Error Correction and Partial Information Rewriting for Flash Memories

Transcription

1 Error Correcion and Parial Informaion Rewriing for Flash Memories Yue Li, Anxiao (Andrew) Jiang, and Jehoshua Bruck Texas A&M Universiy, College Saion, TX 7783, USA California Insiue of Technology, Pasadena, CA 95, USA {yli, Absrac This paper considers he parial informaion rewriing problem for flash memories In his problem, he sae of informaion can only be updaed o a limied number of new saes, and errors may occur in memory cells beween wo adjacen updaes We propose wo coding schemes based on he models of rajecory codes The bounds on achievable code raes are shown using polar WOM coding Our schemes generalize he exising rewriing codes in muliple ways, and can be applied o various pracical scenarios such as file ediing, log-based file sysems and file synchronizaion sysems I INTRODUCTION Flash memories are asymmeric in he sense ha programming a cell from 0 o is efficien while bringing a cell from o 0 needs o erase a whole block of cells, which is cosly and degrades he qualiy of cells Moreover, o achieve higher sorage densiy, he geomery of flash chips coninuously shrink, making daa more prone o noise This paper sudies codes ha combine error correcion and rewriing for miigaing he endurance and he reliabiliy issues in flash memories A basic way o model flash memories is hrough wrie-once memory (WOM), where a cell level can be increased from low o high bu no vice versa The design of WOM codes has been exensively sudied in lieraure This includes linear codes, abular codes, codes based on projecive geomery, cose coding ec High-rae codes have been developed [9], and codes ha achieve capaciy were proposed very recenly [] [8] To make he WOM model more pracical, he sudy on error correcing WOM (EC-WOM) codes has been iniiaed Codes ha correc a few errors (eg,, or 3) have been proposed [0] [] More recenly, WOM codes ha correc an arbirary number of errors were developed [3] [6] This paper follows he parial informaion rewriing model of rajecory codes [5], where he curren informaion can be changed o a limied number of new saes during each updae Definiion (Parial Rewriing) Le G(V, E) be a direced general rewriing graph ha is srongly conneced Le each verex v V denoe a message M {0, } log V and le π : {0, } log V V be a one-o-one mapping defined by π(m) v Le each edge e E denoe he change beween he messages allowed by each updae Le D be he maximum ou degree of each verex, where D Parial rewriing sores a sequence of N messages (M 0,, M N ) such ha (a) π(m j ) V for j {0,,, N } (b) (π(m j ) π(m j+ )) E for j {0,,, N } This work was suppored in par by he NSF Gran CCF-79 and a gran from Inellecual Venures The model of parial rewriing can be found in many pracical sorage applicaions such as file ediing, log-based file sysems and file synchronizaion sysems In such applicaions, daa end o be frequenly updaed while each updae only makes small changes on he daa Parial rewriing increases he number of block erasures in flash memories and degrades memory performance Noe ha a noiseless channel is assumed in he sudy of rajecory code for parial rewriing [5] The conribuions of his paper are general coding schemes for parial rewriing wih noise, where errors from a binary symmeric channel may occur in cells beween wo adjacen updaes We propose wo specific consrucions based on rajecory codes We show he bounds on achievable code raes based on our previous work on polar EC-WOM codes [6] Our work generalize he exising rewriing codes in muliple ways II PRELIMINARY CONCEPTS This secion inroduces he noisy WOM model, which is an insance of he noisy parial rewriing model in his paper I hen revisis he polar EC-WOM codes [6] and he rajecory codes [5], which he codes of his paper are mainly based on A The Model of Rewriing wih Noise A code for rewriing and error correcion consiss of encoding funcions E, E,, E and decoding funcions D, D,, D Le here be N binary cells Le [z] {,,, z} for ineger z For i [N] and j [], le s i,j, s i,j {0, } denoe he level of he i-h cell immediaely before and afer he j-h wrie, respecively The WOM consrain requires for all i and j, s i,j s i,j Le c i,j {0, } denoe he level of he i-h cell a any ime afer he j-h wrie and before he (j + )-h wrie, when reading of he message M j can happen The error c i,j s i,j {0, } is he error in he i-h cell caused by he binary symmeric channel denoed by BSC(p e ) wih error probabiliy p e (Here is an XOR funcion) For j [], he encoding funcion E j changes he cell levels from s j = (s,j, s,j,, s N,j ) o s j = (s,j, s,j,, s N,j ) given he iniial cell sae s j and he message o sore M j (Namely, E j (s j, M j ) = s j ) When he reading of M j happens, he decoding funcion D j recovers he message M j given he noisy cell sae c j = (c,j, c,j,, c N,j ) (Namely, D j (c j ) = M j ) For j [], define insananeous rae of he j-h wrie as R j Mj N, where M j is he number of bis in M j The sumrae is defined as R sum j [] R j When here is no noise, he maximum sum-rae (ie capaciy) of WOM is known o be log ( + ) bis per cell However, for he noisy WOM described above, he exac capaciy is sill largely unknown [] //$300 0 IEEE 087

2 B Polar EC-WOM Codes Polar EC-WOM codes combine rewriing and error correcion wih efficien encoding and decoding algorihms [6] The scheme exends he consrucions by Burshein and Srugaski [], which is based on polar codes and achieves he capaciy of noiseless WOM wih echniques relaed o lossy source coding [7] In [], a WOM channel is designed for each wrie such ha he WOM consrain is saisfied and he capaciy of he channel can mach he insananeous rae A polar code is hen consruced for each WOM channel and used for encoding The polar WOM code was exended o correc errors [6] The code in [6] is shown o have good performance when he frozen ses of he polar codes respecively consruced for he encoding channel and he decoding channel have subsanial overlapping, he exisence of which is analyzed boh analyically and experimenally C Trajecory Codes Trajecory codes are rewriing codes ha are asympoically opimal for noiseless parial rewriing [5] Given he rewriing graph G(V, E), le L = V, divide a group of n binary cells ino C subgroups For i {0,,, C }, le he i-h subgroup have n i cells and be referred o as regiser r i, namely, n = C i=0 n i A regiser sores a -wrie WOM code Le s j = (s,j, s,j,, s n,j ) and s j = (s,j, s,j,, s n,j) be he cell saes immediaely before and afer soring M j A rajecory code has C encoders E 0, E,, E C and decoders D 0, D,, D C, supporing N = C message updaes For j {0,,, C }, he encoder E j : {0, } n {0, } log L G(V, E) {0, } n compues he new cell saes from he message, he curren sae and G(V, E) (namely, E j (s j, M j, G(V, E)) = s j) and he decoder D j : {0, } n G(V, E) {0, } log L reads he message M j from he curren cell sae a any ime beween he j-h and he (j + )-h updaes (Namely, D j (s j, G(V, E)) = M j ) The C updaes are performed using a differenial scheme: he firs message M 0 is sored in r 0 To wrie message M, we compue he label {0, } log D of he edge π(m 0 ) π(m ) in G(V, E), and sore in r (Insead of labeling edges globally, each ougoing edge of a verex is given a local label ha coss log D bis, where D is he maximum ou degree) The nex C updaes can be wrien in he same way Afer r C is used, an updae cycle is compleed, and he regiser r 0 will be rewrien wih he new log L-bi message for he nex updae The ieraion coninues unil he las updae is finished The consrucion implies he consrain ha for all j, and for all i such ha he i-h cell belongs o r j mod C, we have s i,j s i,j The code rae of rajecory codes is R C log L n bis/cell () III ERROR CORRECTING TRAJECTORY CODES We sudy he coding problem for join parial rewriing and error correcion, where he parial rewriing model is exended by allowing cell saes o be changed by noise beween wo adjacen updaes In flash memories, he noise is from various sources such as inerference and charge leakage [] A Error Model and WOM Parameers Before we presen he code consrucion, we firs inroduce he relaed model and parameers Le he noise channel for he errors received by a regiser beween wo adjacen updaes (eg he ime period afer soring M 3 and before soring M ) be a binary symmeric channel BSC(p) wih p (0, ) We assume ha errors sar occurring in a regiser afer he regiser is wrien for he firs ime The assumpion is moivaed by pracical flash memories, where he major errors for rewriing are inroduced by cell-o-cell inerference ha happens mainly when cells are being programmed [] Following he model of rajecory codes, he noise channel ha a regiser goes hrough a he ime immediaely before is nex WOM rewrie is BSC(p C ), where p C is he overall error probabiliy of C cascaded BSC(p) compued using p i ( p)i In WOM, i is common o use some parameers o conrol he amoun of informaion ha is wrien in each wrie For j [], le he parameer α i,j be he fracion of cells ha are a sae 0 immediaely before he j-h wrie of he regiser r i s WOM code We have α i,0 = Le he parameer ɛ i,j be he fracion of cells a sae 0 ha will be raised o by he j-h rewrie We have ɛ i, = The parameers of he WOM codes used in our seing of parial rewriing also depends on he error probabiliy p When n i, he values of α i,, α i,,, α i, are compued by α i,j = [α i,j ( ɛ i,j )] p C, where a b a( b) + ( a)b, and he parameers ɛ i,, ɛ i,,, ɛ i, are specified by users B Code Consrucion Our firs consrucion is a naural exension of rajecory codes, where each regiser independenly correcs he errors in i The recovered messages are used by he nex updae We formally presen he consrucion by defining he encoder and he decoder in he following Consrucion For i {0,,, C }, le regiser r i use a -wrie EC-WOM code, correcing he errors from BSC(p (C i) ) Le l = j modc, and le s j be he saes of he n cells righ before he j-h updae, and le s j denoe he cell saes a any ime beween he j-h and he (j + )-h updae (Therefore, s j is he value of s j a a paricular momen) For j = 0,,, C, we have Encoder E j (s j, M j, G(V, E)) = s j If l = 0, rewrie r 0 wih M j Oherwise, do; () Recover message ˆM j = D j (s j, G(V, E)) () Compue he label j s (π( ˆM j ) j π(m j )) E (Here π is specified in Definiion ) (3) Sore he label j in regiser r l using rewriing (ie using he EC-WOM code for r l ) 088

3 3 Decoder D j (s j, G(V, E)) = ˆM j () Decode r 0 and obain he esimaed message ˆM j l Le ˆv j l = π( ˆM j l ) () For k from o l, decode r k and obain he esimaed edge label ˆ j l+k (Noe ha in he rewriing graph G(V, E), he edge from message M j l+k o message M j l+k has he label j l+k ) (3) Compue ˆM j Sar from he verex ˆv j l, raverse G(V, E) along he pah marked by ˆ j l+, ˆ j l+,, ˆ j, which leads o ˆv j Oupu ˆMj = π (ˆv j ) Example 3 We now show a simple example for n = 6 cells (In pracice, he code usually has housands of cells) Le he cells be divided ino C = regisers wih n 0 =, n =, and le =, L = and D = Assume ha beween wo adjacen updaes, an error occurs in each regiser Le he WOM codes of r 0 and r correc and errors, respecively Le he rewriing graph G whose verex and edge ses be defined as V = {v 0, v, v, v 3 }, E = { v 0 v, v () v, v () () v 3, v 3 v 0 }, where he () verex v i represening he symbol i and having wo ougoing edges locally labeled wih and () Le he sequence of messages be (0, 3,, ), which corresponds o he pah () v 0 v 3 v v in he graph Assume ha he changes on he saes of r 0 and r during he updaes are he ones shown in he able below Here j and j + denoe he momens immediaely before and afer he j-h updae, respecively A bi marked wih underlines indicaes an error Noe ha a he momen j =, alhough performing he updae does no require recovery of he messages wrien a momens j = 0 and j =, hose messages can sill be recovered unil he momen j = if needed j r 0 r Commens 0 (0, 0, 0, 0) (0, 0) Iniializaion 0 + (0,, 0, 0) (0, 0) Wroe daa 0 in r 0 (0,,, 0) (0, 0) An error occurs in r 0 + (0,,, 0) (, 0) Decoded r 0, wroe () in r (0, 0,, 0) (0, 0) Errors occur in r 0 and r + (, 0,, 0) (0, 0) Rewroe r 0 o sore 3 (, 0, 0, 0) (0, ) Errors occur in r 0 and r 3 + (, 0, 0, 0) (0, ) Decoded r 0, wroe in r C Analysis of he Correcness of he Consrucion To see he correcness of he coding scheme, we use inducion (Here we assume he number of cells goes o infiniy) Le us assume ha he firs j messages have been sored successfully, and we show ha M j can be recovered reliably a any ime beween he (j )-h and he j-h updae, and he j-h message can be sored successfully Le he index of he regiser be wrien as l = j modc If l = 0, we are a he firs wrie of a new cycle, and do no need o recover M j o sore M j ; if l > 0, we perform he updae by soring he difference j beween M j and M j in r l To do so, we firs recover he value of M j by decoding he regisers r 0, r,, r l which have respecively received errors from he channels BSC(p l ), BSC(p (l ) ),, BSC(p) a he ime of he decoding As heir WOM codes respecively correc errors from BSC(p C ), BSC(p (C ) ),, BSC(p (C l+) ) which are degraded versions of heir curren noise channels, hese regisers can be decoded, oupuing he messages wrien by he las l updaes (which include ˆM j l sored in r 0, and he labels ˆ j l+, ˆ j l+,, ˆ j from r,, r l ) Given G(V, E) we can deermine he value of ˆMj, and furher compue he label j of he edge from π(m j ) o π(m j ) By soring he label j ino r l, he j-h updae succeeds D Code Analysis We analyze he code performance for Consrucion Le L = V For j [], le R i,j > 0 be he achievable insananeous rae of he j-h wrie of he EC-WOM code in r i As each regiser uses a consan-rae WOM code (here regiser r 0 sores log L bis per wrie, and he oher regisers each sores log D bis per wrie), for i [C ] we have n 0 = log L log, n i = D () min j [] R 0,j min j [] R i,j Subsiuing Eq () in Eq () gives he rae of he code R = C min j [] R 0,j + log D log L C i= min j [] R i,j Noe ha he EC-WOM in Consrucion is general To be specific, we can use he polar EC-WOM code in [6] for each regiser, and derived he bounds on R We firs revisi some resuls from [6] ha are needed o derive he bounds o he insananeous raes for he polar EC-WOM code Le he WOM channel used for performing he j-h wrie/encoding of he polar EC-WOM be WOM(α j, ɛ j ) wih he parameers α j and ɛ j, and le he channel of noise in cell saes beween wo adjacen wries be BSC(p e ) Le F WOM(αj,ɛ j) [N] be he frozen se of he polar code consruced for WOM(α j, ɛ j ), and le F BSC(pe) [N] be he frozen se of he code consruced for BSC(p e ) When N, le x j F WOM(αj,ɛ j) F BSC(pe) / F BSC(pe) For j [], he number of bis wrien in he j-h rewrie is M j = F WOM(αj,ɛ j) F WOM(αj,ɛ j) F BSC(pe) = Nα j H(ɛ j ) x j F BSC(pe) = N(α j H(ɛ j ) x j H(p e )) and he number of addiional cells we use o sore he bis in F BSC(p) F WOM(αj,ɛ j) is N addiional,j = N H(pe)( xj) H(p e) Therefore, we ge he insananeous rae for he j-h wrie R j M j N + k= N addiional,j = α j H(ɛ j ) H(p e )x j + H(pe) H(p e) k= ( x k) Lemma [6, Lemma 5] Le 0 < p e α j ɛ j Then x j γ j, where { } αj H( pe α j ) αj H(ɛj) + H(pe) H(αj ɛj) γ j max, H(p e) H(p e) Lemma 5 Le 0 < p e α j ɛ j Then R j [R j, R+ j ], where R j = α j H(ɛ j ) H(p e ) + H(pe) H(p e) k= ( γ k), (3) R + j = α j H(ɛ j ) H(p e )γ j () 089

4 Bound on Achievable Code Rae (bis/cell) Basic LB Basic UB LB D = 3 LB D = 0 UB D = 3 UB D = Fig The lower and upper bounds (marked by LB and UB) on he achievable code raes for differen and D Here log L = 3, C = 8 and p = 0 3 The resuls above can be direcly applied o he codes in Consrucion For i {0,,, C }, le 0 < p (C i) α i,j ɛ i,j, hen R i,j [R i,j, R+ i,j ] where R i,j and R+ i,j are compued wih he righ hand sides of Eq (3) and () by replacing α j, ɛ j and p e wih α i,j, ɛ j and p (C i) Theorem 6 For i {0,,, C }, j [], le 0 < p (C i) α i,j ɛ i,j Then R {R, R + } where R = min j [] R 0,j C + log D log L C i= min j [] R i,j and he upper bound R + can be compued by replacing R 0,j and R i,j in he above equaion wih R+ 0,j and R+ i,j, respecively Figure shows some numerical resuls for he bounds of our code, where for all i, j we le ɛ i,j = /( + j) To show he benefi obained by aking advanage of he parial rewriing consrains, we compare he bounds of our scheme o hose of he basic scheme, which is simply a C-wrie polar EC-WOM code correcing errors from BSC(p) In each rewrie, he basic scheme sores each updaed message using rewriing The resuls sugges our code performs significanly beer han he basic scheme (Noe ha he WOM codes considered in his paper are consan rae codes Given such codes, he bounds in Figure decreases when becomes sufficienly large due o he drop in he insananeous raes) IV A MORE GENERALIZED CODING SCHEME We now discuss a more generalized coding scheme In his scheme, he rajecory codes no only use regisers o sore he changes in he messages, bu can also sore par of he errors found in previous regisers When he error probabiliy of he channel is small, only a small number of addiional cells are needed o sore such error informaion We focus on a specific consrucion in he following A Code Consrucion Le he error-free cell saes of regiser r i (immediaely afer i is wrien) be c 0 i {0, } ni Le he cell saes immediaely before each of he nex C updaes of messages be c i, c i,, cc i According o he error model in Secion III, he error vecor c k i ck+ i conains he errors inroduced by BSC(p) When n i, he vecor c k i ck+ i can be compressed ino n i H(p) bis using lossless source coding The encoder and he decoder for he j-h updae in he new consrucion are defined below Consrucion 7 For i {0,,, C }, le regiser r i use a -wrie EC-WOM code, correcing he errors from BSC(p) For j = 0,,, C, we have Encoder E j (s j, M j, G(V, E)) = s j If l = 0, rewrie r 0 wih M j Oherwise, do; () Recover message ˆM j = D j (s j, G(V, E)) () Compue he label j s (π( ˆM j ) j π(m j )) E (3) Rewrie regiser r j o sore j and he compressed version of he error vecors c 0 l c l, c l c l,, cl 0 c l 0 Decoder D j (s j, G(V, E)) = ˆM j () For k from 0 o l, le he sae of regiser r l k be c k+ l k Using i and he error vecors obained previously from decoding r l k+, r l k+,, r l, we ge c k+ l k k x=0 (cx l k cx+ l k ) = c0 l k (ck l k ck+ l k ) (Noe ha when k = 0, he above equals c l ) Decode he righ hand side of he above equaion, and obain he recorded error vecors abou he firs (l k) regisers c l k 0 c l+ k 0, c l k c l k,, c 0 l k c l k he esimaed message ˆM j l (when k = l) or he esimaed edge label ˆ j k (when k < l) () We now compue ˆM j ; we raverse he graph G(V, E) along he pah marked by he labels ˆ j l+, ˆ j l+,, ˆ j, which leads o verex ˆv j Oupu ˆMj = π (ˆv j ) Example 8 Le n = 0,, C = 3, L = and D = Assume n 0 = 3, n = 3, n =, and ha he WOM code of each regiser correcs error Assume ha beween wo adjacen updaes, an error occurs in each regiser We assume he same rewriing graph as in Example 3, and le (0, 3, ) be he firs hree messages o be sored We only illusrae he updae for he message due o space limiaion Assume he changes in he cell saes during he updaes are as in he able below A ime, errors occur in r 0 and r To perform he updae, we firs decode r, and obain he label () and he decompressed error vecor c 0 0 c 0 = (0, 0, ) for r 0 Given he error vecor and he curren sae c 0, compue c 0 (c 0 0 c 0) = (0, 0, 0) where he middle bi sill conains error Decoding c 0 (c 0 0 c 0) gives he message 0 Given he new message and he recovered label () and he message 0 in r 0, he label is deermined and sored by wriing in r, which complees he updae j r 0 r r 0 (0, 0, 0) (0, 0, 0) (0, 0, 0, 0) 0 + (0,, 0) (0, 0, 0) (0, 0, 0, 0) (0,, ) (0, 0, 0) (0, 0, 0, 0) + (0,, ) (, 0, 0) (0, 0, 0, 0) (0, 0, ) (,, 0) (0, 0, 0, 0) + (0, 0, ) (,, 0) (, 0,, 0) B Analysis of he Correcness of he Code The correcness of Consrucion 7 can be shown by inducion (We again assume he number of cells in each regiser goes o infiniy) Assume he firs j messages have been 090

5 5 sored successfully, and we elaborae on he j-h updae wih l > 0 To perform he updae, we need o recover he message M j so ha he label of he edge from M j o M j can be compued and sored in r l We firs decode r l wih sae c l which received he errors from BSC(p) Since each regiser oleraes errors from BSC(p), r l can be decoded o obain he edge label j (ha specifies he edge connecing M j o M j ) as well as he error vecors c 0 l c l, c l 3 c l 3,, cl 0 c l 0 wih each error vecor being for one of he firs l regisers Nex, we decode r l wih sae c l To do so, we firs use he error vecor obained previously on r l o correc par of he errors by compuing c l (c0 l c l ) The remaining errors can be equivalenly seen as coming from BSC(p), and are hus correcable Decoding hem gives he edge label j as well as he error vecors regarding he previous regisers We coninue he join decoding in he same fashion owards r 0 Thanks o he error vecors from he previous decoding, each regiser needs o correc errors from BSC(p) (insead of BSC(p (C i) ) for i = 0,,, C ) Afer r 0 is decoded, we obain he message M j l and he labels j l+, j l+,, l By raversing G(V, E) along he pah marked by he labels, we recover ˆM j The label j is hen deermined and wrien ino r l C Code Analysis We analyze he code performance of Consrucion 7 The analysis is differen from Consrucion mainly for wo reasons The EC-WOM code of each regiser for he codes of his secion correcs he errors from BSC(p) while each WOM code oleraes differen amoun of noise in he previous consrucion Moreover, since each regiser (besides r 0 ) sores boh error vecors as well as an edge label, he value of n i also depends on n 0, n,, n i We firs derive n i for each r i As r 0 is used in he same way as he previous codes, and r i sores i error vecors and one edge label in each wrie, we have n 0 = log L/ min j [] R 0,j and n i = (log D + H(p) i k=0 n k)/ min j [] R i,j for i [C ] Here he erm H(p) i k=0 n k is he lengh of he compressed error vecors c 0 i c i, c i c i,, ci 0 c i 0 In pracice, each regiser can choose o use he WOM code wih he same parameers o simplify he implemenaion In such cases, (n, n,, n C ) form a geomeric sequence Proposiion 9 For i {,,, C }, le min j [] R i,j be some consan A Then we have n i = (n 0 H(p) + log D)(A + H(p)) i /A i Therefore, he rae of he code in his secion can be compued using Eq () To derive he bounds for Consrucion 7, we apply he same echniques used in Secion III Assume each WOM code in is a polar EC-WOM code which correcs errors from BSC(p) By applying Lemma 5, we show he bounds o he insananeous raes R i,j in he nex lemma Lemma 0 For i {0,,, C } and j [], le 0 < p α i,j ɛ i,j Then we have [ R i,j [R i,j, R+ i,j ], where R i,j = [α i,j H(ɛ i,j ) H(p)]/ + H(p) ] H(p) j= ( γ i,j) and R + i,j = α i,j H(ɛ i,j ) H(p)γ i,j Theorem For all i and j, le 0 < p α i,j [ ɛ i,j Then we have R [R, R + ], where R log = C log L/ L + min j [] R ] 0,j, and R + can be compued by log D+H(p) i k=0 n k i [C ] min j [] R i,j replacing R (0, j) and R (i, j) in R above wih R + (0, j) and R + (i, j), respecively Figure shows he numerical resuls ha compare he bounds of Consrucion and Consrucion 7 on parameers ha are common for flash memories (eg message lengh > 000 bis) The bounds for he codes in his secion are Bound on Achievable Code Rae (bis/cell) Cons LB C=5 Cons UB C=5 Cons 7 LB C=5 Cons 7 UB C= Cons LB C=5 Cons UB C=5 Cons 7 LB C=5 Cons 7 UB C=5 Fig The bounds o he achievable raes of he wo consrucions on differen and C Here log L = 3, p = 0 3, and ɛ i,j = + j igher han hose of he previous consrucion When is sufficienly large, all bounds will decrease due o he decrease of he minimum insananeous raes However, he bounds of he codes in his secion decrease more slowly This is because in he firs consrucion, he WOM code of r i needs o olerae he errors from BSC(p (C i) ) Is error raes become much higher han wha he codes in his secion needs o olerae (which is BSC(p)) when C becomes large Therefore, he minimum insananeous raes of he WOM codes in he previous scheme decrease faser when increases han hose of he codes in his secion do REFERENCES [] D Burshein and A Srugaski, Polar wrie once memory codes, IEEE Trans IT, vol 59, no 8, pp , 03 [] Y Cai, E F Harasch, O Mulu, and K Mai, Error paerns in MLC NAND flash memory: Measuremen, characerizaion, and analysis, in Proceedings of he Conference on Design, Auomaion and Tes in Europe, 0, pp 5 56 [3] E En Gad, Y Li, J Kliewer, M Langberg, A Jiang, and J Bruck, Polar coding for noisy wrie-once memories, in Proc ISIT, 0 [] C Heegard, On he capaciy of permanen memory, IEEE Trans Informaion Theory, vol 3, no, pp 3, January 985 [5] A Jiang, M Langberg, M Schwarz, and J Bruck, Trajecory codes for flash memory, IEEE Trans IT, vol 59, no 7, pp 530 5, 03 [6] A Jiang, Y Li, E En Gad, M Langberg, and J Bruck, Join rewriing and error correcion in wrie-once memories, in Proc ISIT, 03, pp [7] S Korada and R Urbanke, Polar codes are opimal for lossy source coding, 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