On the Capacity of Bounded Rank Modulation for Flash Memories

Transcription

1 O the Capacty of Bouded Rak Modulato for Flash Meores Zhyg Wag Electrcal Egeerg Departet Calfora Isttute of Techology Pasadea, CA 925, USA Eal: Axao (Adrew) Jag Coputer Scece Departet Texas A&M Uversty College Stato, TX 77843, USA Eal: Jehoshua Bruck Electrcal Egeerg Departet Calfora Isttute of Techology Pasadea, CA 925, USA Eal: Abstract Rak odulato has bee recetly troduced as a ew forato represetato schee for flash eores Gve the charge levels of a group of flash cells, sortg s used to duce a perutato, whch tur represets data Motvated by the lower sortg coplexty of saller cell groups, we cosder bouded rak odulato, where a sequece of perutatos of gve szes are used to represet data We study the capacty of bouded rak odulato uder the codto that perutatos ca overlap for hgher capacty I INTRODUCTION Flash eory s a portat o-volatle storage techology of wde applcatos I flash eores, floatg-gate cells use ther charge-levels to store data [2] For hgher capacty, ult-level cells (MLCs) wth a creasg uber of levels are beg developed To crease a cell level, charge s jected to the cell by the Fowler-Nordhe tuelg echas or the hotelectro jecto echas Ths prograg process s teratve to avod over-jecto To lower ay cell level, oe ust erase a whole cell block (typcally 52K cells) ad reprogra the startg at the lowest level Ths asyetrc property caused by block erasure s a proet feature of flash eores ad presets a bottleeck of flash eores ters of speed ad relablty There has bee a uber of recet works usg the forato theoretc approach to develop ew storage schees for flash eores They clude codg schees for rewrtg data [] [4] [5] [6] [0], codes for correctg lted-agtude errors [3], ad the ew rak odulato schee for effcet ad relable cell prograg ad data storage [7] [8] I ths paper, we focus o ad exted the rak odulato schee Rak odulato s a ew data represetato schee that uses the relatve order of cell levels to represet data [7] [8] Let (c, c 2,, c ) deote the charge levels of cells, where each c (for ) s a aalog uber ad = j, c = c j Let I(c, c 2,, c ) = (a, a 2,, a ) be a fucto that duces fro the charge levels a perutato, where for =, 2,,, a = { j a j a, j =, 2,, } For exaple, f = 4 ad (c, c 2, c 3, c 4 ) = (02, 03, 2, 05), the the duced perutato s (a, a 2, a 3, a 4 ) = (, 2, 4, 3) A group of cells ca store log 2 (!) bts of forato Sce rak odulato uses perutatos to represet data, the charge levels ca take aalog values stead of dscrete values, akg the prograg process uch ore robust to over-jecto ad the stored data ore robust to asyetrc errors I ths paper, we study the capacty of rak odulato wth bouded perutato szes To duce a perutato fro a group of cells, a sortg algorth of coplexty O( log ) s eeded Reducg the sortg coplexty s portat for the effcet hardware pleetato of rak odulato To study the capacty uder ths costrat, we propose a dscrete odel Noralze the gap betwee the u ad axu charge levels of the eory to, ad let δ deote the u charge dfferece to dstgush two levels The the largest possble sze for a perutato s D = δ + However, practce the perutato sze should be saller tha D ot oly to reduce the sortg coplexty, but also to ake cell prograg effcetly pleetable I ths paper, we let D deote the gve perutato sze (whch s also the uber of cells a group), ad study the achevable capacty Each cell level s deoted by a teger the set {, 2,, D} It should be oted that these D dscrete ubers do ot ea that practce the charge levels are to be dscrete stead of aalog They are used to derve the theoretcal capacty uder the cosdered costrats Whe ore costrats are troduced, the odel ca certaly be geeralzed A portat observato s that by allowg cell groups to have overlaps (e, shared cells), the capacty ca be proved I ths paper, we study ths odel, ad explore the correspodg capacty We preset coputatoal techques ad bouds for capacty ad copare the capactes of dfferet schees Due to the lted space, we skpped soe detals ths paper Iterested readers please refer to [] II BOUNDED RANK MODULATION I ths secto, we defe the basc cocepts of bouded rak odulato For coveece, tegers a b, defe [a, b] = {a, a +,, b}

2 Let ad D be tegers such that D A block s a set of cells whose levels are fro the set [, D] ad are all dstct Let (c, c 2,, c ) deote those cell levels The by defto, c [, D] for [, ] ad = j, c = c j For coveece, we call (c, c 2,, c ) a block, too, ad call I(c, c 2,, c ) the duced perutato (I s as defed the prevous secto) If a block B duces a perutato P, the B s called a realzato of P Note that a perutato ay have ultple realzatos For exaple, f = 6 ad P = (, 4, 3, 2), the both (, 6, 4, 3) ad (2, 5, 4, 3) are realzatos of P Let (c, c 2,, c ) be the levels of cells Let v < be a teger ad for coveece, let ( v)/( v) be a teger as well For =, 2,, v, let B deote the block (c ( )()+, c ( )()+2,, c ( )()+ ) Note that the last v cell levels of B are also the frst v cell levels of B +, so we say these two blocks overlap by v We say (c, c 2,, c ), or (B, B 2,, B ( v)/() ), s a cell-level sequece cosstg of blocks that overlap by v For =, 2,, v, let the levels B be all dstct ad P = I(B ) The the sequece duces v perutatos (P, P 2,, P ( v)/() ), called the duced perutato sequece Ad we call (B, B 2,, B ( v)/() ) ts realzato Aga, a perutato sequece ay have ultple realzatos Defto ( BOUNDED RANK MODULATION C(,, D, v)) I a bouded rak odulato (BRM) code C(,, D, v), every codeword s a perutato sequece (P, P 2,, P ( v)/() ) that has at least oe realzato Let C(,, D, v) deote the uber of codewords code C The, the capacty of the code s log C(,, D, v) cap(c) = l I geeral, allowg overlap betwee perutatos ca crease capacty Whe there s o overlap (e, v = 0), the BRM code has capacty Whe v > 0, the capacty ay crease because every perutato cosues just v cells o average III BRM CODE WITH ONE OVERLAP AND CONSECUTIVE LEVELS I ths secto, we study a specal BRM code that allows effcet coputato of ts capacty Frst, we preset a coputatoal ethod based o costraed systes Detaled deftos are show [] Sce c [, D] for [, ], the BRM code s a costraed syste Let G = (V, E, L) be a deterstc labeled graph represetg C(,, D, v), where V, E ad L are the state set, the edge set, ad the edge labelg, respectvely L(u, v) = l s deoted by u l v, l S (the syetrc group) If A, A 2,, A k are the adjacecy atrces of the rreducble copoets G, the cap(c(,, D, v)) = ax k log λ(a ) v where λ(a) s largest postve egevalue of A [9] Exaple 2 A BRM code C(, 2, 3, ) ca be represeted by the deterstc graph G Fgure (a) Each state represets the curret cell level S 2 = {2, 2}, V = {, 2, 3}, ad E = {(, + ) =, 2} {(, ) = 2, 3} The labelg s L(, + ) = 2, =, 2 ad L(, ) = 2, = 2, 3 For exaple, the path alog the states, 2, 3, ad 2 s a realzato of the perutato sequece (2, 2, 2) The adjacecy atrx of G s A = By (), the capacty s log(λ(a)) = 05 Notce Exaple 2, every block B = (c, c + ) cossts of two cosecutve tegers, e, c c + = If we geeralze ths dea to arbtrary D 2 but keep = 2, ad v =, we get the costraed syste Fgure (b), ad the capacty s log(2 cos( D+ π )) [9] We ow forally defe ths type of BRM code Defto 3 ( BRM CODE WITH ONE OVERLAP AND CONSECUTIVE LEVELS C I (,, D, )) For the BRM code C I (,, D, ), every codeword (P, P 2,, P ( )/( ) ) eeds to satsfy the followg addtoal costrat: the codeword has a realzato (B, B 2,, B ( )/( ) ) such that for =, 2,,, the cell levels the block B for a set of cosecutve ubers That s, f B = (c, c 2,, c ), the {c, c 2,, c } = [ j= c j, ax j= c j ] I a labeled graph for C I (,, D, ), each state correspods to the charge level of a overlapped cell, so there are D states,, 2,, D Ad each edge represets a perutato a block (c,, c ) The frst (or last) dgt a edge labelg correspods to the tal (or teral) state of the edge Let (a,, a ) = I(c,, c ), the k, l [, ], c k c l = a k a l For exaple, the labeled graph for C I (, 3, 4, ) s show Fgure (c) The costructo of the adjacecy atrx for code C I (,, D, ) s preseted the followg theore () Theore 4 The adjacecy atrx A = (A j ) for C I (,, D, ) has A j = ( 2)! { j,, j, D +, D j +, D + } (2) f j, ad A j = 0 otherwse

3 D (a) (b) (c) Fg Labeled graphs for C I (a) C I (, 2, 3, ); (b) C I (, 2, D, ) ad D s arbtrary; (c) C I (, 3, 4, ) Proof: A j dcates the uber of perutatos wth c =, c = j For fxed a ad a, there are ( 2)! choces for (a 2,, a ) Notce j oly f a a = c c [, ] So {(a, a )} j, f j [, ] Ad {(a, a )} = 0 otherwse If [, ], k c k = c (a ) = a +, whch ples a [, ], or {a } = Ad we have slar results for other values of Hece, {a } = {, D +, D +, } Ths arguet also works for the teral state j Therefore, f j, the A j = ( 2)! {(a, a )} = ( 2)! { j,, j, D +, D j +, D + } log λ(a) The capacty of C I s cap(c I ) = Soe values of cap(c I ) ad the capacty of the o-overlap code C(,, D, 0) (for coparso) are show Fgure 2 Cap ooverlap Fg 2 Capacty for C I (stars) ad for the o-overlap code (sold le) The stars each vertcal le correspod to the sae perutato sze, ad D =, +,, + 4 fro botto to top It s clear that the capacty of C I (,, D, ) creases wth D Ad f D, cap(c I (,, D, )), whch s larger tha the capacty of the o-overlap code We ow preset a ore geeral result Theore 5 For ay 2 ad D + 2, cap(c I (,, D, )) > cap(c(,, D, 0)) Proof: Notce cap(c(,, D, 0)) = /, D, so we eed to prove cap(c I (,, + 2, )) > / Whe = 2, 3, the theore s trval Whe 4, D = + 2, by (2), A s ( 2)! (+2) (+2) Let B = A, I be the detty atrx ad x be ( 2)! a deterate varable det(b xi) = 0 ples ( x 3) 3 (x 2 + x ) f (x) = 0, where f (x) = x 3 + (3 8)x 2 + (7 0)x Thus λ(b) s the largest postve root of f (x) It ca be prove that λ(b) > 3 6, ad λ(a) > (3 6)( 2)! Now log λ(a) we are left to show > log(3( 2)( 2)!), whch s equvalet to 3 ( 2) ( 2)! ( ) ( ) Notce e, ad Strlg s Approxato, ( ) 3e 2π( ) thus 3 ( 2) ( 2)! ( ) 2e IV BRM CODE WITH ONE OVERLAP We ow cosder the geeral BRM code wth oe overlap, C(,, D, ), whch does ot have the addtoal costrat of code C I (,, D, ) The cell levels of a block, {c,, c }, ca be ay set Q such that Q {, 2,, D} ad Q = The labeled graph H geerated s ot deterstc geeral However, we are able to fd a deterstc graph G that s equvalet to H [9] Here s a exaple Exaple 6 The labeled graph H of C(, 2, 4, ) s show Fgure 3 (a) Ths s ot deterstc sce state has 3 outgog edges labeled 2 Let G be the deterstc represetato of C, the the states V(G) are subsets of l v f j v, l j The resultg graph G s show Fgure V(H) Ad for u, v V(G), u u ad 3 (b) States {2}, {3}, {, 3}, etc, have oly outgog edges, so ther capactes are 0 Therefore the rreducble copoet of G axzg λ(a ) s as Fgure 3 (c) Hece by () cap(c(, 2, 4, )) = log λ(a ) = 0879 I geeral, suppose the deterstc graph G represets C(, 2, D, ), ad A s the adjacecy atrx

4 Fg 3 Labeled graphs for C(, 2, 4, ) (a) Labeled graph; (b) deterstc graph; (c) rreducble graph for the rreducble copoet of G that has the largest egevalue The λ(a ) s the largest postve root of x D + 2x D = 0 It ca be see that cap(c) teds to faster tha cap(c I ) fro the followg table: D cap(c I (, 2, D, )) cap(c(, 2, D, )) The costructo the above exaple ca be aturally exteded to the case > 2 Ecoder/decoder for BRM codes ca be costructed usg sldg-block fte-state perutato ecoder/decoder, cell-level ecoder/decoder, ad flash prograg/readg Ad the ecodg rate ca be arbtrarly close to the capacty For exaple, a rate 3 : 4 blockdecodable ecoder ca be costructed for C(, 2, 4, ) More detals are show [] V LOWER BOUND FOR CAPACITY I ths secto, we preset a lower boud to the capacty of the BRM code To derve ths, we frst preset a ew for of rak odulato called the star BRM A Star BRM A Star BRM code uses + v cells For coveece, let be a ultple of v v of these + v cells are called achors, ad we deote ther cell levels by (l, l 2,, l v ) The other cells are called storage cells, ad we deote ther cell levels by c, c 2,, c For =, 2,, v, l [, D]; for =, 2,,, c [, D] We call (l, l 2,, l v, c, c 2,, c ) a cell-level sequece For =, 2,,, defe block B as (l, l 2,, l v, c ( )()+, c ( )()+2,, c () ) These blocks share the achor cells For =, 2,,, we requre that the cell levels B are all dfferet, ad let P = I(B ) B s a realzato of P Aga, a perutato sequece (P, P 2,, P /() ) ay have ultple realzatos Defto 7 ( STAR BRM CODE S(,, D, v)) I a Star BRM code S(,, D, v), every codeword s a perutato sequece (P, P 2,, P /() ) that has at least oe realzato Let S(,, D, v) deote the uber of codewords code S The, the capacty s log S(,, D, v) cap(s) = l + v To derve the capacty of Star BRM, we frst show how the achors (l, l 2,, l v ) affect the perutato sequeces For fxed (l, l 2,, l v ), defe Z(l, l 2,, l v ) as the total uber of perutatos that ca be duced by the cell levels (l, l 2,, l v, c, c 2,, c ), where the cell levels are all dfferet ad all belog to the set [, D] Whe we perute the v achor levels, the value of Z(l, l 2,, l v ) reas the sae For exaple, whe v = 3 ad D = 6, Z(2, 3, 6) = Z(3, 2, 6) = Z(6, 2, 3) So wthout loss of geeralty, assue l < l 2 < < l v Let β(l, l 2,, l v ) deote the uber of solutos for the varables x, x 2,, x v+ such that () v+ = x = v; (2) x [0, l ], x [0, l l ] for [2, v], ad x v+ [0, D l v ] Lea 8 Gve D > v, we have Z(l, l 2,, l v ) = ( v)! β(l, l 2,, l v ) Sketch of the proof: A perutato duced by (l, l 2,, l v, c, c 2,, c ) ca be uquely detered by the relatve order of the v cell levels (c, c 2,, c ) ad ther relatve values copared to l, l 2,, l v Lea 9 Z(l, l 2,, l v ) s axzed whe the ubers the followg set dffer by at ost oe: {l, D l v } {l l = 2, 3,, v} (Every uber the above set s ether D v v+ or D v v+ ) Please see [] for detaled proofs of Lea 8 ad 9 Let l < l 2 < < l v be the v achor levels that satsfy the codto Lea 9 ad Z = Z(l, l 2,, l v) Z ca be coputed usg a algorth of te coplexty O(D 2 ) (see []) The followg theore presets the capacty of the Star BRM Theore 0 The capacty of S(,, D, v) s cap(s) = log Z v

5 Sketch of the proof: S(,, D, v) s o less tha the uber of codewords duced by (l, l 2,, l v, c, c 2,, c ), or (Z ) O the other had, by Lea 9, S(,, D, v) s o ore tha (Z ) tes the uber of choces for (l, l 2,, l v ), or v!( D v ) Therefore, log Z log S(,,D,v) cap(s) = l +v log(v!( l D v )(Z ) ) log Z = So the theore s proved The above proof leads to the followg corollary Corollary The Star BRM code S(,, D, v) acheves ts capacty eve f the v achor cell levels are fxed as (l, l 2,, l v) The capacty of the Star BRM code S(,, D, v) s o-decreasg D However, whe D = ( v + )v + ( v), the capacty reaches ts axu value Further creasg D wll ot crease the capacty That s because whe D ( v + )v + ( v), Z reaches ts axu value!/v! B Lower Boud for The Capacty of BRM We ow derve a lower boud for the capacty of the bouded rak odulato code C(,, D, v) Theore 2 For the BRM code C(,, D, v), whe 2v, ts capacty cap(c) log Z + log v! + log( 2v)! 2( v) Proof: Let S(,, D, v) be a Star BRM code such that every codeword has a realzato whch the v achors are (l, l 2,, l v) By Corollary, S acheves capacty For a codeword s S, let (l, l 2,, l v, c, c 2,, c ) = (B, B 2,, B /() ) be ts realzato For =, 2,, /( v), correspodg to block B, we buld two blocks B ad B of legth as follows Say B = (l, l 2,, l v, c, c 2,, c ) The frst v cell levels of B take values fro the set {l, l 2,, l v} (we have v! choces), ad the ext v cell levels of B are the sae as (c, c 2,, c ) The frst v cell levels of B overlap the last v cell levels of B We pck 2v D 2v values dfferet fro the frst v ad the last v cell levels of B, ad assg the to the ext 2v cell levels of B (we have ( 2v)! choces) The fal v cell levels of B take values aga fro the set {l, l 2,, l v} The we costruct a cell-level sequece (B, B, B 2, B 2,, B /(), B /() ), where every two adjacet blocks overlap by v Correspodg to every codeword s S, there are at least (v!( 2v)!) such cell-level sequeces, whch we deote by Q s No two cell-level sequeces Q s duce the sae perutato sequece Ad whe s = s, every par of cell-level sequeces fro Q s ad Q s, respectvely, also duce dfferet perutato sequeces Besdes, every cell-level sequece costructed above duces a codeword the code C(2 + v,, D, v) So correspodg to the S(,, D, v) codewords of the Star BRM code S(,, D, v), we ca fd at least S(,, D, v) (v!( 2v)!) codewords of the BRM code C(2 + v,, D, v) So the capacty of code C(,, D, v) s cap(c) log S(,,D,v) +(log v!+log( 2v)!) l 2+v = log Z +log v!+log( 2v)! So the theore s proved 2() Corollary 3 Let C(,, D, v) be a BRM code, ad let S(,, D, v) be a Star BRM code The, whe 2v, cap(c) 2 cap(s) I partcular, f v > or > 2v, cap(c) > 2 cap(s) Defe A k = ( k )k! =!/( k)! Suppose < 2v ad v = k( v) + s, where k N + ad s v Let r = v s Defe a costat M = A s (A 2() s) ) k ( v)! Slar to Theore 2, we have the followg lower boud for the BRM code whe < 2v (see [] for proof) Theore 4 For the BRM code C(,, D, v), whe < 2v ad D + r, ts capacty cap(c) log(z M r!) + r ACKNOWLEDGMENT Ths work was supported part by the NSF CAREER Award CCF , the NSF grat ECCS , ad the Caltech Lee Ceter for Advaced Networkg REFERENCES [] V Bohossa, A Jag ad J Bruck, Buffer codes for asyetrc ult-level eory, Proc IEEE ISIT, 2007, pp [2] J E Brewer ad M Gll, Novolatle eory techologes wth ephass o flash, Chapter 2, Wley-IEEE, 2007 [3] Y Cassuto, M Schwartz, V Bohossa ad J Bruck, Codes for asyetrc lted-agtude errors wth applcato to ultlevel flash eores, Proc IEEE ISIT, 2007 [4] H Fucae, Z Lu ad M Mtzeacher, Desgg floatg codes for expected perforace, Proc of the 46th Aual Allerto Coferece, 2008 [5] A Jag, V Bohossa ad J Bruck, Floatg codes for jot forato storage wrte asyetrc eores, Proc IEEE ISIT, 2007, pp [6] A Jag ad J Bruck, Jot codg for flash eory storage, Proc IEEE ISIT, 2008, pp [7] A Jag, R Mateescu, M Schwartz ad J Bruck, Rak odulato for flash eores, Proc IEEE ISIT, 2008, pp [8] A Jag, M Schwartz ad J Bruck, Error-correctg codes for rak odulato, Proc IEEE ISIT, 2008, pp [9] B H Marcus, R M Roth ad P H Segel, A troducto to codg for costraed systes, 5th Edto, 200, avalable at http : //wwwathubcca/ arcus/hadbook/dexhtl [0] E Yaakob, A Vardy, P H Segel ad J Wolf, Multdesoal flash codes, Proc of the 46th Aual Allerto Coferece, 2008 [] Z Wag, A Jag, ad J Bruck, O the capacty of bouded rak odulato for flash eores, avalable at http : //wwwparadsecaltechedu/etrhtl