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 95, 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 95, 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 [] For hgher capacty, ult-level cells (MLCs) wth a creasg uber of levels are beg developed To crease a cell level, charge s ected to the cell by the Fowler-Nordhe tuelg echas or the hotelectro ecto echas Ths prograg process s teratve to avod over-ecto To lower ay cell level, oe ust erase a whole cell block (typcally 5K 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,, c ) deote the charge levels of cells, where each c (for ) s a aalog uber ad =, c = c Let I(c, c,, c ) = (a, a,, a ) be a fucto that duces fro the charge levels a perutato, where for =,,,, a = { a a, =,,, } For exaple, f = 4 ad (c, c, c 3, c 4 ) = (0, 03,, 05), the the duced perutato s (a, a, a 3, a 4 ) = (,, 4, 3) A group of cells ca store log (!) 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-ecto 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 {,,, 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, provde ecodg ad decodg techques that acheve ay rate saller tha the capacty, ad copare the capactes of dfferet schees II BOUNDED RANK MODULATION I ths secto, we defe the basc cocepts of bouded rak odulato For coveece, for ay two

2 tegers a, b such that a b, we defe [a, b] = {a, a +,, b} 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,, c ) deote those cell levels The by defto, c [, D] for [, ] ad =, c = c For coveece, we call (c, c,, c ) a block, too, ad call I(c, c,, 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, ), the both (, 6, 4, 3) ad (, 5, 4, 3) are realzatos of P Let (c, c,, c ) be the levels of cells Let v < be a teger ad for coveece, let ( v)/( v) be a teger as well For =,,, v, let B deote the block (c ( )()+, c ( )()+,, 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,, c ) s a cell-level sequece that cossts of blocks that overlap by v, whch we ay also deote by (B, B,, B ( v)/() ) For =,,, v, let the levels B be all dstct The the sequece duces ( v)/( v) perutatos (P, P,, P ( v)/() ), where P = I(B ) for =,,, v We call (P, P,, P ( v)/() ) the duced perutato sequece, ad call (B, B,, 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,, P ( v)/() ) that has at least oe realzato (The eag of the paraeters,, D, v s as preseted above) 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, log! v = 0), the BRM code has capacty Whe v > 0, the capacty ay crease because every perutato cosues ust v cells o average III BRM CODE WITH ONE OVERLAP AND CONSECUTIVE LEVELS I ths secto, we study a specal for of BRM code that allows effcet coputato of ts capacty Frst, we preset a coputatoal ethod based o costraed systes Sce c [, D] for [, ], the BRM code s a costraed syste over the alphabet S (the syetrc group o the set [, ]) Defe a labeled graph G = (V, E, L) to be a drected graph wth a state set V, a edge set E V V ad a edge labelg L : E S For (u, v) E, L(u, v) = l s deoted by u l v G represets C f the set of all fte sequeces obtaed fro readg the labels of paths G equals the set of the codewords of C If the outgog edges of each state are labeled dstctly, the G s deterstc Ad G s rreducble f u, v V, there s a path fro u to v Defe A V V as the adacecy atrx of G, where A uv equals the uber of edges fro u to v I addto, suppose a deterstc graph G represets C(,, D, v) ad A, A,, A k are the adacecy 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 A BRM code C(,, 3, ) ca be represeted by the graph G Fgure (a) Each state represets the level of the curret cell S = {, }, the states are V = {,, 3}, ad the edges are E = {(, + ) =, } {(, ) =, 3} The labelg s defed by L(, + ) =, =, ad L(, ) =, =, 3 For exaple, the path alog the states,, 3, ad s a realzato of the perutato sequece (,, ) G s deterstc ad rreducble Hece, the adacecy atrx of G s A = By (), the capacty s log(λ(a)) = 05 Notce Exaple, the labelg L s essetally the raks of the tal ad teral states of a edge Also otce that every block B = (c, c + ) cossts of two cosecutve tegers, e, c c + = If we expad the dea of Exaple to arbtrary D but keep =, ad v =, we wll get the costraed syste Fgure (b) The adacecy atrx s A = 0 0 D D The capacty s log λ(a) = log( cos( D+ π )) [9] We ow forally defe ths type of costraed BRM code Defto 3 ( BRM CODE WITH ONE OVERLAP AND CONSECUTIVE LEVELS C I (,, D, )) For ()

3 3 D (a) (b) (c) Fg Labeled graphs for C I (a) C I (,, 3, ); (b) C I (,, D, ) ad D s arbtrary; (c) C I (, 3, 4, ) the BRM code C I (,, D, ), every codeword (P, P,, P ( )/( ) ) eeds to satsfy the followg addtoal costrat: the codeword has a realzato (B, B,, B ( )/( ) ) such that for =,,,, the cell levels the block B for a set of cosecutve ubers That s, f B = (c, c,, c ), the {c, c,, c } = [ = c, ax = c ] I a labeled graph for C I (,, D, ), each state correspods to the charge level of a overlapped cell, so there are D states,,,, 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 sce each block has cosecutve ubers, k, l [, ], c k c l = a k a l () f, A = ( )! {(a, a )} = ( )! {, {a }, {a } } = ( )! {,,, D +, D +, D + } Ad A = 0 otherwse 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 3 5 For exaple, the labeled graph for C I (, 3, 4, ) s show Fgure (c) The costructo of the adacecy atrx for code C I (,, D, ) s preseted the followg theore Cap 5 05 ooverlap Theore 4 The adacecy atrx A = (A ) for C I (,, D, ) has A = ( )! {,,, D +, D +, D + } (3) f, ad A = 0 otherwse Proof: A dcates the uber of perutatos wth c =, c = For fxed a ad a, there are ( )! choces for (a,, a ) Notce oly f a a = c c [, ] So {(a, a )}, f [, ] Ad {(a, a )} = 0 otherwse If [, ], the by (), k c k = c (a ) = a +, whch ples a [, ], or {a } = Slarly, f [D +, D], we wll get a [ D +, ], or {a } = D + For [D +, ], a [ D +, ], or {a } = D + Ad f [, D + ], the a [, ], or {a } = Hece, {a } = {, D +, D +, } Ths arguet also works for the teral state Therefore, Fg 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 the code C I (,, D, ) creases wth D Ad f D, log! cap(c I (,, D, )), whch s larger tha the capacty of a o-overlappg code C(,, D, 0) We ow preset a ore geeral result Theore 5 For ay ad D +, cap(c I (,, D, )) > cap(c(,, D, 0)) Proof: Notce cap(c(,, D, 0)) = log!/, D If we proved cap(c I (,, +, )) > log!/, the ths theore s proved Whe =, 3, cap(c I (,, 4, )) = 0694 > log!/ = 05 ad cap(c I (, 3, 5, )) = 00 > log 3!/3 = 0867

4 Whe 4, D = +, by (3), A s ( )! (+) (+) By (), t s ecessary to fd λ(a) Let B = ( )! A, I be the detty atrx ad x be a deterate varable det(b xi) = 0 ples ( x 3) 3 (x + x ) f (x) = 0, where f (x) = x 3 + (3 8)x + (7 0)x Thus λ(b) s the largest postve root of f (x) Notce x > λ(b), f (x) < 0, but f (3 6) = 3( + 5) > 0, 4 So λ(b) > 3 6, ad λ(a) > (3 6)( )! Now we are left to show log λ(a) > log(3( )( )!) log! whch s equvalet to 3 ( ) ( )! Notce ( ) ( ) 4, e, ad Strlg s Approxato,! π(/e), thus 3 ( ) ( )! ( ) = 3 ( )( )! ( ) 3 ( )! e 3 π( ) ( ) e e ( ) 3 π( ) e e Thus the proof s copleted 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, ) I ths case, the cell levels of a block, {c,, c }, ca be ay set Q such that Q {,,, 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 (Lea [9]) Here s a exaple Exaple 6 The labeled graph H of C(,, 4, ) s show Fgure 3 (a) Ths s ot deterstc sce state has 3 outgog edges labeled Let G be the deterstc represetato of C, the the states V(G) are subsets of V(H) Ad for u, v V(G), u l v f v, u l ad So the resultg graph G s as show Fgure 3 (b) States {}, {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) By ()we ca the get cap(c(,, 4, )) = log λ(a ) = 0879 > cap(c I (,, 4, )) = 0694 I geeral, suppose the deterstc graph G represets C(,, D, ), ad A s the adacecy atrx for the rreducble copoet of G that has the largest egevalue The λ(a ) s the largest postve root of x D + x D = 0 Coparg cap(c I ) ad cap(c), we have Fgure 4 It ca be see that cap(c) teds to faster tha cap(c I ), sce t akes better use of the levels provded Cap Code I Code II D Fg 4 Capacty for C The sold ad dashed les show capacty for C I ad C, respectvely The costructo the above exaple ca be aturally exteded to the case > Ecoders ad decoders for BRM codes ca be costructed as fgure 5 I the ecodg process, the put forato sequece (x 0, x, ) s frst ecoded to a perutato sequece, (P 0, P, ), satsfyg the axu cell level costrat, whch s further apped to a cell-level sequece, (c, c, ) At last, the cell-level sequece s prograed to flash eory Ad the decoder reverses ths process by readg the cell levels, forg a perutato sequece, ad at last retrevg the forato Perutato ecoder/decoder Let C be a costraed syste wth a represetato G, ad p,q be postve tegers The q-th power of C, C q, s represeted by the labeled graph G q, wth the sae set of states as G ad edges/labelgs correspodg to paths/labelgs of legth q G A fte-state ecoder wth rate p : q s a lossless labeled graph H such that H G q ad each state of H has out-degree p We ca the assg p put tags (or p bary forato bts) to the outgog edges of each state A ecoder s (,a)-sldg-block decodable f the -th put tag

5 Fg 3 Labeled graphs for C(,, 4, ) (a) Labeled graph; (b) deterstc graph; (c) rreducble graph Iforato Sequece x 0,x,x, Perutato Ecoder (Fte-State Ecoder) Perutato Sequece P 0,P,P, Cell Level Ecoder (a) Cell Level Sequece c,c,c 3, Flash Prograg Prograed Logc Array Prograed Logc Array Flash Readg Cell Level Sequece c,c,c 3, Cell Level Decoder (-Sortg) Perutato Sequece P 0,P,P, Perutato Decoder (Sldg-Block Decoder) Iforato Sequece x 0,x,x, (b) Fg 5 (a)ecoder ad (b) decoder for BRM codes a put tag sequece s uquely detered by the q- block labelg sequece P q, Pq +,, Pq,, Pq +a The followg theore states that the capacty of ay costraed syste s always achevable [9] Theore 7 Let C(,, D, v) be a costraed syste ad p/q < ( v)cap(c(,, D, v)) for postve tegers p ad q The there exsts a sldg-block decodable fte-state ecoder wth rate p : q ad perutato ecodg rate p/(q( v)) Theore 7 ca be proved by explct costructos of ecoders, such as the state-splttg algorth [9] Ad a sldg-block decoder s essetally a appg fro ( + a + ) q-block labelgs to a p-block bary put tag Notce the decodg process, decodg delay ad error propagato s cotrolled wth + a + q-blocks, whch depeds o the costructo of the ecoder/decoder Exaple 8 Cotug Exaple 6, take p = 3 ad q = 4, the p/q < cap(c(,, 4, )) = 0879 The 4-th power of C(,, 4, ) has adacecy atrx A 4 = After deletg States 3 ad 4, ad soe edges fro the graph, we get a fte-state ecoder Fgure IV (a), whch has out-degree p = 8 for each state, ad each labelg block has sze q = 4 The otato u x3 /P 4 v eas the 4-block labelg P 4 s assged the bary put tag x 3 For coveece, we deote the perutato (, ) by, ad (, ) by 0 the labelg After ergg the States ad 6, t s further splfed as Fgure IV (b) Let State be the tal state for the ecodg Dvde the put bary forato bts to blocks of 3, ad for ay x 3 = (x 3, x 3+, x 3+ ), ecode t as the correspodg 4-block labelg, P 4 = (P 4, P 4+, P 4+, P 4+3 ) Notce that the ecodg process, each 4-block labelg correspods to oly oe put tag, depedet of startg state Therefore, we ca costruct a (0, 0)- sldg-block decoder: for each receved perutato block P 4 decode t to the uque forato block x 3 = (x 3, x 3+, x 3+ ), whch equals to (P 4+, P 4+, P 4+3 ) f P 4 = P 4+ ad equals to (P 4+3, P 4+3, P 4+3 ), otherwse Cell-level ecoder/decoder Before prograg to flash eores, we ust frst ecode the perutato sequece (P 0, P, ) to a cell-level sequece (c, c, ), such that t duces ths perutato sequece ad does ot exceed the axu level The followg costructo provdes such a ecodg Let P = (a, a,, a ) ad deote c ( )+ by c, for 0, =,,, The for BRM codes

6 000/000 0/00 00/00 000/000 0/0 0/00 0/00 000/0 00/00 0/00 00/00 0/00 00/00 /0 00/00 /0 0/0 00/000 00/ / /00 5 0/0 0/00 0/00 00/00 00/00 00/00 /000 /0 00/000 0/0 00/000 (a) 000/000 0/00 000/0 00/00 000/000 0/00 0/00 0/00 00/00 00/00 00/00 0/00 /0 0/00 00/00 00/00 /0 /000 00/000 0/0 00/000 0/0 5 0/0 00/000 (b) Fg 6 Rate 3 : 4 fte-state ecoder for C(,, 4, ) (a) Labeled subgraph wth out-degree 8, ad (b) splfed ecoder wth oe overlap, c s the cell level of a, for 0, =,,,, ad c = c, for Costructo 9 Let (P 0,, P, ) be the put perutato sequece of the cell-level ecoder The the followg assget of c 0, c0,, c0,, c, c,, c, defes a cell-level ecoder { c 0 a = 0, D + a 0, f a 0 < a0 f a0 > a0 For, ax(a, c + a a ), f a < a, a < a ax(a, a ), c = f a > a, a < a (D + a, D + a ), f a < a, a > a (D + a, c + a a ), f a > a, a > a For 0 ad =,,, (4) c + a a, c = f a < a, a < a or a > a, a > a c + + a a, f a < a, a > a or a > a, a < a (5) Whe =, usg ad 0 to represet the perutatos (, ) ad (, ), respectvely, the above costructo s reduced to {, f P0 = c = 0, f P 0 = 0 ad for, c + f P =, P = f P c + = = 0, P = D f P =, P = 0 c f P = 0, P = 0 Oe ca check that f a perutato sequece satsfes the costrats for C(,, D, ), e, there are at ost D zeros (or oes) betwee ay two successve oes (or zeros), the the geerated cell-level sequece realzes ths sequece ad has cell levels [, D] We ow show that Costructo 9 geerates a celllevel realzato for each codeword BRM code Theore 0 Let (P 0, P,, P ) be a codeword C(( ) +,, D, ) ad C = (c 0, c0,, c0,, c, c,, c, c ) be the cell-level sequece geerated Costructo 9 The C s a realzato of P = (P 0, P,, P ) I partcular, each cell level rages betwee ad D Proof: Fro (5) t s clear that (c, c,, c, c+ ) has raks P = (a, a,, a ) We are left to show that C rages betwee ad D Assue (d 0, d0,, d0,, d, d,, d, d+ s a arbtrary realzato of P ad d D for =,,, = 0,,, For c, = 0,,,, we wll prove a stroger codto: ) a c d, a c, f a < a d c a + D, c a + D, f a > a (6) The equaltes cotag a are ot cosdered whe = 0 Notce the cell-level sequece (d,, d, d+ ) duces (a,, a, a ) Hece, for, k such that a a k, we have a a k d d k (7) Let a = a, a k =, ad a =, a k = a, ad we get a d a + D (8) Slarly, sce (d,, d, d ) duces (a,, a, a ), we have a d a + D Suppose (6) holds, the a c a + D D ad a c a + D D Ad by (5), ether a = a + a a c = c + a a (a + D ) + a a = a + D D or a c = c + + a a a + D D, whch would coplete the proof we wll prove (6) by ducto For the base case( = 0), f a 0 < a0, the c 0 = a0 d0 Ad f a0 > a0, the d 0 a0 + D = c0 Now suppose (6) holds for 0 If a < a ad a < a, the by ducto, a c d, a c (9)

7 Now ether a (9) c + a a a = c (8) d (9) d + a or a c + a a = c a (7) d (9) ad a c a + a = c Therefore, (6) holds for c ths case If a > a ad a < a, the by (8) ether a a = c d or a a = c d Therefore, (6) holds ths case, too Ad we ca prove by slar arguets that (6) s true for the other two cases, thus coplete ths proof The cell-level decoder s a -sorter that orders every -cell-level tuple assocated wth the perutatos The coplexty s log Flash prograg/readg To avod over-prograg, the prograg of a cell-level sequece to flash eores s operated fro the lowest to the hghest rak Such a prograg ethod wll lead to a delay of the worst case However, sce for BRM code wth oe overlap, oly c, 0 are cotaed two perutatos, we oly eed to obta correct orderg for {c, c+ } ad {c, c,, c, c+ }, for all 0 Ad orderg of other sets of cell levels are ot used the code Costructo follows the above dea ad has saller delay tha, for suffcetly log Costructo we frst defe a wrtg operato for c, deoted by Op(), as follows ) If, copare c ad c If c < c, wrte to flash {c =,,, c < c } fro low to hgh level Otherwse, wrte {c =,,, c < c < c } fro low to hgh level ) If c < c+, wrte to flash {c =,,, c < c } fro low to hgh level If c > c+, wrte {c =,,, c+ < c < c } fro low to hgh level 3) Wrte c so that t has a hgher charge level tha all the cells wrtte steps ad 4) If, ad c > c, the wrte {c =,,, c > c } fro low to hgh level Ad f c > c +, wrte to flash {c = } fro low to hgh level,,, c > c I steps,, ad 4, the oly requreet s that the charge level of each prograed cell s hgher tha the prevously wrtte oe Now startg fro = 0, f c < c+,do Op(), ad update wth + Otherwse, fd the sallest e, such that c +e < c +e+, do Op( + e), Op( + e ),, Op(), ad the update wth + e + The above procedure clearly realzes the gve perutato sequece, ad oreover has a worst-case delay D( ) I the worst case, c = D, c+ = D,, c +D =, ad c < c < D, for soe, hece oe has to receve c,, c, c,, c,, c+d to wrte c For BRM code wth perutatos of sze, the prograg process reduces to operatos o c oly, for 0, wth axu delay D Flash readg ca be sply realzed by sequetally readg off cell charge levels fro flash Notce that both the cell-level ecoder ad flash prograg has rate, so the BRM code ecoder has rate p/(q( v)) by Theore 7, whch ca be arbtrarly close to the capacty We wll ow gve a exaple of the coplete ecodg/decodg process for BRM code C(,, 4, ) The ecoder has rate 075 ths exaple Exaple Ecodg: suppose the forato sequece s (x 0 x ) = ( ) The followg Exaple 8, the ecoded perutato sequece s (P 0 P 5 ) = ( ), ad the state trasto the fte-state ecoder s State Usg Costructo 9, we get the ecoded cell-level sequece (c c 6 ) = ( ) Ad at last t ca be prograed to flash the order: (c, c 4, c 3, c, c 6, c 5, c 9, c 8, c 7, c 0, c, c 3, c, c 5, c 4 ) Decodg: assue we are to decode the flash cells prograed above Frst read the cells sequetally ad the copare c + ad c +, for 0 Decode P as f c + < c +, as 0 f c + > c + Thus we get (P 0 P 5 ) = ( ), ad by the decodg schee Exaple 8, we get (x 0 x ) = ( ) V LOWER BOUND FOR CAPACITY I ths secto, we preset a lower boud to the capacty of the BRM code To derve ths boud, 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,, l v ) The other cells are called storage cells, ad we deote ther cell levels by c, c,, c For =,,, v, l [, D]; for =,,,, c [, D] For =,,,, we defe block B to be these cell levels: (l, l,, l v, c ( )()+, c ( )()+,, c () ) We ca see that these blocks share the sae v cells, aely, the achor cells For =,,,, we requre that the cell levels the block B are all dfferet, ad we use P to deote the perutato duced by B B s a realzato of P Aga, a

8 perutato sequece (P, P,, P /() ) ay have ultple realzatos Defto 3 ( STAR BRM CODE S(,, D, v)) I a Star BRM code S(,, D, v), every codeword s a perutato sequece (P, P,, P /() ) that has at least oe realzato (The eag of the paraeters,, D, v s as preseted above) Let S(,, D, v) deote the uber of codewords code S The, the capacty of the code s log S(,, D, v) cap(s) = l + v To derve the capacty of Star BRM, we frst show how the achor levels (l, l,, l v ) affect the perutato sequeces Defe Z(l, l,, l v ) as the total uber of perutatos that ca be duced by the cell levels (l, l,, l v, c, c,, c ), where the cell levels are all dfferet ad all belog to the set [, D] (Here (l, l,, l v ) are fxed, ad the v cell levels (c, c,, c ) ca vary ad therefore ca have (D v)! cobatos Soe of the duce the (D )! sae perutato) It ca be observed that whe we perute the v achor levels (l, l,, l v ), the value of Z(l, l,, l v ) reas the sae For exaple, whe v = 3 ad D = 6, Z(, 3, 6) = Z(3,, 6) = Z(6,, 3) So wthout loss of geeralty (WLOG), we assue l < l < < l v Gve (l, l,, l v ), let β(l, l,, l v ) deote the uber of solutos for the varables x, x,, x v+ that satsfy the followg two codtos: () v+ = x = v; () x [0, l ], x [0, l l ] for [, v], ad x v+ [0, D l v ] Lea 4 Gve D > v, we have Z(l, l,, l v ) = ( v)! β(l, l,, l v ) Proof: Gve the achor cell levels (l, l,, l v ), a perutato duced by (l, l,, l v, c, c,, c ) ca be uquely detered by the followg two steps: () detere the relatve order of the v cell levels (c, c,, c ) (that s, whch cell level s the hghest, secod hghest, ad so o aog the); () detere how ay cell levels aog (c, c,, c ) are below l, or betwee l ad l, or betwee l ad l 3,, or above l v Step has ( v)! choces, ad step has β(l, l,, l v ) choces So the cocluso holds Lea 5 Z(l, l,, l v ) s axzed whe the ubers the followg set dffer by at ost oe: {l, D l v } {l l =, 3,, v} (That s, every uber the above set s ether D v v+ or D v v+ ) Proof: By Lea 4, axzg Z(l, l,, l v ) s equvalet to axzg β(l, l,, l v ) Defe α = l, α = l l for [, v], ad α v+ = D l v Suppose there exsts = such that α α + WLOG, let < Let x, x,, x v+ be varables satsfyg these two codtos: () v+ k= x k = v; () x k [0,α k ] for k [, v + ] The uber of such solutos s β(l, l,, l v ) Now, let us fx the values of x,, x, x +,, x, x +,, x v+ ( a vald soluto), ad see how ay dfferet values x ca take (Note that the value of x s detered by x ) Let z = D k {,,,+,,, +,,v+} x k = x + x Let γ(z) deote the uber of values x ca take The costrats are 0 x α, 0 z x α If z α, γ(z) = α + α z + ; f α z < α, γ(z) = α + ; f z < α, γ(z) = z + So f we crease α by oe ad decrease α by oe, γ(z) wll ot decrease although the values α,α,,α v+ wll becoe ore eve So gve a sequece (l, l,, l v ), we ca chage t that way to a sequece that satsfes the codto the lea, wthout decreasg β(l, l,, l v ) It s easy to see that whe α,α,,α v+ dffer by at ost oe, o atter what ther order s, β(l, l,, l v ) s the sae (whch s the axu value of β(l, l,, l v )) Let (l, l,, l v) be the v achor levels that satsfy the codto Lea 5 It axzes the value of Z(l, l,, l v ) For coveece, we assue that l < l < < l v It s very sple to fd these v values For coveece, we use Z to deote Z(l, l,, l v), ad use β to deote β(l, l,, l v) The values of β ad Z ca be coputed effcetly by the followg dyac prograg algorth of te coplexty O(D ) Let α = l, α = l l for [, v], ad α v+ = D l v Let w(, ) deote the uber of solutos for x, x,, x such that k= x k = ad x k [0,α k ] for k =,, The algorth s as follows: () w(, ) = α k=0 w(, k) Also, w(, ) = 0 f < 0, w(, ) = f 0 α, ad w(, ) = 0 f > α ; () β = w(v +, D v), ad Z = ( v)!β The followg theore presets the capacty of the Star BRM Theore 6 The capacty of the Star BRM code S(,, D, v) s cap(s) = log Z v log Z Proof: We frst show that cap(s) There are v!( D v ) ways to assg values to (l, l,, l v ), whch we deote by W = {w, w,, w v!( D v ) } We call (l, l,, l v, c, c,, c ) the cell-level sequece For =,,, v!( D v ), let γ, deote the

9 axu set of cell-level sequeces satsfyg two codtos: () They all assg w to (l, l,, l v ); () The perutatos duced by the are all dstct By the defto of Z(l, l,, l v ), every block ca duce Z(l, l,, l v ) perutatos Sce there are /( v) blocks, we get γ, = (Z(w )) By Lea 5, Z(w ) Z Sce every codeword of S has at least oe realzato soe γ,, S(,, D, v) =,,,v!( D v ) γ, v!( D v )(Z ) So cap(s) = l log S(,,D,v) l log(v!( D v )(Z ) ) = log Z +v log Z We ow show that cap(s) Say that (l, l,, l v) = w for soe Every codeword of S has at ost oe cofgurato γ,, so S(,, D, v) γ, log S(,,D,v) So cap(s) = l +v = log γ l, log(z = l ) So the theore s proved The above proof leads to the followg corollary log Z Corollary 7 The Star BRM code S(,, D, v) acheves ts capacty eve f the v achor cell levels are fxed as (l, l,, 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 8 For the BRM code C(,, D, v), whe v, ts capacty cap(c) log Z + log v! + log( v)! ( v) (As preseted prevously, Z s a value detered by the paraeters, D, v) Proof: Let S(,, D, v) be a Star BRM code wth a addtoal costrat: every codeword of S has a realzato whch the v achor levels are (l, l,, l v) By Corollary 7, S acheves capacty For coveece, assue /( v) s a teger Let (P, P,, P /() ) be a codeword S, ad let (l, l,, l v, c, c,, c ) = (B, B,, B /() ) be ts realzato For =,,, /( v), correspodg to block B, we buld two blocks B ad B of legth as follows Say B = (l, l,, l v, c, c,, c ) The frst v cell levels of B take values fro the set {l, l,, l v} (we have v! choces here), ad the ext v cell levels of B are the sae as (c, c,, c ) The frst v cell levels of B overlap the last v cell levels of B For the ext v cell levels of B, we frst pck v D v values dfferet fro the frst v ad the last v cell levels of B, the let the v cell levels take oly those v values (we have ( v)! choces here) The fal v cell levels of B take values aga fro the set {l, l,, l v} The we costruct a cell-level sequece (B, B, B, B,, B /(), B /() ), where every two adacet blocks overlap by v Correspodg to every codeword s S, there are at least (v!( v)!) such cell-level sequeces, whch we deote by Q s It s sple to see that o two cell-level sequeces Q s duce the sae perutato sequece O the other sde, whe s = s, every par of cell-level sequeces fro Q s ad Q s, respectvely, also duce dfferet perutato sequeces (To see that, let us call the par of cell-level sequeces q ad q Replace all ther overlappg cell levels by (l, l,, l v), ad get two ew cell-level sequeces p ad p The codewords s ad s are subsequeces of I(p) ad I(p ), respectvely Sce s = s, I(p) = I(p ) So I(q) = I(q )) We ca also see that every cell-level sequece costructed above duces a codeword the code C( + 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!( v)!) codewords of the BRM code C( + v,, D, v) So the capacty of code C(,, D, v) s cap(c) log S(,,D,v) +(log v!+log( v)!) l +v = log S(,,D,v) log v!+log( v)! l + = () cap(s) log v!+log( v)! + = log Z +log v!+log( v)! () () So the theore s proved Corollary 9 Let C(,, D, v) be a BRM code, ad let S(,, D, v) be a Star BRM code The, whe v, cap(c) cap(s) I partcular, whe v > or > v, cap(c) > cap(s) We ow preset a lower boud for the case < v Defe A k = ( k )k! =!/( k)!, whch s the uber of ways to arrage k eleets postos Suppose < v ad v = k( v) + s, where k N + ad s v Let r = v s (So 0 r v ) Defe a costat M = A s (A () s) ) k ( v)! We have the followg lower boud for the BRM code whe < v

10 Theore 0 For the BRM code C(,, D, v), whe < v ad D + r, ts capacty cap(c) log(z M r!) + r Proof: Use the otatos Theore 8 For each block B, =,,, /( v), we frst costruct k + blocks B (), B (),, B (k+) Ad the buld a cell-level sequece q = (B (), B(),, B(k+),, B () /(), B() /(),, B (k+) /() ) of legth = (k + ) + v = (+r) + v, such that every two adacet blocks overlap by v Defe B (k+) 0 = (,,, v, l, l l,, l v) The the frst v cell levels of B () are the sae as the last v cell levels B (k+), =,, /( v), ad the frst v cell levels of B ( ) are exactly the last v cell levels of B ( ), =,, k + So we are left to buld the last v cell levels of B ( ) Assg (c, c,, c ) to the last v cell levels of B () For B (), the (v + )-th through the (v + r)-th cell levels take r D ubers fro [, D] that are dfferet fro B We have r! choces here Ad the last s cell levels are assged s values fro {l, l l,, l v} that are dfferet fro the last (k )( v) + s cell levels of B (k+) Thus detcal levels a block are avoded ad we have A s choces here For the last v cell levels of B ( ), = 3,, k +, we pck v values fro {l, l l,, l v} that are ot the last ( v)( 3) + s cell levels of B ( ) or the last (k + )( v) + s cell levels of B (k+) There are A () s choces for each = 3,, k + Fally, the last v cell levels of B (k+) are chose fro {l, l l,, l v} such that they are dfferet fro the last ( v)(k ) + s cell levels of B (k+), whch results ( v)! choces Notce q s a vald cell-level sequece of C(,, D, v) as each cell level s o ore tha D ad ay block q has dfferet levels For each codeword s S, there are at least M r! such cell-level sequeces, deoted by Q s Also otce that slar to Theore 8, ether two cell-level sequeces Q s or two cell-level sequeces dstct Q s ad Q s duce the sae perutato sequece for C(,, D, v) Moreover, the uber of dstct perutato sequeces C(,, D, v) s at least s S Q s S(,, D, v) (M r!) Therefore, cap(c) l log S(,,D,v) (M r!) = () log S(,,D,v) l + log(m r!) (+r) +r = ()cap(s) +r + log(m r!) +r = log(z M r!) +r Thus we have proved the theore Corollary Let C(,, D, v) be a BRM code, ad S(,, D, v) be a Star BRM code The f < v ad D + r, cap(c) > ( v)cap(s) + r = cap(s) k + VI CONCLUDING REMARKS The questo of what overlap provdes the hghest capacty for a gve perutato sze ad a gve axu level s partally dscussed ths paper Deote ths optal overlap by v (D) The followg observatos show the two extree cases ad the result of Theore 5: ) Whe D =, cap(c(,,, v)) = log()! Therefore, cap(c(,,, 0)) > cap(c(,,, )) > > cap(c(,,, )) We ca coclude that v () = 0 ) Whe D =, t s clear that log(!/v!) cap(c(,,, v)) = The cap(c(,,, 0)) < cap(c(,,, )) < < cap(c(,,, )), whch ples that v ( ) = 3) Whe D +, cap(c(,, D, )) > cap(c(,, D, 0)) So the optal overlap for D + satsfes v (D) The optal overlap values for + D < are ot thoroughly exaed, whch ca be our future work drecto Besdes, for ay rate o ore tha the capacty, the exact fors of the perutato ecoders ad decoders addto to ther effcecy ad coplextes are stll left to be worked o I addto, a geeralzed BRM code ca be vewed as a set of cells, aog whch we choose subsets of sze ad for perutatos All cell levels are o ore tha D ad two subsets ay overlap Uder ths fraework, how to ake choces of the subsets so as to optze the capacty s stll a ope proble I suery, ths paper used the tools of labeled graphs to fd the capactes of BRM codes wth oe overlap I partcular, t showed that f two extra charge levels are gve, oe ca use the by way of overlap ad acheve hgher capacty tha o-overlap codes I addto, star BRM code s troduced to obta a lower boud for the capacty of BRM codes REFERENCES [] V Bohossa, A Jag ad J Bruck, Buffer codes for asyetrc ult-level eory, Proc IEEE ISIT, 007, pp [] J E Brewer ad M Gll, Novolatle eory techologes wth ephass o flash, Chapter, Wley-IEEE, 007 [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, 007 [4] H Fucae, Z Lu ad M Mtzeacher, Desgg floatg codes for expected perforace, Proc of the 46th Aual Allerto Coferece, 008

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