A High-Rate MSR Code With Polynomial Sub-Packetization Level

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1 A High-Rae MSR Code Wih Polynomial Sub-Packeizaion Level Birenjih Sasidharan, Gaurav Kumar Agarwal, and P Vijay Kumar Deparmen of Elecrical Communicaion Engineering, Indian Insiue of Science, Bangalore biren, agarwal, vijay@eceiiscernein arxiv: v1 [csit] 27 Jan 2015 Absrac We presen a high-rae (n,k,d = n 1)-MSR code wih a sub-packeizaion level ha is polynomial in he dimension k of he code While polynomial sub-packeizaion level was achieved earlier for vecor MDS codes ha repair sysemaic nodes opimally, no such MSR code consrucion is known In he low-rae regime (i e, raes less han one-half), MSR code consrucions wih a linear sub-packeizaion level are available Bu in he high-rae regime (i e, raes greaer han one-half), he known MSR code consrucions required a sub-packeizaion level ha is exponenial in k In he presen paper, we consruc an MSR code for d = n 1 wih a fixed rae R = 1, 2, achieveing a sub-packeizaion level α = O(k ) The code allows help-by-ransfer repair, i e, no compuaions are needed a he helper nodes during repair of a failed node Index Terms Disribued sorage, regeneraing codes, subpackeizaion, msr I INTRODUCTION In a disribued sorage sysem, he daa file comprising of B daa symbols drawn from a finie field F q, is encoded using an error-correcing code of block lengh n and he code symbols are sored in n nodes of he sorage nework A naive sraegy aimed a achieving resilience agains node failures is o sore muliple replicas of he same daa Given he massive amoun of daa being sored, sophisicaed codes such as Reed- Solomon (RS) codes wih low sorage overhead are being employed in pracice However, he amoun of daa download required o repair a single node-failure is quie large for he RS codes The framework of regeneraing codes was inroduced in [1] o address his problem In an (n, k, d)-regeneraing code, a file comprised of B symbols from a finie field F q is encoded ino a se of nα coded symbols and hen sored across n nodes in he nework wih each node soring α coded symbols The parameer α is ermed as he sub-packeizaion level of he code A daa collecor can download he daa by connecing o any k nodes In he even of node failure, node repair is accomplished by having he replacemen node connec o any d nodes and download β α symbols from each node wih α dβ < B The quaniy dβ is ermed he repair bandwidh Here one makes a disincion beween funcional and exac repair By funcional repair (FR), i is mean ha a failed node will be replaced by a new node This research is suppored in par by he Naional Science Foundaion under Gran No and in par by he join UGC-ISF Research Gran No 1676/14 Birenjih Sasidharan would like o acknowledge he suppor of TCS Research Scholar Programme Fellowship such ha he resuling nework coninues o saisfy he daa collecion and node-repair properies defining a regeneraing code An alernaive o funcion repair is exac repair (ER) under which one demands ha he replacemen node sore precisely he same conen as he failed node A cu-se bound based on nework-coding conceps, ells us ha given a code parameer se (n,k,d), he maximum possible size of a daa file under FR, is upper bounded [1] by k B min{α,(d l+1)β} (1) l=1 The above bound is igh since he exisence of codes achieving his bound has been esablished using nework-coding argumens relaed o mulicasing For fixed values of (n,k,d,b), he bound in (1) characerizes a radeoff beween α and β, referred o as he Sorage-Repair Bandwidh radeoff The wo exremal poins in he radeoff are respecively, he minimumsorage regeneraing (MSR) and minimum bandwidh regeneraing (MBR) poins which correspond o he poins a which he sorage and repair bandwidh are respecively minimized A MBR poin, we have ( ) k α = dβ, B = kα β, (2) 2 and a MSR poin, we have α = (d k +1)β, B = kα (3) I is proved ha MSR and MBR poins are achievable by ER codes as well The focus of he curren paper is on ER MSR codes and for convenience we simply refer o hem as MSR codes A MSR Codes The MSR codes can be considered as codes over a vecor alphabef q α wih dimensionk Since hey olerae any(n k) node-erasures, and hey have a file size ofb = kα, MSR codes are Maximum-Disance-Separable (MDS) codes over he vecor alphabe F q α The combinaion of hese wo properies is herefore called he MDS propery of MSR codes On he oher hand, MSR codes in addiion o being vecor MDS codes can repair a failed node wih he leas possible repair bandwidh The consrucion of MSR codes is a well-sudied problem in lieraure In [2], a framework o consruc MSR codes is provided for d 2k 2 In [3], high-rae MSR codes wih

2 parameers (n,k = n 2,d = n 1) are consruced using Hadamard designs In [4], high-rae MSR codes, known as zigzag codes, are consruced for d = n 1; here efficien node-repair is guaraneed only in he case of sysemaic nodes This was subsequenly exended o include he repair of pariy nodes as well in [5] A consrucion for MSR codes wih d = n 1 2k 1 using echniques of inerference alignmen is presened in [6] and [7] In [8], auhors showed he exisence of MSR codes for any value of (n,k,d) B Our Approach On Sub-packeizaion and Conribuions A parameer of ineres for MSR codes is he amoun of sub-packeizaion (α) required for a given value of (n, k, d) The MSR consrucions known as zigzag codes ha allow arbirarily high raes required a sub-packeizaion level ha is exponenial in k Laer in [9], a vecor MDS codes ha repair sysemaic nodes was consruced achieving α = r k r+1 where r := n k Recenly in [10], anoher vecor MDS code ha repairs sysemaic nodes opimally was proposed saisfying an addiional propery known as access-opimaliy The consrucion required α = r k r In [11], auhors derived a lower bound on he sub-packeizaion in erms of k, and r as given below: 2log 2 α(log ( r α+1)+1 k r 1) Earlier in [12], auhors consruced a vecor MDS code wih rae R = 2 3, requiring an α ha is polynomial in k They could also achieve polynomial α for any fixed rae in he regime 2 3 R 1 However, hese codes were also limied by he fac ha opimal repair was feasible for sysemaic codes alone Quie similar o he approach in [12], we also resric our focus o he family of MSR codes wih a fixed rae R = 1, 2 I is worhwhile o remark a his poin ha he family of Produc-Marix MSR codes [2] wih rae resriced by R 1 2 required only a linear sub-packeizaion level In he presen paper, we consruc a (n,k,d = n 1)-MSR code wih a fixed rae R = 1 where 2 is an ineger parameer The code will have α = ( k ) To he bes of our knowledge, hese are he firs MSR consrucions ha achieve a sub-packeizaion level ha is polynomial in k These codes are help-by-ransfer codes, by which we mean ha he helper nodes need no do any compuaion during he repair of a failed node II MSR CODE CONSTRUCTION FOR RATE= 1 In his secion, we provide he consrucion for MSR codes wih a rae, R = 1 for some posiive ineger The consrucion is described for a paricular example of = 3, and subsequenly generalized A Code Consrucion for R = 2 3 We have an auxiliary parameer q = p m for some prime p, and m a posiive ineger 1 Then he code has parameers n = 3q, k = 2q, d = (n 1), α = q 3 1 The auxiliary parameer akes values from a finie-field, hough i is sufficien o work wih a finie-ring This does no cause any lack of generaliy in he principles used for he consrucion A codeword of an MSR code can be reaed as an array of size (α n) We firs inroduce an indexing for he rows and columns (nodes and columns are ofen used inerchangeably) of he codeword array Le F q = {0,1,,q 1} denoe a finie field of size q, and a 2-uple (i,θ),i {1,2,3}, θ F q is used o index he columns The rows are indexed by elemens (x,y,z) from F 3 q where x,y,z F q Thus C(x,y,z;(i,θ)) represens one code symbol from he codeword array a he inersecion of he row (x,y,z) and he node (i,θ) In order o describe he code, we firs inroduce he following noaion n a 1 a 2 a n := c i a i, c i 0, i [n] (4) i=1 o denoe a linear combinaion involving each of he scalars in {a 1,a 2,,a n } wih non-zero coefficiens The noaion is oblivious o he paricular choice of non-zero coefficiens in he linear combinaion The code is described by q 4 pariycheck consrains Throughou his paper, he symbol is used wih a differen meaning The erms wihin a are no conneced by he binary operaor +, bu by he operaor as defined in (4) For every (x,y,z) F 3 q, C(x,y,z;(1,θ)) C(x,y,z;(2,θ)) C(x,y,z;(3,θ)) = 0, (5) C(x,y,z;(1,x)) C(x,y,z;(2,y)) C(x,y,z ;(3,z)) C(x,y,z;(1,θ)) C(x,y,z;(2,θ)) C(x,y,z;(3,θ)) = 0, F q (6) The pariy-check consrain in (5) is referred o as he rowpariy, and he ha in (6) is referred o as he -pariy I can be observed ha he firs hree erms in he -pariy equaions are enries ha do no belong o he(x, y, z)-row These enries are referred o as he shifed enries Wha remains is he idenificaion of coefficiens in hese pariy-check consrains so ha he MDS propery holds Insead of consrucing hese coefficiens explicily, we will show in Sec II-A2 ha such coefficiens indeed exis in a sufficienly large field Therefore, he descripion of he code is complee wih (5), (6) In he zigzag code [4], pariy symbols are caegorized ino wo ypes, namely row-pariies and zigzag pariies The row pariies are made up of message symbols from he same row of he codeword array Bu he zigzag pariies are made up of message symbols belonging o various rows such ha one message symbol is picked per column In our consrucion also, every pariy-check consrain corresponding o 0 involves shifed enries ha do no belong o he row under consideraion In his manner, our consrucion is of a similar flavor as ha in [4] Bu he major difference of our consrucion from he zigzag consrucion lies in he symmery

3 of he pariy-check consrains I also differs in he fac ha wo symbols of he same column can be involved in he same pariy-check consrain in he case of 0 Such an approach was earlier adoped in [10] 1) Opimal Repair of a Failed Node : Wihou loss of generaliy, assume ha he node (1,θ 0 ) failed We download symbols belonging he rows Γ = {(θ 0,y,z) y,z F q } Clearly Γ = q 2 Thus we have {C(θ 0,y,z;(i,θ)) i = 1,2,3,θ θ 0,y,z F q } The rows are seleced such ha x = θ 0, because he firs coordinae of he index of he node is 1 If he firs coordinae had been 2 or 3, we would have fixed y = θ 0 or z = θ 0 respecively All he code symbols C(θ 0,y,z;(1,θ 0 )), y,z F q are repaired using he row-pariies Hence we have all he symbols belonging o rows in Γ from all he n nodes Nex, le us wrie he equaion for -pariy, F q corresponding o an arbirary row (θ 0,y,z) H C(θ 0,y,z;(1,θ 0 )) C(θ 0,y,z;(2,y)) C(θ 0,y,z ;(3,z)) C(θ 0,y,z;(1,θ)) C(θ 0,y,z;(2,θ)) C(θ 0,y,z;(3,θ)) = 0 (7) Excep he erm C(θ 0,y,z;(1,θ 0 )), all oher symbols involved in (7) are known o us Thus C(θ 0,y,z;(1,θ 0 )) can be repaired for all choices of y,z By making use of all he -pariies, we can hus repair all he remaining symbols in he node (1,θ 0 ) The oal number of symbols downloaded per node is β = q 2 = q3 q = α d k +1, and hus he repair is bandwidh-opimal 2) The MDS Propery: In his secion, we will show ha we can find an assignmen of coefficiens o he row-pariies and -pariies such ha he code saisfies he MDS propery We sar wih saing a useful fac Lemma 21: Le H be a ((n k) n)-pariy-check marix of a linear code C If S [n], S = (n k) is such ha rank(h S ) = (n k), hen i is possible o decode every codeword of C accessing symbols belonging o locaionss c = [n]\s Based on he pariy-consrains in (5), (6), we will deermine he srucure of he pariy-check marix H Firs, we vecorize he codeword array node-by-node so ha he firs α = q 3 columns of H represen he firs node, he second q 3 columns represen he second node and so on The group of q 3 columns associaed wih a node is referred o as a hick column The pariy-check marix hus obained will be of size (q 4 3q 4 ) wih n hick columns each conainingαhin columns In order o describe he suppor and hereby he srucure of H, we will for a momen assume ha all he coefficiens are se o 1 This marix is denoed by H s, and is given by H s = J +E, where he marices J and E are given in (8) and (9) The equaion (8) also illusraes he fac ha he rows of J can be decomposed ino blocks of size q 3, each corresponding o pariy-check consrains wih a fixed The firs se of q 3 pariy-check consrains correspond o row-pariies possibly associaed wih = 0 In (8), (9), I q 3,0 q 3 respecively represen ideniy and allzero marix of size q 3 q 3 The marices {E i δ,θ i {1,2,3},δ F q,θ F q } are made up of 1s and zeros, and represen he shifed enries of he corresponding -pariies The marices J,E and hence H are block marices of size (q 3q) where each block is a square marix of size q 3 We will show ha he MDS propery can be ensured by assigning suiable coefficiens o locaions idenified by he suppor of H s Our mehod is quie similar o he mehod used in [10] By 21, i is sufficien ha H resriced o any (n k) = q hick columns has a rank equal o qα = q 4 Le us assign an indeerminae c o all he locaions deermined by he suppor of E Now consider he square submarix H D obained by resricing H o D [n], D = (n k) hick columns If we assume ha all he coefficiens of J are fixed, he deerminan of H D will be a polynomial in he indeerminae c Le us denoe his polynomial by p D (c) In he following lemma, we prove ha p D (c) can be made a non-zero polynomial for every choice of D [n], D = n k Lemma 22: There exiss an assignmen of coefficiens o J such ha p D (c) is a non-zero polynomial for every choice of D [n], D = n k Proof: Consider a [3q, 2q]-RS code and is pariy-check marix H mds of size (q 3q) Clearly a (q q)-marix obained by resricing H mds o any q columns has full rank Le A B denoe he Kronecker produc of marices A and B If we se J o J 0, J 0 = H mds I q 3, (10) hen we mus have p D (0) evaluaing o a non-zero value for every choice of D [n], D = n k Hence p D (c) mus be a non-zero polynomial for every valid choice of D Henceforh, we assume ha he coefficiens of he polynomials p D (c) are fixed by he coefficiens of J as deermined by Lemma 22 By he srucure of E, i is clear ha Nex, consider he polynomial p(c) = deg(p D (c)) q 4 q 3 (11) D [n], D =k p D (c) (12) Clearly p(c) is no idenically zero, and is degree is upper bounded by ( n k) q 3 (q 1) Hence i is sufficien ha we find an non-zero assignmen c 0 0 for c such ha p(c 0 ) 0 By Combinaorial Nullsellansaz [13], his is possible if we choose he field size greaer han ( n k) q 3 (q 1)+1 Thus we have proved he following heorem Theorem 23: There exiss an assignmen for he coefficiens in he pariy-check consrains in (5), (6) such ha he code described in II-A is an MSR code

4 E = J = = 0 = 1 = q 1 0 q 3 0 q 3 0 q 3 0 q 3 0 q 3 0 q 3 E1,1 1 E1,q 1 E1,1 2 E1,q 2 E1,1 3 E1,q 3 (8) (9) E 1 q 1,1 E 1 q 1,q E 2 q 1,1 E 2 q 1,q E 3 q 1,1 E 3 q 1,q Using he consan c 0 guaraneed in he proof of Thm 23, and J 0 in (10), he pariy-check marix H of he MSR code akes he form H = J 0 +c 0 E (13) B Code Consrucion for R = 1, 2 The principle of he consrucion is elucidaed in he las secion compleely, and he generalizaion o he case of rae R = 1, 2 is sraighforward For an auxiliary parameer q = p m for some prime p, and m a posiive ineger, he code consrucion has parameers n = q, k = ( 1)q, d = (n 1), α = q A 2-uple (i,θ),i {1,2,,}, θ F q is used o index he columns The rows are indexed by elemens (x 1,x 2,,x ) from F q where x j F q Thus C(x 1,x 2,,x ;(i,θ)) represens one code symbol from he codeword array a he inersecion of he row (x 1,x 2,,x ) and he node (i,θ) The code is described by q +1 pariy-check consrains For every x = (x 1,x 2,,x ) F q, C(x;(1,θ)) C(x;(2,θ)) C(x;(2,θ)) C(x;(,θ)) = 0, F q (15) The pariy-check consrain in (14) is referred o as he rowpariy, and he ha in (15) is referred o as he -pariy As in he special case of = 3 described in Sec II-A, he firs erms in he -pariy equaions are enries ha do no belong o he (x 1,x 2,,x )-row These enries are referred o as he shifed enries Exisence of coefficiens for pariy-check equaions ha ensure MDS propery follows in he same line as ha in Sec II-A2 Wha remains is o presen a repair sraegy ha is bandwidh-opimal 1) Opimal Repair of a Failed Node : Assume ha he node(i 0,θ 0 ) failed We download symbols belonging he rows Γ = {x x j F q, j i 0, x i0 = θ 0 } Clearly Γ = q 1 Thus we have {C(x;(i,θ)) (i,θ) (i 0,θ 0 ),x Γ} All he code symbols C(x;(i 0,θ 0 )), x Γ are repaired using he row-pariies Then we have all he symbols belonging o rows in Γ from all he n nodes Nex, le us wrie he equaion for -pariy, F q corresponding o an arbirary row x Γ C(x 1,x 2,,x ;(1,x 1 )) C(x 1,,x i0,,x ;(i 0,θ 0 )) C(x 1,x 2,,x ;(,x )) C(x;(j,θ)) = 0,j [] (16) Excep he erm C(x 1,,x i0,,x ;(i 0,θ 0 )), all oher symbols involved in (16) are known o us By varying, we can hus repair all he remaining symbols in he node (i 0,θ 0 ) C(x;(,θ)) = 0, (14) The oal number of symbols downloaded per node is β = q 1 = q α = C(x 1,x 2,,x ;(1,x 1 )) C(x 1,x 2,,x ;(2,x 2 )) q d k +1, C(x 1,x 2,,x ;(,x )) and hus he repair is bandwidh-opimal C(x;(1,θ)) REFERENCES [1] A Dimakis, P Godfrey, Y Wu, M Wainwrigh, and K Ramchandran, Nework coding for disribued sorage sysems, IEEE Trans Inf Theory, vol 56, no 9, pp , Sep 2010 [2] K V Rashmi, N B Shah, and P V Kumar, Opimal Exac- Regeneraing Codes for Disribued Sorage a he MSR and MBR Poins via a Produc-Marix Consrucion, IEEE Trans Inf Theory, vol 57, no 8, pp , Aug 2011 [3] D Papailiopoulos, A Dimakis, and V Cadambe, Repair Opimal Erasure Codes hrough Hadamard Designs, IEEE Trans Inf Theory, vol 59, no 5, pp , 2013 [4] I Tamo, Z Wang, and J Bruck, Zigzag codes: MDS array codes wih opimal rebuilding, IEEE Trans Inform Theory, vol 59, no 3, pp , 2013

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