Constrained Codes for Phase-change Memories

Transcription

1 2010 IEEE Iformaio Theory Workshop - ITW 2010 Dubli Cosraied Codes for Phase-chage Memories Axiao Adrew Jiag Compuer Sciece ad Eg. Dep. Texas A&M Uiversiy College Saio, TX ajiag@cse.amu.edu Jehoshua Bruck Elecrical Egieerig Deparme Califoria Isiue of Techology Pasadea, CA bruck@calech.edu Hao Li Compuer Sciece ad Eg. Dep. Texas A&M Uiversiy College Saio, TX hao@cse.amu.edu Absrac Phase-chage memories PCMs are a impora emergig o-volaile memory echology ha uses amorphous ad crysallie cell saes o sore daa. The cell saes are swiched usig high emperaures. As he semi-sable saes of PCM cells are sesiive o emperaures, scalig dow cell sizes ca brig sigifica challeges. We cosider wo poeial hermal-based ierferece problems as he cell desiy approaches is limi, ad sudy ew cosraied codes for hem Fig. 1. The rasiios amog q cell levels. Here q = 4. The forward ad backward edges represe he SET ad RESET operaios, respecively. I. INTRODUCTION Phase-chage memories PCMs are a impora emergig o-volaile memory NVM echology. Is basic sorage ui, a PCM cell, has a leas wo saes: he amorphous sae ad he crysallie sae. To achieve higher sorage capaciy, mulilevel cells MLCs are beig developed, where addiioal parially crysallie saes are used [1]. We model he q 2 saes of a PCM cell by q levels levels 0,1,...,q 1 where level 0 is he amorphous sae, level 1,...,q 2 are he parially crysallie saes, ad level q 1 is he crysallie sae. As a cell becomes more crysallized, is level icreases. The level of a PCM cell is swiched usig high emperaures. A cell ca be heaed by a high cell-melig emperaure abou 600 o C 700 o C o chage o level 0 amorphous sae, or be heaed by a more moderae emperaure o icrease is level i.e., o a more crysallized sae. To model he direc swichig of saes, we use he diagram i Fig. 1 for q = 4 as a example []. We see ha he cell ca be chaged from ay level i {1,2,...,q 1} direcly o level 0 called a RESET operaio, ad from ay level i direcly o level j for 0 i < j q 1 called a SET operaio. However, for 0 < j < i q 1, o chage i from level i o level j, boh he RESET ad SET operaios are eeded. PCMs are uder acive sudy ad developme due o heir very aracive poeials. Compared o he widely used flash memories, PCMs ca poeially scale o much smaller cell sizes ad achieve higher sorage capaciy. They ca also have subsaially beer edurace, daa reeio ad read/wrie speed [1]. However, scalig dow cell sizes ca also brig sigifica challeges, ad solvig hem will be key o he PCM developme [1]. A major ad widely ackowledged challege is he hermal issue, because he amorphous ad parially-crysallie saes are oly semi-sable saes, ad high eviromeal emperaures ca furher crysalize he cell, i.e., uieioally icrease he cell level [1], [5]. We cosider wo poeial challeges whe he PCM s cell desiy scales oward is limi. The firs oe is he hermal crossalk problem, amely, whe a cell is RESET o level 0 by he high melig emperaure, he hea affecs is adjace cell ad makes i furher crysalized [1], [5]. Noe ha his may happe boh whe he adjace cell is o beig programmed ad whe i is beig SET, uless i is already i he fully crysalized sae level q 1. I is because i boh cases, he semi-sable cell sae is sesiive o high emperaures. The secod problem is he local hermal accumulaio problem. Whe cells are repeaedly programmed, he hea ca accumulae i he area [5]. This residual hea ca be a major facor ha limis he wriig badwidh of PCMs, because he wriig accuracy is sesiive o emperaure [1], [5]. Whe he cell desiy scales oward is limi, relaive o he high I/O speed, i ca ake orivial ime for he locally geeraed hea o spread ou uiformly i he memory chip. So if a applicaio repeaedly wries a cluser of adjace cells a very high speed, he accumulaed hea may appear localized [5]. I ha case, i is worh cosiderig wheher here exis schemes ha ca make he hermal accumulaio more balaced. The above problems have bee cosidered i [1], [5] from he device ad sysem perspecives. I his paper, we cosider codig echiques for hese poeial challeges. For he hermal crossalk problem, we use a scheme ha removes he crossalk ierferece, ad he sudy codig echiques ha reduce he programmig cos measured by he umber of RESET operaios. For he local hermal accumulaio problem, we sudy codig echiques ha impose ime ad space cosrais o wriig, o help he hea geeraed by programmig be more balaced spaially. Cosraied codig is a impora area of codig echiques, ad has bee applied successfully o boh mageic recordig ad opical recordig [4]. Compared o coveioal cosraied codig, he codes sudied i his paper are for a differe seig ad someimes require very differe codig /10/$ IEEE

2 echiques. I is also oable ha for PCMs, he cosrais are o limied o jus wihi a codeword. They are iroduced by he differece bewee old codewords ad he ew codewords ha overwrie hem, because oly cells ha are programmed ca geerae hea, which may affec oher cells. The res of he paper is orgaized as follows. I Secio II, we prese symbol-cosraied codes for he hermal crossalk problem. I Secio III, we prese space-ime cosraied codes for he local hermal accumulaio problem. I Secio IV, we prese he cocludig remarks. Due o space limiaio, we leave some deailed aalysis i he full paper [2]. II. SYMBOL-CONSTRAINED CODES I his secio, we sudy codig echiques for he hermal crossalk problem. Le c 1,c 2,...,c be cells. For i {1,2,...,} [], le l i {0,1,...,q 1} deoe he level of c i. I his paper, we cosider he cells as a oedimesioal array. The coceps ca be exeded o higher dimesios, oo. Two cells c i ad c j are eighbors if ad oly if i j = 1. Le γ {1,2,...,q 1} be a parameer. We cosider he followig simple model for hermal crossalk: Whe a cell c i is RESET o level 0, he hermal crossalk from c i a ha mome will icrease is eighborig cell level l j for j = i ± 1 by a mos γ, uless he eighborig cell c j is also beig RESET a ha mome. However, a cell level cao exceed q 1, he sable fullycrysallie sae. Ad a SET operaio does o affec he eighborig cells due o is cosiderably lower emperaure. Le C {0,1,...,q 1} be a code, whose rae RC is defied as log 2 C. Clearly, RC log 2 q. A rewrie is o chage he cell levels from he curre codeword X = x 1,...,x C o a ew codeword Y = y 1,...,y C. For i [], if x i > y i, he he rewrie eeds o RESET c i ad he SET c i if y i > 0. For j = i±1, if c i is RESET ad y j < mi{x j + γ,q 1}, he he rewrie eeds o RESET c j as well, because oherwise he hermal crossalk from c i may make l j greaer ha y j. Therefore, a RESET operaio applied o a cell ca rigger he RESET of is eighborig cell, ad his effec ca propagae o may cells. Le us defie a RESET segme i codeword Y as a maximum ru of symbols y i,y i+1,...,y j where 1 i j such ha: 1 i {i,...,j}, y i < mi{x i +γ,q 1}; 2 i {i,...,j} such ha x i > y i. By our above aalysis, he rewrie mus RESET all he cells i a RESET segme before seig hem. To rewrie cells usig parallel programmig, i is aural o use he followig wo-sep procedure: Firs, RESET all he cells i RESET segmes of he ew codeword; he, SET all he cells whose levels are sill lower ha heir values i he ew codeword. The secod sep has o crossalk effec. Example 1. Le = 11, q = 4 ad γ =. Assume he cells eed o chage from a old codeword 1,,2,2,2,2,2,2,1,1,1 o a ew codeword 0,,2,2,1,2,2,2,,1,2. Firs, we RESET he cells {c 1,c,c 4,c 5,c 6,c 7,c 8 }. Afer his sep, he cell levels will be 0,,0,0,0,0,0,0,l 9,1,1, where l 9 {1,2,}. Sice cell c 8 is RESET, he hermal crossalk from c 8 may make l 9 be greaer ha is origial value 1. The, we SET he cells {c,c 4,c 5,c 6,c 7,c 8,c 9,c 11 }, o icrease he cell levels o 0,,2,2,1,2,2,2,,1,2. We defie he cos of a rewrie operaio as he umber of cells ha are RESET durig rewriig. I Example 1, he cos is 7. The umber of RESETs is a very impora cos measureme because PCM cells have a limied logeviy: PCM cells ca edure abou RESETs or SET-RESET cycles before becomig o-fucioal [1], []. Noe ha for he rewrie, for every cell ha eeds o decrease is level, he whole RESET segme coaiig i is forced riggered o be RESET, oo. This moivaes us o sudy cosraied codes where RESET segmes have limied leghs. Give his cosrai, we seek capaciy-achievig codes. I his paper, we focus o he case γ = q 1 he worscase sceario for hermal crossalk. We defie a usable segme i a codeword X = x 1,...,x as a maximum ru of symbols x i,x i+1,...,x j where 1 i j such ha for i {i,...,j}, x i < q 1. The legh of his usable segme is j i + 1. Whe he memory wries X, a usable segme i i will become a RESET segme if ay of he cells i ha usable segme eeds o decrease is level compared o he old codeword. For a code C, if i all is codewords he usable segmes leghs are a mos k, he durig rewriig, he legh of every RESET segme is a mos k. Defiiio 2. SYMBOL-CONSTRAINED CODES Le k be a posiive ieger. A code C {0,1,...,q 1} is k-limied if i every codeword of C, every usable-segme s legh is a mos k. I is also called a symbol-cosraied code. The k-limied codes are a cosraied sysem S over alphabe Σ {0,1,...,q 1}. A example for q = 4,k = is show i Fig. 2. We see ha i geeralizes he d = 0,k- ru-legh-limied RLL codes [4] from he biary alphabe o he q-ary alphabe. Is Shao capaciy is caps = lim sup 1 log N; S, where N; S is he umber of words of legh i S. We have cosruced a k-limied code for q = 4 ad k = 1, which has a rae 6 : 5 fiie-sae ecoder. Is rae is 1.2 bis/cell, close o he Shao capaciy which ca be show o be Whe he code is used for sorig daa, assumig ha he ipu iformaio bis have a uiform i.i.d. disribuio, for every rewrie, he raio of he average umber of RESET operaios o he umber of iformaio bis is This compares favorably wih he o-codig mehod i.e., sorig 2 bis per cell, which has a higher raio of So symbol-cosraied codes ca reduce RESETs. For more deails of he code, see [2]. We oe ha rewriig codes for reducig he RESET operaios for PCMs have bee sudied i [], where ieresig WOM-like codes have bee used. However, he sudy i [] did o cosider ay hermal ierferece problem. We also sress ha he codes i [] ad he codes we sudy are wo drasically differe approaches. While he codes i [] always RESET

3 Fig. 2. Shao cover of he -limied codes cosraied sysem, for q = 4. TABLE I SHANNON CAPACITY BITS PER CELL OF k-limited CODES q\k all cells a he same ime o ge a fresh sar for rewriig, we use cosraied codes ha are based o local cosrais, ad cells are mos likely RESET i differe rewries. Ad wih our cosraied-codig approach, slide-block decoders ca be buil o locally decode iformaio bis efficiely. The followig heorem preses he Shao capaciy of he symbol-cosraied codes, for arbirary q ad k. Due o he space limiaio, we prese is full proof i [2]. Theorem. Le q 2 ad k 1 be iegers. Le fλ = λ k+2 qλ k+1 + q 1 k+1. The equaio fλ = 0 has a mos hree real-valued soluios, oe of which is q 1. Amog hose real-valued soluios, if q = k + 2, le λ be he soluio wih he greaes absolue value; oherwise, amog he a mos wo real-valued soluios uequal o q 1, le λ be he soluio wih he greaer absolue value. The he Shao capaciy of k-limied codes is log 2 λ bis per cell. Based o Theorem, he Shao capaciy of symbolcosraied codes for differe q ad k are show i Table I. III. SPACE-TIME CONSTRAINED CODES I his secio, we sudy codig echiques for a differe ierferece problem: he local hermal accumulaio problem. I is kow ha whe cells are repeaedly programmed, adjace cells ca be crysallized/disurbed [5]. We seek codes for rewriig daa ha ca balace hea beer. This moivaes us o sudy he space-ime cosraied codes defied below. Le c 1,...,c be PCM cells, whose levels are deoed by l 1,...,l {0,...,q 1}. Le V = {0,1,...,v 1} be a alphabe of size v. The daa sored i he cells akes is value from he alphabe V. A code C is a mappig from he cell levels L l 1,...,l {0,...,q 1} o he daa values V. We allow i o be a may-o-oe mappig isead of a oeo-oe mappig. The code C has wo associaed fucios: a decodig fucio F d ad a updae fucio F u. The decodig fucio F d : {0,...,q 1} V ells us ha he cell levels L represe he daa F d L V. The updae fucio F u : {0,...,q 1} V {0,...,q 1} ells us ha if he old cell levels are L ad we wa o wrie he ew daa s V io he cells, we will chage he cell levels o F u L,s. Clearly, we should have F d F u L,s = s. A rewrie ca chage he daa o ay value i V. Here we do o cosider he hermal crossalk problem. So whe a rewrie chages a old codeword X = x 1,...,x {0,...,q 1} o a ew codeword Y = y 1,...,y, for i [], a cell c i eeds o be programmed oly if x i y i. We defie he rewrie cos as he umber of cells ha are programmed, {i [] x i y i }, which is he Hammig disace bewee X ad Y. To balace programmig-geeraed hea, we sudy he followig code. 1 Defiiio 4. SPACE-TIME CONSTRAINED CODES Le α, β, p be posiive iegers. A code is α, β, p- cosraied if for ay α cosecuive rewries ad for ay segme of β cells amely, c i,c i+1,...,c i+β 1 for some i [] he oal rewrie cos of hose β cells over hose α rewries is a mos p. I is also called a space-ime cosraied code. We oe ha he space-ime cosraied codes are ieresig because alhough he sysem ca keep movig daa ha are frequely rewrie o balace hea, such a approach may cause subsaial overhead for file-sysem/compiler desig ad heir opimizaio. Ad for coe-addressable sysems, where he address of daa is deermied by he coe of he daa e.g., by usig a hash fucio for fas daa rerieval, relocaig daa ca also be very challegig. I his paper, as he sarig poi of udersadig space-ime cosraied codes, we sudy he ime ad space cosrais separaely. A. Time-cosraied Codes We firs sudy ime-cosraied codes wih α 1,β = 1,p = 1. This is he simple case where every cell ca be programmed a mos oce i every α cosecuive rewries. Noe ha he rae of he code C is defied as log 2 v bis per cell. I is easy o see ha a simple idea based o ime divisio ca give us a code of rae log 2 q α bis per cell, as follows: Le = α log q v, ad divide he cells evely io α groups call hem he 0h, 1s, 2d,..., α 1-h cell groups; for i = 1,2,, for he i-h rewrie we wrie he daa io he i mod α-h cell group. Whe which also meas 1 The model ca be geeralized by differeiaig he cos of RESET ad SET operaios. I PCMs, he RESET operaio uses a higher emperaure ha he SET operaio, bu has a shorer duraio of ime.

4 v, he code rae approaches log 2 q α bis per cell. So he quesio is if here exis codes of higher raes. Noe ha he challege for desigig ime-cosraied codes is ha we cao afford o remember for every cell how log ago he cell was programmed for he las ime up o α pas rewries, because ha aloe will cos log 2 α bis of sorage space for every cell. Cosider he case q = 2. So he programmig of cells eeds o be sychroized i some way so ha his iformaio cos ca be reduced. We ow prese a geeral ime-cosraied code cosrucio for q = 2 ha uses he wrie-oce memory WOM codes [6] as sub-codes. Le D be a WOM code ha sores daa of alphabe size w i m cells of q = 2 levels. Deoe he alphabe of he sored daa by W = {0,1,...,w 1}. The code D also has a decodig fucio F dd : {0,1} m W ad a updae fucio F ud : {0,1} m W {0,1} m. WOM codes have a uique propery: wih every rewrie, he cell levels ca oly icrease, o decrease [6]. Le deoe he umber of rewries he code D ca guaraee o suppor. Le he iiial cell levels all be zero. Clearly, due o he uique propery of WOM codes, is a fiie umber. Example 5. Le w = 4, m =, q = 2. Le L l 1,l 2,l {0,1} deoe he hree cell levels. The followig WOM code D was preseed by Rives ad Shamir [6] wih = 2: L F dd L If he = 2 rewries firs wrie he daa as 2, he rewrie i as 1, he code will firs le L be 0,1,0, he chage i o 0,1,1. Le E be a elevaor code ha mimics D bu allows he cell levels o icrease ad decrease i a sychroized way, described as follows. E also sores daa of alphabe size w i m cells of q = 2 levels. Plaily speakig, for he firs α rewries, E rewries daa i he same way as D; he i pushes all he m cell levels o q 1 = 1; for he ex α rewries, E rewries daa by decreasig cell levels, i exacly he opposie way of D; he i pushes all he m cell levels o 0; he he hird bach of α rewries are implemeed i he same way as D agai; ad so o. We ow formally defie he decodig fucio F de ad he updae fucio F ue of E. Le us call a sequece of rewries he 0h, 1s, 2d, rd rewries. For i = 0,1,2..., le L i deoe he cell levels afer he i-h rewrie, ad le e i W deoe he daa ha he i-h rewrie wries io he cells. Clearly, we should have F de L i = e i. The if 0 i mod 2α α 1, F de L i = F dd L i ; oherwise, F de L i = F dd q 1,,q 1 L i. If i 0 mod 2α, F ue L i 1,e i = FuD 0,,0,e i. If 1 i mod 2α α 1, F ue L i 1,e i = F ud L i 1,e i. If i α mod 2α, FuE L i 1,e i = q 1,...,q 1 F ud 0,,0,e i. If α + 1 i mod 2α 2α 1, F ue L i 1,e i = q 1,...,q 1 F ud q 1,,q 1 L i 1,e i. Example 6. Le D be he WOM code i Example 5, ad le E be he elevaor code defied as above. The whe he rewries chage he daa as , he code E chages he cell levels as 0, 0, 0 1,0,0 1,0,1 1, 1, 1 1,1,0 0,1,0 0, 0, 0 0,0,1 0,1,1 1, 1, 1 We ow cosruc he code C ha sores daa of alphabe size v i cells of q = 2 levels. Le v = w, where gcd, α is he greaes commo divisor of ad α. Le = m+α. We see he sored daa as a vecor X = x 1,x 2,...,x {0,1,...,w 1}. For i = 0,1,2, le X i deoe he daa vecor ha he i-h rewrie wries io he cells. We divide he +α cells evely io groups, ad call hem he 0h, 1s,, gcd,α 1-h groups. Every cell group has m cells. We impleme a sequece rewries as follows. For i i = 0,1,2..., le gi =. The for he i-h rewrie, he elemes of X i are, respecively, wrie +α +α io he gi mod -h, gi + 1 mod - h,, gi + 1 mod +α -h cell groups. Afer he rewrie, he daa ca be decoded from hose cell groups as well. Every cell group uses he elevaor code E o rewrie daa. For a cell group, afer i is used for cosecuive rewries, all is cell levels will be pushed o zero or q 1 whe he ex rewrie comes. The i will res for α 1 rewries. We ca see ha every cell group will be programmed i +1 cosecuive rewries, he o programmed for aoher α 1 cosecuive rewries, ad he repea his process. I such a period of +α rewries, he cell levels are eiher all icreasig or all decreasig; sice q = 2, every cell ca be programmed oly oce. So every cell is programmed a mos oce for every α cosecuive rewries. So C is a ime-cosraied code. The oly deail lef o specify is how o kow he value of i mod 2+α whe he i-h rewrie happes, which is eeded i he above codig process. I is used o compue gi ad o impleme he elevaor code. This value ca be obaied by usig a simple couer of 2+α cells of q = 2 levels. Le l 1,l 2,...,l 2+α deoe heir levels. We cyclically program he cells; for every rewrie, we chage he level of oe cell. We see 2+α 1 j=1 l j l 2+α equals i mod 2 + α. So we ca ge he waed value, ad every cell i he couer is programmed exac oce for every 2 + α > α rewries. Le w, fix as a cosa, ad we choose he smalles m such he WOM code D exiss. By he kow resuls o WOM codes [6], whe = 2, m 1.294log 2 w; whe =, m 1.549log 2 w; ; for sufficiely large, m is lim w log 2 log 2 w. The rae of he ime-cosraied code C log 2 v +2+α = lim w = +α log 2 w m. log 2 w m+α By he kow values of log 2 w m [6], we show he rae of C i he followig able, ad compare i wih 1 α he rae of he code usig ime sharig. We see ha he code C ca achieve a higher rae.

5 α /α = rae = of C = = The followig heorem preses a upper boud o he rae of ime-cosraied codes. Due o he space limiaio, we prese is skeched proof. For is full proof, please refer o [2]. Theorem 7. Defie v max as α 1 v max max q 1 i. =1,2,..., α i i=0 The he rae of α,1,1-cosraied codes ha use cells of q levels is upper bouded by log 2 v max / bis per cell. Proof: We prese he skech of he proof here. Give a ime-cosraied code C ha sores daa of alphabe size v i cells of q levels, we greedily selec a sequece of rewries such ha a each rewrie, he umber of programmed cells is locally maximized. For i = 1,2,, le δ i deoe he umber of cells programmed by he i-h rewrie. Cosruc a sequece of posiive iegers α+2,..., 1, 0, 1, 2,,... his way. Firs, le 0 = 1 = = α+2 = 0; he, for i = 1,2,, le i be he smalles ieger such ha v i P i 1 j=i α+1 j k=0 k q 1 k. Based o he greedily chose rewrie sequece, i ca be proved by iducio ha for i = 1,2,..., δ i i. I ca also be show ha for i > j 1, i j. Sice he sequece 1, 2, cao keep sricly moooically icreasig, here mus exis posiive iegers i ad such ha whe i i, i =. Based o he way ha 1, 2, are defied, we see ha he heorem holds. B. Space-cosraied Codes We ow sudy space-cosraied codes wih α = 1, β 1 ad p = 1. This is he simple case where for every segme of β cells amely, c i,c i+1,c i+β 1 for some i [] a rewrie will program a mos oe cell i he segme. Noe ha he code uses cells of q levels o sore daa of alphabe size v. We derive a upper boud for he rae of space-cosraied codes. Le x = x 1,x 2,...,x {0,1,...,q 1} be a vecor ha is o equal o 0,0,...,0. We call x a β- cosraied vecor if for ay wo o-zero eries x i ad x j i x, we have i j β. Le M,β be he se of all β-cosraied vecors. We see ha wih a space-cosraied code, if he curre cell levels are L = l 1,l 2,...,l, a rewrie ca chage i oly o he cell-level saes i he se {L + x x M,β }. For all he eries i he vecor L + x, ake modulo q. Sice he sored daa have v disic values, ad a rewrie ca chage he daa from ay value o ay oher value, we have v M,β +1. So for space-cosraied codes ha use cells of q levels, he code rae is upper bouded by log 2 M,β +1 bis per cell. We ow compue he value of M,β. Whe β, M,β = q 1 because oly oe ery i a vecor x M,β ca be o-zero. Now cosider he case β +1. Le x = x 1,...,x be a geeric vecor i M,β. If x 1 = x 2 = = x β+1 = 0, he here are M β,β q+q 1 ways o choose he values of x 1,...,x β ad x. If oe of he elemes i {x 1,x 2,...,x β+1 } is o zero, he x mus be zero; ad i is o hard o see ha i his case, he umber of choices for x is M 1,β M β,β. So we have he recursio M,β = M β,β q + q 1 + M 1,β M β,β = M 1,β + q 1 M β,β + q 1 Alog wih he β iiial values M,β = q 1 for = 1,2,...,β, we ca use he above recursio o solve for M,β. Noe ha whe q = 2, he vecors i M,β correspod o he codewords of legh i he d = β 1,k = -RLL cosraied sysem [4], excep ha M,β does o coai he all-zero codeword. Therefore, whe q = 2 ad, he rae of he space-cosraied code is upper bouded by he capaciy of he β 1, -RLL cosraied sysem. IV. CONCLUSION I his paper, we cosider hermal ierferece problems for PCMs, which ca be challegig whe he cell desiy scales oward is limi [1], [5]. We cosider he hermal crossalk problem ad he local hermal accumulaio problem, ad propose ew cosraied codes for solvig hem. We have sudied he capaciy of he cosraied codes ad some code cosrucios. For he symbol-cosraied codes, he capaciy ad code cosrucio for small γ correspodig o less serious crossalk bewee cells sill eed o be sudied. For space-ime cosraied codes, he capaciy ad cosrucio of codes wih boh space ad ime cosrais sill eed o be udersood. They remai as our fuure research opics. ACKNOWLEDGMENT This work was suppored i par by he NSF CAREER Award CCF , he NSF gra ECCS , ad by a NSF-NRI award. REFERENCES [1] G. W. Burr e al., Phase chage memory echology, Joural of Vacuum Sciece ad Techology, vol. 28, o. 2, pp , Mar [2] A. Jiag, J. Bruck ad H. Li, Cosraied codes for phase-chage memories, olie: hp://faculy.cse.amu.edu/ajiag/pcm.pdf. [] L. A. Lasras-Moao, M. Fraceschii, T. Mielholzer, J. Karidis ad M. Wegma, O he lifeime of mulilevel memories, i Proc. IEEE Ieraioal Symposium o Iformaio Theory, Seoul, Korea, 2009, pp [4] B. H. Marcus, R. M. Roh ad P. H. Siegel, A iroducio o codig for cosraied sysems, 5h Ediio, 2001, olie: hp:// ca/marcus/hadbook/idex.hml. [5] A. Pirovao e al., Reliabiliy sudy of phase-chage ovolaile memories, IEEE Tras. Device ad Maerials Reliabiliy, vol. 4, o., pp , Sep [6] R. L. Rives ad A. Shamir, How o reuse a wrie-oce memory, Iformaio ad Corol, vol. 55, pp. 1 19, 1982.