0 λ 1 λ 0 λ λ 2 0 λ 2 0 λ2 1 λ 2 0 λ2 λ 0 λ 0 λ 1 λ 0 λ After a sequence of elementary column operatons, M becomes the followng Vandermonde matrx λ 0 λ 1 λ M = λ 2 0 λ 2 1 λ 2 λ 0 λ 1 λ Snce λ s are dstnct, we know M and hence M s non-sngular Therefore, span{e a (0), e a (0) A,, e a (0) A } = span{e a (0), e a (1),, e a ()} Snce S contans e a (0) for all r-ary vector a, we know S + S A + + S A = F l Case 2: [rm, (r + 1)m] Agan, we frst defne some sets of vectors to help wth our arguments B u = { e a : a j = u, a z [0, r 1], z =, j} a =0 C t = { a =0 a j =0 e a : a z [0, r 1], z =, j} Here the sum n B u has fxed values of a j = u and a z, z =, j, and the -th dgt vares n [0, r 1] The sum n C t has fxed values of a z, z =, j, and the -th and j-th dgt both vary n [0, r 1] Then one can check that B u span(p j,u ), C t span(q j ) j [tm + 1, (t + 1)m], t [0, r 1], and j tm = rm The egenspaces of A j are span(p j,u ), u = t and span(q j ) And S = span(q ) = span({b u : u = t} C u) We can see that every vector n B u, u = t and C t s an egenvector of A j j [rm + 1, (r + 1)m], j = The egenspaces of A j are P j,u, u [0, r 1] And S = span(q ) = span{b u : u [0, r 1]} We can see that every vector of B u s an egenvector of A j j tm = rm, t [0, r 1] Then the t-th egenspace of A j s span(q ), whch s equal to S

8 j = Take u=0 e a (u) S for arbtrary a, then e a (u) As = λ s ue a (u) u=0 u=0 Wrtten n a matrx form, we have e a (0) e a (0) A e a (0) A2 e a (0)A λ 0 λ 1 λ = λ 2 0 λ 2 1 λ 2 λ0 λ 1 λ e a (0) e a(1) e a () So smlar to Case 1, we know S + S A + S A spans the entre space F l Agan, ths constructon can be shortened to an optmalaccess code of length rm [5] and an optmal-update code of length m [3], [10], [16] The fnte feld sze of ths code can be bounded by the followng theorem In the followng, we do not assume that the egenvalue of A s, s the s-th power of A 2,, and A 1, s not necessarly dentty Hence, we only assume that A 1,,, A r, share the same egenspaces for all Theorem 7 A fnte feld of sze k r m suffces for the code to be MDS and optmal repar bandwdth Here k = (r + 1)m Proof: Let {λ (s),j } be the j-th egenvalue of A s,, [k], j [0, r 1], s [r] In order to show that the code s MDS, we need to check f all x x submatrces of the followng matrx are nvertble, for all x [1, r] A 1,1 A 1,2 A 1,k A 2,1 A 2,2 A 2,k A r,1 A r,2 A r,k Note that each A s, can be wrtten as V 1 Λ s, V for some dagonal matrx Λ s,, where the rows of V are egenvectors and the dagonal of Λ s, are egenvalues Snce A 1,,, A r, share the same egenvectors V, we can multply V 1 on the rght of the -th block column, and not change the rank of the above matrx: V1 1 Λ 1,1 V2 1 Λ 1,2 V 1 k Λ 1,k V1 1 Λ 2,1 V2 1 Λ 2,2 V 1 k Λ 2,k M = V1 1 Λ r,1 V2 1 Λ r,2 V 1 Λ r,k k Here all λ (s),j are unknowns n the fnte feld F q We are gong to show that f we wrte the determnants of each x x submatrx as a polynomal, and take the product of all these polynomals, then t s an nonzero polynomal Moreover, by Combnatoral Nullstellensatz [1] we can fnd assgnments of the unknowns over a large enough fnte feld, such that ths polynomal s not zero Then we are guaranteed to have all the x x submatrces nvertble In [1] t s proved that f the degree of a polynomal f(x 1,, x s ) s deg( f) = s =1 t, and the coeffcent of s =1 xt s nonzero, then a fnte feld of sze max {t } s suffcent for an assgnment c 1,, c s such that f(c 1,, c s ) = 0 By the symmetry of the λ (s),j, we consder only the degree of λ := λ (1) 1,0 We wll fnd ts maxmum degree n the polynomal of determnants Ths unknown varable only appears n the matrx Λ 1,1 = λ (1) 1,0 I λ (1) 1, I, where I s the dentty matrx of sze r m 1 r m 1 Let B = V1 1 Λ 1,1 Then we know that λ appears only n the frst r m 1 columns of B For the determnant of any x x submatrx of M, only the ones contanng B needs to be consdered, because we are only nterested n the degree of λ Therefore, there are ( k 1 x 1 )( x 1 ) submatrces of sze x x that has λ n ts determnant, and ts degree s r m 1 for each submatrx So the total degree of λ s r m 1 r ( )( ) k 1 r 1 x 1 x 1 x=1 Moreover, we know from the proof of Theorem 6 that optmal repar bandwdth s acheved for the frst systematc node ff the followng matrx s nvertble λ (1) 1,0 λ (1) 1, λ (r) 1,0 λ (r) 1, Hence, we need to multply ts determnant to our polynomal The total degree of λ s 1 + r m 1 r ( )( ) k 1 r 1 x 1 x 1 x=1 ( )( ) m 1 k 1 r 1 = 1 + r x=0 x x ( ) m 1 r 1 < 1 + r (k 1) x x x=0 = 1 + r m 1 k Hence the proof s completed We can see that n the above theorem, for hgh-rate codes the feld sze s expostonal n the number of systematc nodes

9 But we beleve that there s stll a large space to mprove ths bound V CONCLUSIONS In ths paper, we presented a famly of codes wth parameters (n = (r + 1)m + r, k = (r + 1)m, l = r m ) and they are so far the longest hgh-rate MDS code wth optmal repar The codes were constructed usng egenspaces of the codng matrces, such that they satsfy the subspace property Ths property gves more nsghts on the structure of the codes, and smplfes the proof of optmal repar If we requre that the code rate approaches 1, e, r beng a constant and m goes to nfnty, then the column length l s exponental n the code length k However, f we requre the code rate to be roughly a constant fracton, e, m beng a constant and r goes to nfnty, then l s polynomal n k Therefore, dependng on the applcaton, we can see a tradeoff between the code rate and the code length It s stll an open problem what s the longest optmal-repar code one can buld gven the column length l Also, the bound of the fnte feld sze used for the codes may not be tght enough Unlke the constructons n ths paper, the feld sze may be reduced when we assume that the codng matrces do not have egenvalues or egenvectors (are not dagonalzable) These are our future work drectons [13] N B Shah, K V Rashm, P V Kumar, and K Ramchandran, Interference algnment n regeneratng codes for dstrbuted storage: necessty and code constructons, IEEE Trans on Informaton Theory, vol 56, no 4, pp , 2012 [14] C Suh and K Ramchandran, Exact-Repar MDS Code Constructon Usng Interference Algnment, IEEE Trans on Informaton Theory, vol 57, no 3, pp , 2011 [15] C Suh and K Ramchandran, On the exstence of optmal exact-repar MDS codes for dstrbuted storage, Tech Rep arxv: , 2010 [16] I Tamo, Z Wang, and J Bruck, MDS array codes wth optmal rebuldng, n ISIT, 2011 [17] I Tamo, Z Wang, and J Bruck, Zgzag codes: MDS array codes wth optmal rebuldng, Tech Rep arxv: , 2011 [18] I Tamo, Z Wang, and J Bruck, Access vs bandwdth n codes for storage, submtted to ISIT, 2012 Avalable at http : //paradsecaltechedu/etrhtml [19] Z Wang, I Tamo, and J Bruck, On codes for optmal rebuldng access, n Allerton Conference on Control, Computng, and Communcaton, Urbana-Champagn, IL, 2011 [20] Y Wu and A Dmaks, Reducng repar traffc for erasure codng-based storage va nterference algnment, n ISIT, 2009 [21] Y Wu, R Dmaks, and K Ramchandran, Determnstc regeneratng codes for dstrbuted storage, n Allerton Conference on Control, Computng, and Communcaton, Urbana-Champagn, IL, 2007 [22] L Xu, V Bohossan, J Bruck, and D Wagner, Low-densty MDS codes and factors of complete graphs, IEEE Trans on Informaton Theory, vol 45, no 6, pp , Sep 1999 [23] L Xu and J Bruck, X-code: MDS array codes wth optmal encodng, IEEE Trans on Informaton Theory, vol 45, no 1, pp , 1999 REFERENCES [1] N Alon, Combnatoral nullstellensatz, Combnatorcs Probablty and Computng, vol 8, no 1-2, pp 7 29, Jan 1999 [2] M Blaum, J Brady, J Bruck, and J Menon, EVENODD: an effcent scheme for toleratng double dsk falures n RAID archtectures, IEEE Trans on Computers, vol 44, no 2, pp , Feb 1995 [3] V R Cadambe, C Huang, and J L, Permutaton code: optmal exactrepar of a sngle faled node n MDS code based dstrbuted storage systems, n ISIT, 2011 [4] V R Cadambe, C Huang, S A Jafar, and J L, Optmal repar of MDS codes n dstrbuted storage va subspace nterference algnment, Tech Rep arxv: , 2011 [5] V R Cadambe, C Huang, J L, and S Mehrotra, Polynomal length MDS codes wth optmal repar n dstrbuted storage systems, n Proceedngs of 45th Aslomar Conference on Sgnals Systems and Computng, Nov 2011 [6] V R Cadambe, S A Jafar, and H Malek, Mnmum repar bandwdth for exact regeneraton n dstrbuted storage, n Wreless Network Codng Conference (WNC), Jun 2010 [7] P Corbett, B Englsh, A Goel, T Grcanac, S Kleman, J Leong, and S Sankar, Row-dagonal party for double dsk falure correcton, n Proc of the 3rd USENIX Symposum on Fle and Storage Technologes (FAST 04), 2004 [8] A Dmaks, P Godfrey, Y Wu, M Wanwrght, and K Ramchandran, Network codng for dstrbuted storage systems, IEEE Trans on Informaton Theory, vol 56, no 9, pp , 2010 [9] D S Papalopoulos, and AG Dmaks, Dstrbuted storage codes through Hadamard desgns, n ISIT, 2011 [10] D S Papalopoulos, AG Dmaks, and V R Cadambe, Repar Optmal Erasure Codes through Hadamard Desgns, n Allerton Conference on Control, Computng, and Communcaton, Urbana-Champagn, IL, 2011 [11] J S Plank, The RAID-6 lberaton codes, The Internatonal Journal of Hgh Performance Computng and Applcatons, vol 23, pp , Aug 2009 [12] K V Rashm, N B Shah, P V Kumar, and K Ramchandran, Explct Constructon of Optmal Exact Regeneratng Codes for Dstrbuted Storage, n Allerton Conference on Control, Computng, and Communcaton, Urbana-Champagn, IL, 2009