Rank Modulation for Flash Memories

Transcription

1 ISIT 008, Toroto, Caada, July 6 -, 008 Rak Modulatio for Flash Memories Axiao (Adrew) Jiag Robert Mateescu Moshe Schwartz Jehoshua Bruck Departmet of Computer Sciece Califoria Istitute of Techology Electrical ad Computer Egieerig Texas A&M Uiversity 00 E Califoria Blvd, Mail Code 6-9 Be-Gurio Uiversity College Statio, TX 778, USA Pasadea, CA 95, USA Beer Sheva 805, Israel ajiag@cstamuedu {mateescu,bruck}@paradisecaltechedu schwartz@eebguacil Abstract We explore a ovel data represetatio scheme for multi-level flash memory cells, i which a set of cells stores iformatio i the permutatio iduced by the differet charge levels of the idividual cells The oly allowed charge-placemet mechaism is a push-to-the-top operatio which takes a sigle cell of the set ad makes it the top-charged cell The resultig scheme elimiates the eed for discrete cell levels, as well as overshoot errors, whe programmig cells We preset urestricted Gray codes spaig all possible -cell states ad usig oly push-to-the-top operatios, ad also costruct balaced Gray codes We also ivestigate optimal rewritig schemes for traslatig arbitrary iput alphabet ito -cell states which miimize the umber of programmig operatios I INTRODUCTION Flash memory is a o-volatile memory techology that is both electrically programmable ad electrically erasable Its reliability, high storage desity, ad relatively low cost have made flash memory a domiao-volatile memory techology ad a promiet cadidate to replace the wellestablished magetic recordig techology i the ear future The most cospicuous property of flash storage is its iheret asymmetry betwee cell programmig (charge placemet) ad cell erasig (charge removal) While addig charge to a sigle cell is a fast ad simple operatio, removig charge from a sigle cell is very difficult I fact, curret flash memories do ot allow a sigle cell to be erased but rather oly a large block of cells Thus, a sigle-cell erase operatio requires the cumbersome process of copyig a etire block to a temporary locatio, erasig it, ad the programmig all the cells except for the sigle cell to be erased To keep up with the ever-growig demad for deser storage, the multi-level flash cell cocept is used to icrease the umber of stored bits i a cell [] Istead of the usual sigle-bit flash memories, where each cell is i oe of two states (erased/programmed), each multi-level flash cell stores oe of q levels ad ca be regarded as a symbol over a discrete alphabet of size q This is doe by desigig a appropriate set of threshold levels which are used to quatize the charge level readigs to symbols from the discrete alphabet Fast ad accurate programmig schemes for multi-level flash memories are a topic of sigificat research ad desig efforts [9], [5], [] All these ad other works share the attempt to iteratively program a cell to a exact prescribed charge level i a miimal umber of programmig cycles As metioed This work was supported i part by the Caltech Lee Ceter for Advaced Networkig above, flash memory techology does ot support charge removal from idividual cells As a result, the programmig cycle sequece is desiged to cautiously approach the target charge level from below so as to avoid udesired global erases i case of overshoots Cosequetly, these attempts still require may programmig cycles, ad they work oly up to a moderate umber of levels per cell I additio to the eed for accurate programmig, the move to multi-level flash cells also aggravates reliability The same reliability aspects that have bee successfully hadled i sigle-level flash memories may become more proouced ad traslate ito higher error rates i stored data Oe such relevat example is errors that origiate from low memory edurace [], by which a drift of threshold levels i agig devices may cause programmig ad read errors We therefore propose the rak modulatio scheme, whose aim is to elimiate both the problem of overshootig while programmig cells, ad the problem of memory edurace i agig devices I this scheme, a ordered set of multi-level cells stores the iformatio i the permutatio iduced by the charge levels of the cells I this way, o discrete levels are eeded (ie, o eed for threshold levels) ad oly a basic charge-comparig operatio (which is easy to implemet) is required to read the permutatio If we further assume that the oly programmig operatio allowed is raisig the charge level of oe of the cells above the curret highest oe (pushto-the-top), the the overshoot problem is o loger relevat Additioally, the techology may allow i the ear future the decrease of all the charge levels i a block of cells by a costat amout smaller tha the lowest charge level (block deflatio), which would maitai their relative values, ad thus leave the iformatio uchaged This ca elimiate a desigated erase step, by deflatig the etire block wheever the memory is ot i use Oce a ew data represetatio is defied, several tools are required to make it useful I this paper we preset Gray codes that exploit the full represetatioal power of rak modulatio, ad data rewritig schemes Error-correctig codes for rak modulatio are preseted i a compaio paper [8] The origial Gray code [] has bee geeralized i coutless ways, ad has bee used i a wide rage of applicatios Some of the Gray code costructios we describe iduce a simple algorithm for geeratig the list of permutatios Efficiet geeratio of permutatios has bee the subject of much research as described i the geeral survey [0], ad the more /08/$ IEEE 7

2 ISIT 008, Toroto, Caada, July 6 -, 008 specific [] (ad refereces therei) I [] the trasitios we use i this paper are called ested cyclig ad the algorithms cited there produce lists which are ot Gray codes sice some of the permutatios repeat, which makes the algorithms iefficiet We preset a balaced costructio, which is a ew permutatio geeratio algorithm, that optimizes the trasitio step size ad is suitable for block deflatio We also ivestigate rewritig schemes for rak modulatio Sice erasig/reprogrammig cells is expesive, it is very importat to maximize the umber of times data ca be rewritte betwee two erasure operatios [6] For rak modulatio, the key is to miimize the highest charge level of cells We preset two rewritig schemes that are, respectively, optimized for the worst-case ad average-case performace II DEFINITIONS AND BASIC CONSTRUCTION Let S be a state space, ad let T be a set of trasitio fuctios, where every t T is a fuctio t : S S AGray code is a ordered list s, s,,s m of distict elemets from S such that for every i m, s i+ = t(s i ) for some t T Ifs = t(s m ) for some t T, the the code is cyclic If the code spas the etire space S we call it complete Let [] deote the set of itegers {,,, } A ordered set of flash memory cells amed,,,, each cotaiig a distict charge level, iduces a permutatio of [] by writig the cell ames i descedig charge level [a, a,,a ], ie, the cell a has the highest charge level while a has the lowest The state space for the rak modulatio scheme is therefore the set of all permutatios over [], deoted by S We cosider the basic miimal-cost operatio o a give state to be a push-to-the-top, by which a sigle cell has its charge level icreased to become the highest of the set Thus, the set T of miimal-cost trasitios betwee states cosists of fuctios t i, i : t i ([a,,a i, a i, a i+,,a ]) = [a i, a,,a i, a i+,,a ] Throughout this work, our state space S will be the set of permutatios over [], ad our set of trasitio fuctios will be the set T of push-to-the-top fuctios We call such codes legth Rak Modulatio Gray Codes (-RMGC) Example A example of a cyclic ad complete -RMGC is give below The permutatios are the colums beig read from left to right The sequece of trasitios is: t, t, t, t, t, t We will ow show a basic recursive costructio for - RMGCs The resultig codes are cyclic ad complete, ithe sese that they spa the etire state space Our recursio basis is the simple -RMGC: [, ], [, ] Now let us assume we have a cyclic ad complete ( )-RMGC, which we call C,defied by the sequece of trasitios t (), t (),,t (( )!) ad where t (( )!) = t, ie, a push-to-the-top operatio o the secod elemet i the permutatio We further assume that the trasitio t appears This last requiremet merely restricts us to have t used somewhere sice we ca always rotate the set of trasitios to make t be the last oe used at least twice We will ow show how to costruct C, a cyclic ad complete -RMGC with the same property We set the first permutatio of the code to be [,,,], ad the use the trasitios t (), t (),,t (( )! ) to get a list of ( )! permutatios we call the first block of the costructio By our assumptio, the permutatios i this list are all distict, ad they all share the property that their last elemet is (sice all the trasitios use just the firs elemets) Furthermore, sice t (( )!) = t, we kow that the last permutatio geerated so far is [,,,,, ] We ow use to create the first permutatio of the secod block, ad the use t (), t (),,t (( )! ) agai to create the etire secod block We repeat this process times, ie, use the sequece of trasitios t (), t (),,t (( )! ), a total of times to costruct blocks, each cotaiig ( )! permutatios The followig two simple lemmas are give without proof Lemma I ay block, the last elemet of all the permutatios is costat The list of last elemets i the blocks costructed is,,,, The elemet is ever a last elemet Lemma The secod elemet i the first permutatio i every block is Thefirst elemet i the last permutatio i every block is also Combiig the two lemmas above, the blocks costructed so far form a cyclic buot complete -RMGC, that we call C, which may be schematically described as follows (where each box represets a sigle block, ad deotes the sequece of trasitios t (),,t (( )! ) ): t It is ow obvious that C is ot complete because it is missig exactly the ( )! permutatios cotaiig as their last elemet We build a block C cotaiig these permutatios i the followig way: we start by rotatig the list of trasitios t (),,t (( )!) such that its last trasitio is t For coveiece we deote the rotated sequece by τ (),,τ ( )!, where τ ( )! = Thefirst permutatio i the block is [a, a,,a,], ad the last oe is [a,,a, a,]ic we fid a trasitio of the followig form: [a,,a,,a ] [, a,,a, a ]Sucha trasitio must surely exist sice C is cyclic, it cotais permutatios i which is ext to last ad some i which it is ot, it does ot cotai permutatios i which is last, ad so it follows that at some poit i C, the elemet is ext to last ad is the pushed by to the frot At this trasitio we split C ad isert C as follows bellow, where it is easy to see that all trasitios are valid Thus we have created C ad to complete the recursio we have to make sure t appears at least twice, but that is obvious sice the sequece The trasitio must be preset somewhere i the sequece or else the last elemet would remai costat, thus cotradictig the assumptio that the sequece geerates a cyclic ad complete ( )-RMGC 7

3 ISIT 008, Toroto, Caada, July 6 -, 008 a a a a a a a a a a a a a a a a t (),,t (( )! ) cotais at least oe occurrece of t,ad is replicated times, Therefore, we coclude that: Corollary For every there exists a cyclic ad complete -RMGC The -RMGC show i Example is the result of this costructio for = III BALANCED -RMGCS It is sometimes the case that due to precisio costraits i the charge placemet mechaism, the actual possible charge levels i flash memory cells are discrete Thus, we defie the fuctio c i : N N, where c i (p) is the charge level of the i-th cell after the p-th programmig cycle It follows that if we use trasitio t j i the p-th programmig cycle ad the i-th cell is, at the time, j-th from the top, the c i (p) > max k {c k (p )}, ad for k = i, c k (p) =c k (p ) I a optimal settig with o overshoots, c i (p) =max k {c k (p )} + The jump i the p-th roud is defied as c i (p) c i (p ), assumig the i-th cell was the affected oe It is desirable, whe programmig cells, to make the jumps as small as possible We defie the jump cost of a -RMGC as the maximal jump durig the trasitios dictated by the code It is easy to see that the lowest possible jump cost i a complete -RMGC is at leas +, for We call a -RMGC with jump cos + a balaced - RMGC A balaced code is especially suitable if block deflatio is possible (Sectio I) We show a costructio that turs ay ( )-RMGC ito a balaced -RMGC while retaiig properties such as beig cyclic or complete The resultig recursive scheme is a ew permutatio geeratio algorithm, that obeys the geometric costraits of flash memories Theorem 5 Give a cyclic ad complete ( )-RMGC C,defied by the trasitios t i,,t i( )!, the the followig trasitios defie a -RMGC, deoted by C, that is cyclic, complete ad balaced: { t i k/ For k {,,}, t k = +, if k (mod ), otherwise Proof: Let us defie the abstract trasitio t i, i, that pushes to the bottom the i-th elemet from the bottom: ti ([a,,a i, a i+, a i+,,a ]) = [a,,a i, a i+,,a, a i+ ] Because C is cyclic ad complete, usig t i,, t i( )! startig with [a,,a ] produces a complete cycle through S, ad usig them startig with [a,,a ] creates a cycle through all the ( )! permutatios of [] where a is fixed o the first positio Figure Recursive costructio of the balaced -RMGC The ( )! permutatios of [] produced by ti,, t i( )! are also represetatives of the ( )! distict orbits of the permutatios of [] uder the operatio This meas that there are o two permutatios which are cyclic shifts of each other, sice represets a simple cyclic shift whe operated o a permutatio of [] Takig a permutatio of [], the usig the trasitio i+ oce, i, followed by times usig, is equivalet to usig t i Every trasitio of the form i+, i =, moves us to a differet orbit, while the cosecutive executios of geerate all the elemets of the orbit It follows that the resultig permutatios are distict Schematically, the costructio of C based o C is: times times times {}}{{}}{{}}{ i +,,,, i +,,,,, i( )!+,,, }{{}}{{}}{{} ti ti t i( )! The code C is balaced, because i every block of trasitios startig with a i+, i, wehave: the trasitio i+ has a jump of i + ; the followig i trasitios have a jump of +, ad the rest a jump of I additio, because C is cyclic ad complete, it follows that C is also cyclic ad complete We ca use Theorem 5 to recursively costruct all the supportig j-rmgcs, j {,,}, with the basis of the recursio beig the -RMGC: [, ], [, ] Corollary 6 For ay, there exists a cyclic, complete ad balaced -RMGC Example 7 Fig shows the recursive balaced -RMGC The permutatios are represeted as a by ( )! matrix Each row is a orbit geerated by Each colum has the last elemet fixed The trasitios betwee rows occur whe is the top (leftmost) elemet These trasitios are defied recursively, by a balaced -RMGC over the set {,, } (where the top elemet is ow the rightmost oe): [,,],[,,],[,,],[,,],[,,],[,,] They are t, t, t, t, t, t This is the cycle from Example, with relabeled cells, ad startig with the third colum The recursive balaced -RMGC of Theorem 5 is optimal with respect to the followig asymptotic measure I practice, it is importat to optimize the cost of decidig the trasitio that geerates the ext permutatio We defie a step to be a sigle query of the form what is the i-th highest charged 7

4 ISIT 008, Toroto, Caada, July 6 -, 008 cell? If we start with [a,,a ], the a fractio of of the trasitios are, ad they occur wheever the cell a is ot the highest charged oe Of the cases where a is highest charged, by recursio, a fractio of the trasitios are determied by just oe more query, ad so o At the basis of the recursio, permutatios over two elemets require zero additioal queries Thus, the total umber of queries is i= i! i= Sice lim i! =, the asymptotic average umber of steps to geerate the ext permutatio is just IV REWRITING WITH RANK MODULATION CODES I this sectio, we study rewritig data usig the rak modulatio scheme The objective is to miimize the expesive cell erasure operatios, which, i tur, requires us to maximize the umber of times the data ca be modified before a cell erasure becomes ecessary (ie, whe the highest cell charge level reaches the highest allowed value) We firseed to defie a decodig scheme It is ofte the case that the alphabet size used by the user to iput data ad read stored iformatio differs from the iteral represetatio alphabet size I our case, data is stored iterally i oe of differet permutatios Let us assume the user alphabet is Q = {,,, q} A decodig scheme is a fuctio D : S Q mappig iteral states to symbols from the user alphabet Suppose the curret iteral state is s S ad the user iputs a ew symbol α Q Arewritig operatio for α is ow defied as movig from state s S to state s S such that D(s )=α It should be oted that if D(s )=α the s may be equal to s, ie, the rewritig operatio is degeerate ad does othig The cost of the rewritig operatio is the miimal umber of atomic trasitios from T (ie, the umber of push-to-the-top operatios) required to move from state s to state s I the followig sectios, we first preset a decodig scheme that strictly optimizes the rewritig cost for the worst case The, we exted the costructio to optimize the average rewritig cost with costat approximatio ratios A Optimal Decodig Scheme for Rewritig We start by presetig a lower boud o the cost of a sigle rewritig operatio First, we defie a few terms Defie the trasitio graph G =(V, E) as a directed graph with V = S, ie, with vertices represetig the permutatios i S There is a directed edge u v if v = t(u) for some t T, ie, we ca obtai v from u by a sigle push-to-the-top operatio We ca see that G is a regular digraph: every vertex has icomig edges ad outgoig edges For two vertices u, v V, defie the directed distace d(u, v) as the umber of edges i the shortest directed path from u to v Clearly, 0 d(u, v) Give a vertex u V ad a iteger r (here 0 r ), we defie the ball B r (u) as B r (u) ={v V d(u, v) r}, ad defie the sphere S r (u) as S r (u) ={v V d(u, v) =r} Clearly, B r (u) = 0 i r S r (u) We skip the proof of Lemma 8 due to space costrait Iterested readers please see [7] Lemma 8 u V ad 0 r, B r (u) = ( r)! For r, S r (u) = ( r)! ( r+)! Let ρ deote the smallest iteger such that B ρ (u) q Note that ρ is idepedet of u The followig lemma presets a boud o the rewritig cost (Please see [7] for proof) Lemma 9 For ay decodig scheme ad ay curret cell state, there exists α Q such that the cost of a rewritig operatio for α is at least ρ(ρ is as defied above) Next, we preset a optimal code costructio Costructio 0 Divide the states S ito sets, ( ρ)! where two states are i the same set if ad oly if their ρ top-charged cells are the same Amog the sets, choose q sets ad map them to the q symbols of Q arbitrarily The other q sets eed ot represet ay symbol ( ρ)! Example Le = ad q = Sice B (u) =, ρ = Wedividethe = 6 states ito = sets ( ρ)! which iduce the decodig fuctio: {[,,], [,, ]}, {[,, ], [,, ]}, ad{[,, ], [,, ]} The two states i the set are decoded to the same symbol from Q The cost of ay rewrite operatio is at most Sice the top ρ cells of a state uiquely determie the decoded symbol, ay rewritig operatio costs at most ρ trasitios to replace the top ρ cells Thus we have: Theorem The codig scheme i Costructio 0 is optimal i terms of miimizig the worst-case rewritig cost B Optimizig the Average Cost of Rewritig If the probabilities with which the iput symbol takes values from its alphabet are kow, it is also importat to study schemes that optimize the average cost of rewritig Let us assume that for each rewrite, the iput symbol is draw iid from Q = {,,,q} with probability p i to get symbol i Q We study decodig schemes that optimize the average cost of rewritig Depedig o the probabilities {p i },the optimal code may be quite complex we preset below a prefix code that is optimal i terms of its ow desig objective Furthermore, we will prove that whe q /, the prefix code is a -approximatio of ay optimal rak-modulatio solutio We will also show that whe ad q /6, the prefix code is a -approximatio The prefix code cosists of q codewords of variable legths, which represet the q values i Q Each codeword is a prefix of a permutatio from S No codeword is allowed to be the prefix of aother codeword Let a =[a, a,,a i ] be a geeric codeword that represets the value α Q For a state s S,ifa is a prefix ofs the we set D(s) =α Due to the prefix-free property, the decodig fuctio is well-defied A prefix code ca be represeted by a tree First, let us defie a full permutatio tree T where the labels o the paths from its root to leaves are all the permutatios A example is show i Fig (a) The vertices are i layers, with the root i layer 0 ad leaves i layer A prefix code correspods to a subtree C of T (see Fig (b) for a example) Every codeword is mapped to a leaf, ad the codeword is the same as the labels o the path from the root to the leaf 7

5 ISIT 008, Toroto, Caada, July 6 -, 008 (a) (b) () () (,) (,) (,) (,) (,) (,,) (,,) Figure Prefix rak modulatio code for = ad q = 9 (a) The full permutatio tree T (b) A prefix code represeted by a subtree C of T The leaves represet the codewords, which are the labels beside the leaves For i Q, letc i deote the codeword represetig i, ad let c i deote its legth (The codewords i Fig (b) have miimumlegth of admaximum legth of ) Our objective is to miimize the average codeword legth, q i= p i c i, which upper bouds the expected cost of each rewrite The optimal prefix code caot be costructed with a greedy algorithm like the Huffma code ad its extesios, because the vertex degrees i the code tree C are ukow iitially We preset a dyamic programmig algorithm of time complexity O(q ) to costruct the optimal code The algorithm computes a set of fuctios opt i (l, m), for i =,,,, l = 0,,, q, ad m = 0,,,mi{q, /( i)!} We iterpret the meaig of opt i (l, m) as follows We take a subtree of T that cotais the root The subtree has exactly l leaves i the layers i, i +,, Italsohasatmostm vertices i the layer i We let the l leaves represet the l iput values from Q with the lowest probabilities p j : the further the leaf is from the root, the lower the correspodig probability is Those leaves are also l codewords, ad we call their weighted average legth (where the probabilities p j are weights) the value of the subtree The miimum value of such a subtree (amog all such subtrees) is defied to be opt i (l, m) Clearly, the miimum average codeword legth of a prefix code equals opt (q, ) Without loss of geerality, let us assume that p p p q It is easily see that the followig recursio holds: () op (l, m) = ( ) l k= p k for m l > 0; () opt i (0, m) = 0 for i > ; () opt i (l, m) = mi 0 j mi{l,m} {opt i+ (l j, mi{q, (m j)( i)})+ l k=l j+ ip k} for i <, l > 0, m > 0 The last recursio holds because a subtree with l leaves i layers i, i +,, ad at most m vertices i layer i ca have 0,,,mi{l, m} leaves i layer i The algorithm computes op (l, m), op (l, m),, opt (q, ) usig the above recursios Give their values, it is straightforward to determie i the optimal code, how may codewords are i each layer, ad therefore determie the optimal code itself For rewritig based o the code, to chage the stored value to i Q, we simply push the c i cells i its codeword c i to the top sequetially Theorem Whe q /, for every rewrite, the expected rewritig cost of a optimal prefix code is at most three times that of ay rak modulatio code Whe ad q /6, this ratio is at most two Proof: We preset the sketch of the proof for the case q / (The case ad q /6 ca be aalyzed similarly) For the detailed proof, please refer to [7] Let i Q (resp, s i S ) deote the stored data (resp, cell state) at a give momet Let s,, s i, s i+,, s q deote the q cell states whose distace from s i i the trasitio graph, d(s i, s j ), are the smallest oes WLOG, assume that p p i p i+ p q, ad that d(s i, s ) d(s i, s i ) d(s i, s i+ ) d(s i, s q ) To miimize the expected rewritig cost, the ideal solutio is a code that decodes s j as j for j Q Deote by α the expected rewritig cost of this ideal solutio Next, we desig a prefix code B with this property: j Q, ifj = i, its correspodig codeword legth, y j, is at most d(s i, s j ); if j = i, the y j = It ca be proved that such a prefix code B exists Next, let A be a optimal prefix code, ad for j Q, letx j deote the correspodig codeword legth Let β deote the expected rewritig cost of A Bydefiitio, j q p j x j j q p j y j Sicex i = y i, β j q, j =i p j x j j q, j =i p j y j j q, j =i p j d(s i, s j ) = α So the expected rewritig cost of a optimal prefix code is at most three times that of a ideal rak modulatio code V CONCLUSION I this paper, we preset a ovel data storage scheme, rak modulatio, for flash memories We preset several Gray code costructios for rak modulatio, as well as its data rewritig schemes The preseted codig schemes are optimized for cell programmig cost i several differet aspects REFERENCES [] A Badyopadhyay, G J Serrao, ad P Hasler, Programmig aalog computatioal memory elemets to 0% accuracy over 5 decades usig a predictive method, i Proceedigs of the IEEE Iteratioal Symposium o Circuits ad Systems, 005, pp 8 5 [] P Cappelletti ad A Modelli, Flash memory reliability, i Flash Memories, P Cappelletti, C Golla, P Olivo, ad E Zaoi, Eds Kluwer, 999, pp 99 [] B Eita ad A Roy, Biary ad multilevel flash cells, i Flash Memories, P Cappelletti, C Golla, P Olivo, ad E Zaoi, Eds Kluwer, 999, pp 9 5 [] F Gray, Pulse code commuicatio, US Patet 6058, March 95 [5] M Grossi, M Lazoi, ad B Riccò, Program schemes for multilevel flash memories, Proceedigs of the IEEE, vol 9, o, pp 59 60, 00 [6] A Jiag, V Bohossia, ad J Bruck, Floatig codes for joit iformatio storage i write asymmetric memories, i Proc IEEE ISIT, 007, pp [7] A Jiag, R Mateescu, M Schwartz, ad J Bruck, Rak modulatio for flash memories, Califoria Istitute of Techology, Tech Rep, 008 [Olie] Available: [8] A Jiag, M Schwartz, ad J Bruck, Error-correctig codes for rak modulatio, i Proc IEEE ISIT, 008 [9] H Nobukata et al, A -Mb, eight-level NAND flash memory with optimized pulsewidth programmig, IEEE J Solid-State Circuits, vol 5, o 5, pp , 000 [0] C D Savage, A survey of combiatorial Gray codes, SIAM Rev, vol 9, o, pp , 997 [] R Sedgewick, Permutatio geeratio methods, Computig Surveys, vol 9, o, pp 7 6,