} s ) was constructed [7]. C 0 with length n 0 and minimum distance d 0 over F q.

Transcription

1 Ffty-Ffth Annual Allerton Conference Allerton House, UIUC, Illnos, USA October 3-6, 17 Mult-Erasure Locally Recoverable Codes Over Small Felds Pengfe Huang, Etan Yaakob, and Paul H Segel Electrcal and Computer Engneerng Dept, Unversty of Calforna, San Dego, La Jolla, CA 993 USA Computer Scence Dept, Technon Israel Insttute of Technology, Hafa 3, Israel pehuang,psegel@ucsdedu, yaakob@cstechnonacl Abstract Erasure codes play an mportant role n storage systems to prevent data loss In ths work, we study a class of erasure codes called Mult-Erasure Locally Recoverable Codes (ME-LRCs for storage arrays Compared to prevous related works, we focus on the constructon of ME-LRCs over small felds We frst develop upper and lower bounds on the mnmum dstance of ME-LRCs Our man contrbuton s to propose a general constructon of ME-LRCs based on generalzed tensor product codes, and study ther erasure-correctng propertes A decodng algorthm talored for erasure recovery s gven, and correctable erasure patterns are dentfed We then prove that our constructon yelds optmal ME-LRCs wth a wde range of code parameters, and present some explct ME-LRCs over small felds Fnally, we show that generalzed ntegrated nterleavng (GII codes can be treated as a subclass of generalzed tensor product codes, thus defnng the exact relaton between these codes I INTRODUCTION Recently, erasure codes wth both local and global erasurecorrectng propertes have receved consderable attenton [3], [9], [18] [], [], thanks to ther promsng applcaton n storage systems The dea behnd them s that when only a few erasures occur, these erasures can be corrected fast usng only local partes If the number of erasures exceeds the local erasure-correctng capablty, then the global partes are nvoked In ths paper, we consder ths knd of erasure codes wth both local and global erasure-correctng capabltes for a ρ n storage array [3], where each row contans some local partes, and addtonal global partes are dstrbuted n the array The array structure s sutable for many storage applcatons For example, consder a redundant array of ndependent dsks (RAID type of archtecture for sold-state drves (SSDs [3], [8] In ths scenaro, a ρ n storage array can represent a total of ρ SSDs, each of whch contans n flash memory chps Wthn each SSD, an erasure code s appled to these n chps for local protecton In addton, erasure codng s also done across all the SSDs for global protecton of all the chps More specfcally, let us gve the formal defnton of ths class of erasure codes as follows Defnton 1 Consder a code C over a fnte feld F q consstng of ρ n arrays such that: 1 Each row n each array n C belongs to a lnear local code C wth length n and mnmum dstance d over F q Readng the symbols of C row-wse, C s a lnear code wth length ρn, dmenson k, and mnmum dstance d over F q /17/$31 17 IEEE 113 Then, we say that C s a (ρ, n, k; d, d q Mult-Erasure Locally Recoverable Code (ME-LRC Thus, a (ρ, n, k; d, d q ME-LRC can locally correct d 1 erasures n each row, and s guaranteed to correct a total of d 1 erasures anywhere n the array Our work s motvated by a recent work by Blaum and Hetzler [3] In ther work, the authors studed ME-LRCs where each row s a maxmum dstance separable (MDS code, and gave code constructons wth feld sze q maxρ, n usng generalzed ntegrated nterleavng (GII codes [11], [3], [5] Our Defnton 1 generalzes the defnton of the codes n [3] by not requrng each row to be an MDS code There exst other related works The ME-LRCs n Defnton 1 can be seen as (r, δ LRCs wth dsjont repar sets A code C s called an (r, δ LRC [], f for every coordnate, there exsts a punctured code (e, a repar set of C wth support contanng ths coordnate, whose length s at most r + δ 1, and whose mnmum dstance s at least δ Although the exstng constructons [], [] for (r, δ LRCs wth dsjont repar sets can generate ME-LRCs as n Defnton 1, they use MDS codes as local codes and requre a feld sze that s at least as large as the code length A recent work [1] gves explct constructons of (r, δ LRCs wth dsjont repar sets over feld F q from algebrac curves, whose repar sets have sze r + δ 1 = q or r + δ 1 = q + 1 Partal MDS (PMDS codes [] are also related to but dfferent from ME-LRCs n Defnton 1 In general, an ME-LRC s not a PMDS code whch needs to satsfy more strct requrements A ρ n array code s called an (r; s PMDS code f each row s an [n, n r, r + 1] q MDS code and whenever any r locatons n each row are punctured, the resultng code s also an MDS code wth mnmum dstance s + 1 The constructon of (r, s PMDS codes for all r and s wth feld sze O(n ρn was known [6] More recently, a famly of PMDS codes wth feld sze O(maxρ, n r+s s was constructed [7] To the best of our knowledge, however, the constructon of optmal ME-LRCs over any small feld (eg, the feld sze less than the length of the local code, or even the bnary feld has not been fully explored and solved The goal of ths paper s to study ME-LRCs over small felds We propose a general constructon based on generalzed tensor product codes [16], [4], whch were frst utlzed n [1] to construct bnary sngle-erasure LRCs [9], [1], [14], [15], [19], [] The contrbutons of ths paper are: 1 We extend our prevous constructon n [1] to the scenaro of mult-erasure LRCs over any feld As a result, the

2 constructon n [1] can be seen as a specal case of the constructon proposed n ths paper In contrast to [3], our constructon does not requre feld sze q maxρ, n, and t can even generate bnary ME-LRCs We derve upper and lower bounds on the mnmum dstance of ME-LRCs For d d, we show that our constructon can produce optmal ME-LRCs wth respect to (wrt the new upper bound on the mnmum dstance 3 We present an erasure decodng algorthm and ts correspondng correctable erasure patterns, whch nclude the pattern of any d 1 erasures We show that the ME-LRCs from our constructon based on Reed-Solomon (RS codes are optmal wrt certan correctable erasure patterns 4 So far the exact relaton between GII codes [3], [3], [5] and generalzed tensor product codes has not been fully nvestgated We prove that GII codes are a subclass of generalzed tensor product codes As a result, the parameters of a GII code can be obtaned by usng the known propertes of generalzed tensor product codes The remander of the paper s organzed as follows In Secton II, we study feld sze dependent upper and lower bounds for ME-LRCs In Secton III, we propose a general constructon of ME-LRCs The erasure-correctng propertes of these codes are studed and an erasure decodng algorthm s presented In Secton IV, we study optmal code constructon and gve several explct optmal ME-LRCs over dfferent felds In Secton V, we prove that GII codes are a subclass of generalzed tensor product codes Secton VI concludes the paper Due to space constrants, we omt some detaled proofs, whch can be found onlne n the longer verson of ths paper [13] Throughout the paper, we use the notaton [n] to denote the set 1,,n For a length-n vector v over F q and a set I [n], the vector v I denotes the restrcton of the vector v to coordnates n the set I, and w q (v represents the Hammng weght of the vector v over F q The transpose of a matrx H s wrtten as H T For a set S, S represents the cardnalty of the set A lnear code C over F q of length n, dmenson k, and mnmum dstance d wll be denoted by C =[n, k, d] q or [n, k, d] q for smplcty For a code wth only one codeword, the mnmum dstance s defned as II UPPER AND LOWER BOUNDS FOR ME-LRCS In ths secton, we derve feld sze dependent upper and lower bounds on the mnmum dstance of ME-LRCs The upper bound obtaned here wll be used to prove the optmalty of our constructon for ME-LRCs n the followng sectons Now, we gve an upper bound on the mnmum dstance of a (ρ, n, k; d, d q ME-LRC, by extendng the shortenng bound for LRCs n [5] Let d (q opt [n, k] denote the largest possble mnmum dstance of a lnear code of length n and dmenson k over F q, and let k (q opt [n, d] denote the largest possble dmenson of a lnear code of length n and mnmum dstance d over F q Lemma For any (ρ, n, k; d, d q ME-LRC C, the mnmum dstance d satsfes d d (q opt [ρn xn, k xk ], (1 mn x k k 1, x Z and the dmenson satsfes k mn x k k 1, x Z xk + k (q opt [ρn xn, d], ( where k = k (q opt [n, d ] An asymptotc lower bound for ME-LRCs wth local MDS codes was gven n [1] Here, by smply adaptng the Glbert- Varshamov (GV bound [1], we have the followng GV-lke lower bound on ME-LRCs of fnte length wthout specfyng local codes Lemma 3 A (ρ, n, k; d, d q ME-LRC C exsts, f d ( ( d ρ(n log q j= (n 1 j (q 1 j 1 (q 1 = ( < q ρ(n log q d j= ( n 1 (q 1 j k j (3 III ME-LRCS FROMGENERALIZED TENSOR PRODUCT CODES: CONSTRUCTION AND DECODING Tensor product codes, frst proposed by Wolf n [4], are a famly of bnary error-correctng codes defned by a partycheck matrx that s the tensor product of the party-check matrces of two consttuent codes Later, they were generalzed n [16] In ths secton, we frst ntroduce generalzed tensor product codes over F q Then, we gve a general constructon of ME-LRCs from generalzed tensor product codes The mnmum dstance of the constructed ME-LRCs s determned, a decodng algorthm talored for erasure correcton s proposed, and correspondng correctable erasure patterns are studed A Generalzed Tensor Product Codes over F q We start by presentng the tensor product operaton of two matrces H and Let H be the party-check matrx of a code wth length n and dmenson n v over F q The matrx H can be consdered as a v (row by n (column matrx over F q or as a 1 (row by n (column matrx of elements from F q v Let H be the vector H =[h 1 h h n ], where h j, 1 j n, are elements of F q v Let be the partycheck matrx of a code of length l and dmenson l λ over F q v We denote by h 11 h 1l =, h λ1 h λl where h j, 1 λ and 1 j l, are elements of F q v The tensor product of the matrces and H s defned as 114 h H TP = 11 H h 1l H H = h λ1 H h λl H,

3 where h j H =[h j h 1 h j h h j h n ], 1 λ and 1 j l, and the products of elements are calculated accordng to the rules of multplcaton for elements over F q v The matrx H TP wll be consdered as a vλ n l matrx of elements from F q, thus defnng a tensor product code over F q We provde an example to llustrate these operatons Example 1 (cf [4] Let be the followng party-check matrx over F 4 for a [5, 3, 3] 4 code where α s a prmtve element of F 4, [ ] = α α α α α α α 1 α Let H be the followng party-check matrx over F for a [3, 1, 3] Hammng code, [ ] H = Representng the elements of F 4 as α = [ ] [ ] 1, α = 11, and = [ 1 [ ], we have ], α 1 = H TP = H [ α = α 1 α α α 1 α α α 1 α α α 1 α α α 1 α α α 1 α α 1 α α α α α 1 Usng the same symbol-to-bnary vector mappng, we represent H TP over F as H TP = , whch defnes a bnary [15, 11, 3] code Our constructon of ME-LRCs s based on generalzed tensor product codes [16] Defne the matrces H and for = 1,,,μ as follows The matrx H s a v n matrx over F q such that the (v 1 + v + + v n matrx H 1 H B = s a party-check matrx of an [n, n v 1 v v, d ] q code C, where d 1 d d The matrx s a λ l matrx over F q v, whch s a party-check matrx of an [l, l λ, δ ] q v code C We defne a μ-level generalzed tensor product code over F q as a lnear code havng a party-check matrx over F q n the form of the followng μ-level tensor product structure H = 1 μ H H 1 H H μ As the matrx H TP, each level n the matrx H s obtaned by operatons over F q and ts extenson feld We denote ths ] code by C μ GTP Its length s n t = n l and the dmenson s k t = n t μ =1 v λ By adaptng Theorem n [16] from the feld F to F q,we drectly have the followng theorem on the mnmum dstance of C μ GTP over F q Theorem 4 The mnmum dstance d t of a generalzed tensor product code C μ GTP over F q satsfes d t mnδ 1, δ d 1, δ 3d,,δ μd μ 1, d μ Proof: See the longer verson [13] B Constructon of ME-LRCs Now, we present a general constructon of ME-LRCs based on generalzed tensor product codes Constructon A Step 1: Choose v n matrces H over F q and λ l matrces over F q v, for = 1,,,μ, whch satsfy the followng two propertes: 1 The party-check matrx 1 = I l l, e, an l l dentty matrx The matrces H (or B, 1 μ, and j, j μ, are chosen such that d μ δ j d j 1, for j =,3,,μ Step : Generate a party-check matrx H over F q accordng to (4 wth the matrces H and, for = 1,,,μ The constructed code correspondng to the party-check matrx H s referred to as C A Theorem 5 The code C A s a (ρ, n, k; d, d q ME-LRC wth parameters ρ = l, n = n, k = n l μ =1 v λ, d = d 1, and d = d μ Proof: Accordng to Constructon A, the code parameters ρ, n, k, and d can be easly determned In the followng, we prove that d = d μ Snce δ 1 = ( 1 s the dentty matrx and d μ δ d 1 for all =,3,,μ, from Theorem 4, d d μ Now, we show that d d μ For = 1,,,μ, let H = [h 1 (,,h n (] over F q v, and let [h 11 (,,h λ 1 (]T over F q v be the frst column of Snce the code wth party-check matrx B μ has mnmum dstance d μ, there exst d μ columns of B μ, say n the set of postons J = b 1, b,,b d μ, whch are lnearly dependent; that s, j J α j h j ( =, for some α j F q, for all = ( 1,,,μ Thus, we have j J α j h p1 (h j ( = h p1 ( j J α j h j ( =, for p = 1,,,λ and = 1,,,μ That s, the columns n postons b 1, b,,b d μ of H are lnearly dependent C Erasure Decodng and Correctable Erasure Patterns (4 We present a decodng algorthm for the ME-LRC C A from Constructon A, talored for erasure correcton The decodng algorthm for error correcton for generalzed tensor product codes can be found n [16] Let the symbol? represent an erasure and e denote a decodng falure The erasure decoder D A : (F q? nl 115

4 C A e for an ME-LRC C A conssts of two knds of component decoders D and D for = 1,,,μ descrbed below a Frst, the decoder for a coset of the code C wth partycheck matrx B, = 1,,,μ, s denoted by D :(F q? n (F q? j=1 v j (F q? n whch uses the followng decodng rule: for a length-n nput vector y, and a length- j=1 v j syndrome vector s wthout erasures, f y agrees wth exactly one codeword c C + e on the entres wth values n F q, where the vector e s a coset leader determned by both the code C and the syndrome vector s, e, s = eb T, then D (y, s =c ; otherwse, D (y, s =y Therefore, f the length-n nput vector y s a codeword n C + e wth d 1 or less erasures and the syndrome vector s s not erased, then the decoder D can return the correct codeword b Second, the decoder for the code C wth party-check matrx, = 1,,,μ, s denoted by D : (F q v? l (F q v? l whch uses the followng decodng rule: for a length-l nput vector y,fy agrees wth exactly one codeword c C on the entres wth values n F q v, then D (y =c ; otherwse, D (y =y Therefore, f the length-l nput vector y s a codeword n C wth δ 1 or less erasures, then the decoder D can successfully return the correct codeword The erasure decoder D A for the code C A s summarzed n Algorthm 1 below Let the nput word of length n l for the decoder D A be y =(y 1, y,,y l, where each component y (F q? n, = 1,,l The vector y s an erased verson of a codeword c =(c 1, c,,c l C A Algorthm 1: Decodng Procedure of Decoder D A Input: receved word y =(y 1, y,,y l Output: codeword c C A or a decodng falure e 1 Let s 1 j =, for j = 1,,,l ĉ =(ĉ 1,,ĉ l = (D 1 (y 1, s1 1,,D 1 (y l, s1 l 3 Let F = j [l] :ĉ j contans? 4 For =,,μ If F =, do the followng steps; otherwse go to step 5 ( (s 1,,s l =D ĉ 1 H T,,ĉ l H T ĉ j = D (ĉ j, (s 1 j,,s j for j F; ĉ j remans the same for j [l]\f Update F = j [l] :ĉ j contans? end 5 If F =, let c = ĉ and output c; otherwse return e In Algorthm 1, we use the followng rules for operatons whch nvolve the symbol?: 1 Addton +: for any element γ F q?, γ+? =? Multplcaton : for any element 116 γ F q?\, γ? =?, and? = 3 If a length-n vector x, x (F q? n, contans an entry?, then x s consdered as the symbol? n the set F q n? Smlarly, the symbol? n the set F q n? s treated as a length-n vector whose entres are all? To descrbe correctable erasure patterns, we use the followng notaton Let w e (v denote the number of erasures? n the vector v For a receved word y =(y 1, y,,y l, let N τ = y m : w e (y m d τ, 1 m l for 1 τ μ Theorem 6 The decoder D A for a (ρ, n, k; d, d q ME-LRC C A can correct any receved word y that satsfes the followng condton: N τ δ τ+1 1, 1 τ μ, (5 where δ μ+1 s defned to be 1 Proof: See the longer verson [13] The followng corollary follows from Theorem 6 Corollary 7 The decoder D A for a (ρ, n, k; d, d q ME-LRC C A can correct any receved word y wth less than d erasures Proof: See the longer verson [13] IV OPTIMAL CONSTRUCTION AND EXPLICIT ME-LRCS OVER SMALL FIELDS In ths secton, we study the optmalty of Constructon A, and also present several explct ME-LRCs that are optmal over dfferent felds A Optmal Constructon We show how to construct ME-LRCs whch are optmal wrt the bound (1 by addng more constrants to Constructon A To ths end, we specfy the choce of the matrces n Constructon A Ths specfcaton, referred to as Desgn I, s as follows 1 H 1 s the party-check matrx of an [n, n v 1, d 1 ] q code whch satsfes k (q opt [n, d 1 ]=n v 1 B μ s the party-check matrx of an [n, n μ =1 v, d μ] q code wth d (q opt [n, n μ =1 v ]=d μ 3 The mnmum dstances satsfy d μ d 1 4 s an all-one vector of length l over F q v, e, the party-check matrx of a party code wth mnmum dstance δ =, for all =,,μ Theorem 8 The code C A from Constructon A wth Desgn I s a (ρ = l, n = n, k = n l v 1 l μ = v ; d = d 1, d = d μ q ME-LRC, whch s optmal wth respect to the bound (1 Proof: From Theorem 5, the code parameters ρ, n, k, d, and d can be determned We have k = k (q opt [n, d 1 ]= n v 1 Settng x = l 1, we get d mn x k k 1 d (q =d (q d (q opt [ρn xn, k xk ] opt [ln (l 1n, k (l 1k ] opt [n, n μ =1 v ]=d μ

5 Ths proves that C A acheves the bound (1 B Explct ME-LRCs from Constructon A Our constructon s very flexble and can generate many ME-LRCs over dfferent felds In the followng, we present several examples 1 ME-LRCs wth local extended BCH codes over F From the structure of BCH codes [1], there exsts a chan of nested bnary extended BCH codes: C 3 =[ m, m 1 3m,8] C =[ m, m 1 m,6] C 1 =[ m, m 1 m,4] Let the matrces B 1, B, and B 3 be the party-check matrces of C 1, C, and C 3, respectvely Example For μ = 3, n Constructon A, we use the above matrces B 1, B, and B 3 We also choose and H 3 to be the all-one vector of length l over F m From Theorem 5, the correspondng (ρ, n, k; d, d ME- LRC C A has parameters ρ = l, n = m, k = m l (m + 1l m, d = 4, and d = 8 Ths code satsfes the requrements of Desgn I Thus, from Theorem 8, t s optmal wrt the bound (1 sngly-extended RS code, namely H 1 = 1 α α n 1 α d 1 α (n (d 1 For =, 3,, μ, we choose H to be 1 α d 1 1 α (n (d 1 1 H =, 1 α d α (n (d where d 1 < d < < d μ We also requre that d μ δ = d μ d 1 = d = = d d μ 1, =,,μ and δ > δ 3 > > δ μ For =,3,,μ, let be the party-check matrx of an [l, l δ + 1, δ = d μ d 1 ] q v MDS code, whch exsts whenever l q v, where v = d d 1 Note that for an MDS code wth mnmum dstance, the code length can be arbtrarly long ME-LRCs wth local algebrac geometry codes over F 4 We use a class of algebrac geometry codes called Hermtan codes [6] to construct ME-LRCs From the constructon of Hermtan codes [6], there exsts a chan of nested 4-ary Hermtan codes: C H (1 = [8, 1, 8] 4 C H ( = [8,, 6] 4 C H (3 = [8, 3, 5] 4 C H (4 = [8, 4, 4] 4 C H (5 = [8, 5, 3] 4 C H (6 = [8, 6, ] 4 C H (7 =[8, 7, ] 4 Now, let the matrces B 1, B, B 3, and B 4 be the partycheck matrces of C H (4, C H (3, C H (, and C H (1, respectvely Let, =, 3, 4, be the all-one vector of length l over F 4 Example 3 For μ =, n Constructon A, we use the above matrces B 1, B, and From Theorem 5, the correspondng (ρ, n, k; d, d 4 ME-LRC C A has parameters ρ = l, n = 8, k = 4l 1, d = 4, and d = 5 For μ = 3, n Constructon A, we use the above matrces B 1, B, B 3,, and H 3 From Theorem 5, the correspondng (ρ, n, k; d, d 4 ME-LRC C A has parameters ρ = l, n = 8, k = 4l, d = 4, and d = 6 For μ = 4, n Constructon A, we use the above matrces B, = 1,,4, and j, j =, 3, 4 From Theorem 5, the correspondng (ρ, n, k; d, d 4 ME-LRC C A has parameters ρ = l, n = 8, k = 4l 3, d = 4, and d = 8 All of the above three famles of ME-LRCs over F 4 are optmal wrt the bound (1 3 ME-LRCs wth local sngly-extended Reed-Solomon codes over F q Let n q and α be a prmtve element of F q We choose H 1 to be the party-check matrx of an [n, n d 1 + 1, d 1 ] q 117 Example 4 We use the above chosen matrces H and H for Constructon A The correspondng (ρ, n, k; d, d q ME- LRC C A has parameters ρ = l, n = n, k = (n d 1 + 1l μ = ( d μ d 1 1(d d 1, d = d 1, and d = d μ; the feld sze q satsfes q maxq, n, where q = max =,,μ 1 d l d 1 When μ = and d 1 < d d 1, the correspondng (ρ, n, k; d, d q ME-LRC C A has parameters ρ = l, n = n, k =(n d 1 + 1l (d d 1, d = d 1, and d = d ; the feld sze q needs to satsfy q n Snce C A satsfes the requrements of Desgn I, from Theorem 8, t s optmal wrt the bound (1 The followng theorem shows that the μ-level ME-LRC C A constructed n Example 4 s optmal n the sense of possessng the largest possble dmenson among all codes wth the same erasure-correctng capablty Theorem 9 Let C be a code of length ln and dmenson k over F q Each codeword n C conssts of l sub-blocks, each of length n Assume that C corrects all erasure patterns satsfyng the condton n (5, where δ τ = d μ d τ 1 for τ μ Then, we must have dmenson k (n d 1 + 1l μ = ( d μ d 1 1(d d 1 Proof: The proof s based on contradcton Let each codeword n C correspond to an l n array We ndex the coordnates of the array row by row from number 1 to ln Let I 1 be the set of coordnates defned by I 1 = ( 1n + j : δ 1 < l, 1 j d 1 1 For τ μ, let I τ be the set of coordnates gven by I τ =

6 ( 1n + j : δ τ+1 1 < δ τ 1, 1 j d τ 1, where δ μ+1 s defned to be 1 Let I be the set of all the coordnates of the array By calculaton, we have I\(I 1 I I μ =(n d 1 + 1l μ = ( d μ d 1 1(d d 1 Now, assume that k > (n d 1 + 1l μ = ( d μ d 1 1(d d 1 Then, there exst at least two dstnct codewords c and c n C that agree on the coordnates n the set : I\(I 1 I I μ We erase the values on the coordnates n the set : I 1 I I μ of both c and c Ths erasure pattern satsfes the condton n (5 Snce c and c are dstnct, ths erasure pattern s uncorrectable Thus, our assumpton that k > (n d 1 + 1l μ = ( d μ d 1 1(d d 1 s volated Remark 1 The constructon by Blaum and Hetzler [3] based on GII codes cannot generate ME-LRCs constructed n Examples and 3 For the ME-LRC n Example 4, snce the local code s the sngly-extended RS code, the constructon n [3] can also be used to produce an ME-LRC that has the same code parameters ρ, n, k, d and d as those of the ME- LRC C A from our constructon However, the constructon n [3] requres the feld sze q to satsfy q maxl, n, whch n general s larger than that n our constructon V RELATION TO GENERALIZED INTEGRATED INTERLEAVING CODES Integrated nterleavng (II codes were frst ntroduced n [11] as a two-level error-correctng scheme for data storage applcatons, and were then extended n [3] and more recently n [5] as generalzed ntegrated nterleavng (GII codes for mult-level data protecton The man dfference between GII codes and generalzed tensor product codes s that a generalzed tensor product code over F q s defned by operatons over the base feld F q and also ts extenson feld, as shown n (4; n contrast, a GII code over F q s defned over the same feld F q As a result, generalzed tensor product codes are more flexble than GII codes, and generally GII codes cannot be used to construct ME-LRCs over very small felds, eg, the bnary feld The goal of ths secton s to study the exact relaton between generalzed tensor product codes and GII codes We wll show that GII codes are n fact a subclass of generalzed tensor product codes The dea s to reformulate the partycheck matrx of a GII code nto the form of a party-check matrx of a generalzed tensor product code Establshng ths relaton allows some code propertes of GII codes to be obtaned drectly from known results about generalzed tensor product codes We start by consderng the II codes, a twolevel case of GII codes, to llustrate our dea A Integrated Interleavng Codes We follow the defnton of II codes n [11] Let C, = 1,, be[n, k, d ] q lnear codes over F q such that C C and d > d 1 An II code C II s defned as follows: C II = c =(c, c 1,,c m 1 : c C 1, < m, m 1 (6 and α b c C, b =,1,,γ 1, = where α s a prmtve element of F q and γ < m q 1 Accordng to the above defnton, t s known that the party-check matrx of C II s [ ] H II = I H1, (7 Γ H where [ denotes ] the Kronecker product The matrces H 1 H1 and over F H q are the party-check matrces of C 1 and C, respectvely, the matrx I over F q s an m m dentty matrx, and Γ over F q s the party-check matrx of an [m, m γ, γ + 1] q code n the followng form α α m 1 Γ = 1 α α (m 1 (8 1 α (γ 1 α (γ 1(m 1 Remark The party-check matrx H II over F q n (7 of C II s obtaned by operatons over the same feld F q In contrast, the party-check matrx H over F q n (4 of a generalzed tensor product code s obtaned by operatons over both the base feld F q and ts extenson feld Remark 3 In general, the codes C 1 and C n (6 are chosen to be RS codes [11] If C 1 and C are chosen to be bnary codes, then m can only be m = 1 To see the relaton between II codes and generalzed tensor product codes, we reformulate H II n (7 nto the followng form, by splttng the rows of H, I H1 Γ H (1 H II = Γ H (, (9 Γ H (k 1 k where the matrx H 1 over F q s the party-check matrx of C 1, and s treated as a vector over the extenson feld F q n k 1 here; correspondngly, the matrx I s treated as an m m dentty matrx over F q n k 1 For1 k 1 k, H ( over F q represents the th row of H, and Γ over F q s the matrx n (8 Now, referrng to the matrx n (4, the matrx n (9 can be nterpreted as a party-check matrx of a (1 + k 1 k -level generalzed tensor product code over F q Thus, we conclude that an II code s a generalzed tensor product code Usng the propertes of generalzed tensor product codes, we can drectly obtan the followng result, whch was proved n [11] n an alternatve way

7 Lemma 1 The code C II s a lnear code over F q of length N = nm, dmenson K =(m γk 1 + γk, and mnmum dstance D mn(γ + 1d 1, d Proof: For 1 k 1 k, let the followng partycheck matrx H 1 H (1 H ( defne an [n, k 1, d, ] q code It s clear that d 1 d,1 d, d,k1 k = d From the propertes of generalzed tensor product codes, the redundancy s N K = nm K = (n k 1 m + γ(k 1 k ; that s, the dmenson s K = k 1 (m γ+k γ Usng Theorem 4, the mnmum dstance s D mn d 1 (γ + 1, d,1 (γ + 1,,d,k1 k 1(γ + 1, d,k1 k = mn (γ + 1d 1, d B Generalzed Integrated Interleavng Codes Wth the smlar dea used n the prevous subsecton, we contnue our proof for GII codes We use the defnton of GII codes from [5] for consstency Let C, =,1,,γ, be [n, k, d ] q codes over F q such that C s = = C s 1 +1 C s 1 = = C s +1 (1 C 1 = = C 1 C, where = and s = γ The mnmum dstances satsfy d d 1 d γ A GII code C GII s defned as: C GII = c =(c, c 1,,c m 1 : c C, < m, m 1 and α b c C γ b, b =, 1,, γ 1, = (11 where α s a prmtve element of F q and γ < m q 1 Let us frst defne some matrces whch wll be used below Let the matrx I over F q be an m m dentty matrx Let H over F q be [ the party-check ] matrx of C For1 j s, H let the matrx over F H q represent the party-check j matrx of C j, where H j = H 1 \ H \ 1 H j \ j 1 For any j, let matrx Γ(, j;α over F q be the partycheck matrx of an [m, m ( j + 1, j + ] q code n the followng form 1 α α (m 1 1 α +1 α (+1(m 1 Γ(, j;α = (1 1 α j α j(m Now, accordng to the defnton n (11, usng the matrces ntroduced above, the party-check matrx of C GII s I H Γ(, s s 1 1;α Hs H GII = Γ( s s 1, s s 1;α Hs 1, Γ( s, s 1 1;α H Γ( s 1, s 1;α H1 (13 whch can be transformed nto the form of I H Γ(, s 1;α H1 \ Γ(, H GII = s 1 1;α H \ 1 (14 Γ(, s s 1;α Hs 1 \ s Γ(, s s 1 1;α Hs \ s 1 To make a connecton between GII codes and generalzed tensor product codes, we further reformulate the matrx H GII n (14 as follows, I Γ(, s 1;α Γ(, s 1;α Γ(, s 1 1;α Γ(, s 1 1;α H GII = Γ(, s s 1;α Γ(, s s 1;α Γ(, s s 1 1;α Γ(, s s 1 1;α H H1 \ (1 H1 \ (k k 1 H \ 1 (1 H \ 1 (k 1 k, Hs 1 \ s (1 Hs 1 \ s (k s k s 1 Hs \ s 1 (1 Hs \ s 1 (k s 1 k s (15 where, n the frst level, the matrx H over F q s treated as a vector over the extenson feld F q n ko, and correspondngly the matrx I s treated as an m m dentty matrx over F q n ko For1 x s and 1 y k x 1 k x, H x \ x 1 (y over F q represents the yth row of the matrx H x \ x 1 Now, referrng to the matrx n (4, the matrx n (15 can be seen as a party-check matrx of a (1 + k k s -level generalzed tensor product code over F q As a result, we can drectly obtan the followng lemma, whch was also proved n [5] n a dfferent way Lemma 11 The code C GII s a lnear code over F q of length N = nm, dmenson K = γ x=1 k x +(m γk = s j=1 ( j j 1 k j +(m γk, and mnmum dstance D mn (γ + 1d, (γ 1 + 1d 1,,(γ s 1 + 1d s 1, d s

8 Proof: For 1 x s and 1 y k x 1 k x, let the followng party-check matrx H H 1 \ (1 H 1 \ (k k 1 H x \ x 1 (1 H x \ x 1 (y defne an [n, k x 1 y, d x,y] q code, so we have d x 1 d x,1 d x, d x,k x 1 k x = d x From the propertes of generalzed tensor product codes, t s easy to obtan the dmenson K = s j=1 ( j j 1 k j +(m γk From Theorem 4, the mnmum dstance satsfes D mn (γ + 1d, (γ + 1d 1,1,,(γ + 1d 1,k k 1 1, (γ 1 + 1d 1,,,(γ s 1 + 1d s 1, (γ s 1 + 1d s,1,,(γ s 1 + 1d s,k s 1 k s 1, d s = mn (γ + 1d, (γ 1 + 1d 1,,(γ s 1 + 1d s 1, d s Remark 4 In some pror works, we fnd that generalzed tensor product codes are called generalzed error-locaton (GEL codes [4], [17] Recently, n [5], the smlarty between GII codes and GEL codes was observed However, the exact relaton between them was not studed In [5], the author also proposed a new generalzed ntegrated nterleavng scheme over bnary BCH codes, called GII-BCH codes These codes can also be seen as a specal case of generalzed tensor product codes VI CONCLUSION In ths work, we presented a general constructon for ME- LRCs over small felds Ths constructon yelds optmal ME- LRCs wth respect to an upper bound on the mnmum dstance for a wde range of code parameters Then, an erasure decoder was proposed and correspondng correctable erasure patterns were dentfed ME-LRCs based on Reed-Solomon codes were shown to be optmal among all codes havng the same erasure-correctng capablty Fnally, generalzed ntegrated nterleavng codes were proved to be a subclass of generalzed tensor product codes, thus gvng the exact relaton between these two codes REFERENCES [1] A Barg, I Tamo, and S Vladut, Locally recoverable codes on algebrac curves, arxv preprnt arxv: , 16 [] M Blaum, J L Hafner, and S Hetzler, Partal-MDS codes and ther applcaton to RAID type of archtectures, IEEE Trans Inf Theory, vol 59, no 7, pp , 13 [3] M Blaum and S R Hetzler, Integrated nterleaved codes as locally recoverable codes: propertes and performance, Internatonal Journal of Informaton and Codng Theory, vol 3, no 4, pp , 16 [4] M Bossert, H Greßer, J Maucher, and V V Zyablov, Some results on generalzed concatenaton of block codes, n Proc Internatonal Symposum on Appled Algebra, Algebrac Algorthms, and Error-Correctng Codes Sprnger, 1999, pp [5] V Cadambe and A Mazumdar, An upper bound on the sze of locally recoverable codes, n Proc IEEE NetCod, 13, pp 1 5 [6] G Cals and O O Koyluoglu, A general constructon for PMDS codes, IEEE Communcatons Letters, vol 1, no 3, pp , 17 [7] R Gabrys, E Yaakob, M Blaum, and P H Segel, Constructons of partal MDS codes over small felds, n Proc IEEE ISIT, 17 [8] G A Gbson, Redundant Dsk Arrays: Relable, Parallel Secondary Storage MIT Press, 199 [9] P Gopalan, C Huang, H Smtc, and S Yekhann, On the localty of codeword symbols, IEEE Trans Inf Theory, vol 58, no 11, pp , 1 [1] S Goparaju and R Calderbank, Bnary cyclc codes that are locally reparable, n Proc IEEE ISIT, June 14, pp [11] M Hassner, K Abdel-Ghaffar, A Patel, R Koetter, and B Trager, Integrated nterleavng-a novel ECC archtecture, IEEE Trans Magn, vol 37, no, pp , 1 [1] P Huang, E Yaakob, H Uchkawa, and P H Segel, Bnary lnear locally reparable codes, IEEE Trans Inf Theory, vol 6, no 11, pp , Nov 16 [13] P Huang, E Yaakob, and P H Segel (17 Mult-erasure locally recoverable codes over small felds [Onlne] Avalable: [14] P Huang, E Yaakob, H Uchkawa, and P H Segel, Cyclc lnear bnary locally reparable codes, n Proc IEEE ITW, 15, pp 1 5 [15], Lnear locally reparable codes wth avalablty, n Proc IEEE ISIT, 15, pp [16] H Ima and H Fujya, Generalzed tensor product codes, IEEE Trans Inf Theory, vol 7, no, pp , Mar 1981 [17] J Maucher, V V Zyablov, and M Bossert, On the equvalence of generalzed concatenated codes and generalzed error locaton codes, IEEE Trans Inf Theory, vol 46, no, pp , [18] F Ogger and A Datta, Self-reparng homomorphc codes for dstrbuted storage systems, n Proc IEEE INFOCOM, 11, pp [19] D S Papalopoulos and A G Dmaks, Locally reparable codes, n Proc IEEE ISIT, 1, pp [] N Prakash, G Kamath, V Laltha, and P Kumar, Optmal lnear codes wth a local-error-correcton property, n Proc IEEE ISIT, July 1, pp [1] R Roth, Introducton to Codng Theory Cambrdge Unversty Press, 6 [] I Tamo and A Barg, A famly of optmal locally recoverable codes, IEEE Trans Inf Theory, vol 6, no 8, pp , Aug 14 [3] X Tang and R Koetter, A novel method for combnng algebrac decodng and teratve processng, n Proc IEEE ISIT, 6, pp [4] J Wolf, On codes dervable from the tensor product of check matrces, IEEE Trans Inf Theory, vol 11, no, pp 81 84, Apr 1965 [5] Y Wu, Generalzed ntegrated nterleaved codes, IEEE Trans Inf Theory, vol 63, no, pp , Feb 17 [6] K Yang and P V Kumar, On the true mnmum dstance of Hermtan codes, n Codng theory and algebrac geometry Sprnger, 199, pp ACKNOWLEDGMENT Ths work was supported by NSF Grants CCF and CCF , BSF Grant 15816, and Western Dgtal Corporaton 113