Coding for Write lstepup Memories


 Quentin Andrew Weaver
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1 Codig for Write lstepup Memories Yeow Meg Chee, Ha Mao Kiah, A. J. Ha Vick Va Khu Vu, ad Eita Yaakobi Departmet of Idustrial Systems Egieerig ad Maagemet, Natioal Uiversity of Sigapore, Sigapore School of Physical ad Mathematical Scieces, Nayag Techological Uiversity, Sigapore Istitute of Digital Sigal Processig, Uiversity of DuisburgEsse, Duisburg, Germay Computer Sciece Departmet, Techio Israel Istitute of Techology, Haifa, Israel s: Abstract I this work, we propose ad study a ew class of obiary rewritig codes, called write lstepup memories (WlM codes. From a iformatiotheoretic poit of view, this codig scheme is a geeralizatio of obiary writeoce memories (WOM codes. From a practical poit of view, this codig scheme ca be used ot oly to icrease the lifetime of flash memories but also mitigate their overshootig problem. We first provide a exact formula for the capacity regio ad the maximum sumrate of WlM codes. Lastly, we preset several explicit costructios of highrate WlM codes with efficiet ecodig/decodig algorithms. I. INTRODUCTION Writeoce memory (WOM is a biary iformatio storage medium where the state of each cell ca be chaged from 0 to 1 but ot vice versa. I 198, Rivest ad Shamir [1] studied a codig scheme to reuse writeoce memory which is kow i literature as WOM codes. Later, Fiat ad Shamir [] studied a codig scheme for geeralized writeoce memory where each cell has q levels that ca oly be icreased. This codig scheme is called obiary WOM codes. I the 1980 s ad 1990 s, a few differet models of WOM codes were studied due to its beauty from a iformatiotheoretical poit of view ad its applicatios i puch cards ad optical disks [] [6]. Furthermore, there are several versios of WOM codes, icludig write uidirectioal memory [7], write isolated memory [8], ad defective chael [9], [10]. Recetly, WOM codes have gaied sigificat attetio thaks to their applicatios i flash memories [11] [17]. Flash memory, iveted i 1984 by Masuoka [18], has become a popular ovolatile storage techology due to its high capacity, low power cosumptio, ad low cost. However, flash memory still faces several major challeges such as limited lifetime ad oisy programmig. The fudametal data storage elemet i flash memories is a floatiggate (FG trasistor, kow as a cell. I a sigle level cell (SLC, each cell has two levels ad thus is able to store a sigle bit, ad i a multilevel cell (MLC, each cell has q > levels ad is able to store log q bits. While it is possible to icrease a cell level by ijectig a appropriate charge amout ito the cell, it is impossible to decrease a cell level without first erasig a whole block of cells. This erasure operatio is ot oly expesive i terms of time but also reduces the lifetime of flash memories. Each block of cells i a flash memory ca be erased a limited umber of times. To improve the lifetime of flash memory, it is possible to use WOM codes to write multiple messages before the whole block is erased. This work was doe while Y. M. Chee was with the School of Physical ad Mathematical Scieces, Nayag Techological Uiversity, Sigapore. O top the limited lifetime of flash memory, it is also challegig to precisely program the cells. Oe of the mai obstacles i achievig accurate programmig is overshootig. Whe a large amout of charge is ijected ito a cell, it may be over charged tha ecessary ad thus a error occurs. Oe approach to overcome this difficulty is by usig codig schemes such as asymmetric errorcorrectig codes [19] ad rakmodulatio codes [0]. The goal of this paper is to iitiate the study of a codig scheme which combats both limitededurace ad overshootig i flash memory. Oe approach to meet this challege is by usig errorcorrectig codes together with WOM codes as was studied i [1], [1], []. However, i order to directly reduce the overshootig problem, this work proposes to use a geeralizatio of obiary WOM codes where each cell has q levels that ca oly be icreased by at most some l < q levels. These codes are called write lstepup memories (WlM codes. Similarly to WOM codes, there are four models of WlM codes which deped upo whether the ecoder ad the decoder are iformed or uiformed with the previous state of the memory [3] ad each model ca be ivestigated i two cases: ɛerror ad zeroerror. I this paper, we focus o the more practical model where the ecoder is iformed ad the decoder is uiformed for the zeroerror case. Although this paper is the first to study WlM codes, there are several closely related models such as codes for eduracelimited memories (ELM [3] ad writecostraied memories (WCM codes [4]. Results o the exact capacity regio ad the maximum sumrate of WlM codes i some cases ca be accomplished usig kow techiques. However, we are ot aware of ay explicit costructio of highrate WlM codes i the literature. Before we preset these results, we itroduce some ecessary otatios ad defiitios. A. Notatios ad Defiitios I this sectio, we defie WlM codes formally ad preset several related defiitios that will be used throughout the paper. For a positive iteger a, the set {0,..., a 1 is defied by [a]. A vector c [q] will be called a cellstate vector. The complemet of a biary vector c is deoted by c. That is c + c = 1, where 1 is the all oes vector. For two vectors x = (x 1,..., x ad y = (y 1,..., y, x y if ad oly if x i y i for all 1 i. The vector c = max{x, y is defied by c i = max{x i, y i for all 1 i. A vector p = (p 1,..., p is called a probability vector if 0 p i 1 for all 1 i ad i=1 p i = 1. Uless stated otherwise, all logarithms i this paper are take accordig to base. For each probability vector p, we defie the etropy fuctio H(p = i=1 p i log p i. Whe = ad p = (1 p, p, the
2 etropy fuctio is h(p = p log p (1 p log(1 p. We first recall the defiitio of obiary WOM codes. Defiitio 1. A [, t; M 1,..., M t ] q qary twrite WOM code is a codig scheme comprisig of qary cells ad is defied by t pairs of ecodig ad decodig maps E j ad D j for 1 j t. For the map E j, its image is Im(E j for 1 j t. By defiitio, Im(E 0 = {(0,..., 0. For 1 j t, the ecodig ad decodig maps are defied as follows. (1 E j : [M j ] Im(E j 1 [q 1], ( such that for all (m, c [M j ] Im(E j 1 it holds that E j (m, c c. D j : Im(E j [M j ], such that D j (E j (m, c = m for all (m, c [M j ] [q 1]. Next, qary twrite WlM codes are defied. Defiitio. A [, t, l; M 1,..., M t ] q qary twrite WlM code is a codig scheme comprisig of qary cells ad is defied by t pairs of ecodig ad decodig maps E j ad D j for 1 j t. For the map E j, its image is Im(E j for 1 j t. By defiitio, Im(E 0 = {(0,..., 0. For 1 j t, the ecodig ad decodig maps are defied as follows. (1 E j : [M j ] Im(E j 1 [q 1], ( such that for all (m, c [M j ] Im(E j 1 it holds that c = (c 1,..., c = E j (m, c, where c i c i [l] for all 1 i. D j : Im(E j [M j ], such that D j (E j (m, c = m for all (m, c [M j ] [q 1]. The followig defiitios apply both for WOM ad WlM codes. The rate o the jth write is the ratio betwee the umber of writte bits ad the umber of cells, that is, log Mj R j =. The sumrate is the sum of all rates o t writes, that is, R sum = t j=1 R j. I qary twrites WlM code, a rate tuple (R 1,..., R t is said to be achievable if for ay ɛ > 0, there exists a [, t, l; M 1,..., M t ] q qary twrite WlM code log Mj such that R j ɛ for all 1 j t. The capacity regio of qary twrites WlM is the set of all achievable rate tuples ad is deoted by C q,t,l, ad the maximum sumrate is deoted by R q,t,l. From the defiitios above, we observe that WOM codes are a special case of WlM codes whe l = q 1. There are several more kow families of codes i the literature which are closely related to WlM code ad are reviewed ext. B. Related Work First, we recall that the proposed WlM codes are a geeralizatio of obiary WOM codes where every icrease i cell level is at most some value l. Gabrys et al. [5] studied a relevat model where every cell level icrease is at least l. WlM codes are also closely related to the recetly studied edurace limited (ELM codes [3]. I ELM, each cell has two states which ca be chaged at most some b times. For every cell, its coutvector is a vector which idicates the umber of times each cell chaged its state. Whe a cell chages its state i ELM, the correspodig coordiate i the coutvector is icreased by oe. Hece, this coutvector is similar to a word i WlM code for l = 1. Yet, these codes are still ot idetical ad furthermore the coutvector is ot always available i ELM. There are a few differet models i ELM which deped upo whether the ecoder ad decoder are iformed or uiformed. For the formal defiitios of these models, we refer to the referece [3]. Let Ct,q 1 EIA:DI ad C EIA:DU t,q 1 be the capacity regio of (q 1chage twrite ELM codes i the EIA : DI ad EIA : DU model, respectively. Accordig to these defiitios, we ca already obtai the followig result o the capacity regio of qary twrite WlM code with l = 1. Propositio 3. C EIA:DI t,q 1 C q,t,1 C EIA:DU t,q 1. Furthermore, it is kow from [3] that C EIA:DI C EIA:DU t,q 1 = t,q 1 = Ĉt,q 1, where the regio Ĉt,q 1 is defied recursively { as follows. Ĉ t,q 1 = (R 1,..., R t R 1 h(p, p [0, 1], for j t, R j p R j + (1 p R j, (R,..., R t Ĉt 1,q ad (R,..., R t Ĉt 1,q 1 where Ĉt,0 =, ad for all q 1 t 1, we set Ĉ t,q 1 = Ĉ t,t = [0, 1] t. Therefore, for qary twrite WlM whe l = 1, the capacity regio is C q,t,1 = Ĉt,q 1 ad the maximum sumrate is R q,t,1 = log q 1 ( t i, [3]. Recetly, Kobayashi et al. [4] studied a codig scheme for writecostraied memories (WCM which is a WOM code with costs o the statetrasitios. I this codig scheme, each cellstate trasitio from level i to level j has a cost, deoted by c(i j. Usig this otatio, if we assig c(i j = 1 for j i [l] ad c(i j = otherwise, the we ca obtai WlM codes. The work of [4] exteded the results of Fu ad Vick [6] to obtai the capacity regio of WCM codes. However, it is ot possible to explicitly derive the capacity regio C q,t,l usig the expressio from [4] ad costructig explicit capacityachievig codes still remais a iterestig challege which is addressed i the paper. II. THE CAPACITY OF qary WlM I this sectio, we preset the capacity regio ad the maximum sumrate of WlM codes. The followig rate tuples regio is defied recursively, { Ĉ q,t,l = (R 1,..., R t p = (p 0,..., p l is a probability vector R 1 h(p, for j t, R j p i Rj, i (R, i..., Rt i Ĉt 1,q 1 i,l for 0 i l. The ext theorem establishes our result o the capacity of WlM. The proof will appear i the log versio of the paper. Theorem 4. For all q, t, l, C q,t,l = Ĉq,t,l. Accordig to the result i Theorem 4, the maximum sumrate of qary twrite WlM codes is derived. For all t, q, ad l, let ( t B(q, t, l = (i 1,...,i l : l j=1 j ij q 1 (t l j=1 i j, i 1,..., i l,.
3 Theorem 5. For all t, q, ad l, R q,t,l = log B(q, t, l. Proof: Let B q,t,l be the set of all legtht vectors i [l+1] t such that their Leeweight is at most q 1, that is, { t B q,t,l = x = (x 1,..., x t 0 x i l ad x i q 1. The size of the set B q,t,l is B q,t,l = B(q, t, l = (i 1,...,i l : l j=1 j ij q 1 ( t (t l j=1 i. j, i 1,..., i l To prove the theorem, we first show that R q,t,l log B(q, t, l by usig the size of B q,t,l. Let C be a [, t, l; M 1,..., M t ] q qary twrite WlM code. For ay vector of t messages (m 1,..., m t [M 1 ] [M t ] that is writte to the memory usig the code C, we assig a t matrix A m1,...,m t such that every row i the matrix is the icrease i the levels of all cells. Note that every etry i the array is a iteger i [l + 1]. Moreover, the sum of all etries i each colum of the array is at most q 1 sice the highest level of each cell is q 1. Hece, every colum i A m1,...,m t is a vector i B q,t,l. Hece, there are at most B(q, t, l distict possible arrays, so we deduce that R q,t,l log B(q, t, l. Next, it is show that this upper boud is tight, that is, there is a WlM code whose sumrate achieves the upper boud. The proof holds by iductio. It is straightforward to verify that it holds for t = 1. Assume that it holds for t 1 ad its correctess will be proved for t. From Theorem 4 ad the iductio assumptio, there exists a WlM code of sumrate approachig t t R j = h(p + p i Rj i j=1 = = By choosig we get j=1 (p i log 1/p i + p i R q i,t 1,l p i (log 1/p i + log B(q 1 i, t 1, l. p i = B(q i, t 1, l B(q, t, l (log 1/p i + log B(q i, t 1, l = log B(q, t, l for all 0 i l. Hece, t R j = p i log B(q, t, l = log B(q, t, l, j=1 ad the theorem is prove. Remark 1. qary WOM codes could be viewed as a special case of qary WlM codes whe l = q 1. However, whe l = 1, each time the cells are updated i WlM, we oly write a biary vector. I the case l = 1, we observe that the maximum sumrate of qary twrite WlM codes is R q,t,1 (q 1 log(t + 1 whe t q. Comparig to the maximum sumrate of biary t write WOM codes, which is log(t + 1, R q,t,1 is (q 1 times larger. Furthermore, the maximum sumrate of qary twrite WOM codes is log ( ( q 1+t q 1. We observe that log q 1+t q 1 (q 1 log(t + q 1 R q,t,1 whe q is give ad t teds to ifiity. III. CONSTRUCTIONS I this sectio, we preset costructios of qary twrite WlM codes. We study here the case of q = 3, t = 3 ad l = 1, while the extesio to other parameters is left for future work. The weight of a biary vector x = (x 1,..., x [] is defied to be w(x = i=1 x i ad its support set is deoted by supp(x. Let us first remid the defiitio of the covergece rate of rewritig codes, as was defied i [17]. The covergece rate of a costructio of rewritig codes is the miimum legth (ɛ i order to be ɛclose to a rate tuple (R 1,..., R t or a sumrate R. More specifically, it is said that a costructio approaches the rate tuple or sumrate with polyomial, expoetial rate if (ɛ is polyomial, expoetial i 1/ɛ, respectively. Next, a special family of twowrite biary WOM code, called highweight twowrite biary WOM code, is preseted ad will be used i our costructio. Defiitio 6. A [, ; M 1, M ] (w 1, w highweight twowrite biary WOM code is a codig scheme comprisig of biary bits. It cosists of two pairs of ecodig ad decodig maps (E 1, D 1 ad (E, D which are defied as follows: (1 E 1 : [M 1 ] [] ad D 1 : Im(E q,1 [M 1 ] such that for all m 1 [M 1 ], it holds that E 1 (m 1 = c [] ad w(c = w 1. Furthermore, D 1 (E 1 (m 1 = m 1. ( E : [M ] Im(E 1 [] ad D : Im(E [M ] such that for all (m, c [M ] Im(E 1, it holds that E (m, c = c c ad w(c w. Furthermore, D (E (m, c = m. I a similar way, (w 1, w costatweight twowrite biary WOM codes are defied if o the secod write, w(c = w. Without the weight costrait o the two writes, we obtai the classical twowrite biary WOM codes. For w 1 = (1 p 1 ad w = (1 p 1 p, where 0 p 1, p 1, it is possible to show that a rate tuple (R 1, R, where R 1 = h(p 1 ad R = p 1 h(p, is achievable. For example, a determiistic costructio of these codes ca be obtaied usig Shpilka s techiques [14], however with expoetial covergece rate. This techique ca also be exteded for costatweight twowrite biary WOM codes with the same covergece rate. Highweight twowrite biary WOM codes will be a importat compoet code i the followig costructio of threewrite terary WlM code with l = 1. Costructio 7. Give p 1, p [0, 1], assume the followig codes exist: Let C 1 (p 1 be a [,, M 1,p1, M,p1 ] twowrite biary WOM code such that o the first write w(c = p 1. The two pairs of ecodig/decodig maps are (E 1,p1, D 1,p1 ad (E,p1, D,p1. Let C (p 1, p be a [,, M1,p hr, M,p hr ] (w 1, w highweight twowrite biary WOM code such that w 1 = (1 p 1 ad w = (1 p 1 p. The two pairs of ecodig/decodig maps are (E1,p hr, D1,p hr ad (E,p hr, D,p hr. The proposed [, 3, 1; M 1, M, M 3 ] 3 threewrite terary WlM code is defied usig the three pairs of ecodig/decodig maps as follows. First write: The idea is to ecode a message as a codeword of legth with weight w 1 = p 1. Hece, the pair of ecoder/decoder o the first write of threewrite terary WlM
4 code is the same as the pair of ecoder/decoder o the first write of twowrite biary WOM code. That is, (E 1, D 1 = (E 1,p1, D 1,p1. So, M 1 = M 1,p1 ad the rate is R 1 = h(p 1. Secod write: Let c 1 = (c 1,1, c 1,,..., c 1, be the cellstate vector after the first write ad c 1 = c 1 = (c 1,1,..., c 1, be its complemet. Let M = M,p1 M,p hr. For each m [M ], we ca determie the uique pair (m,p1, m,p such that m,p1 [M, p 1 ] ad m,p [M,p hr ]. Now, we are ready to defie the ecoder E : [M ] Im(E 1 [3] o the secod write. For each (m, c 1 [M ] Im(E 1, we ca determie E (m, c 1 = c = (c,1, c,,..., c, i two steps as follows. Step 1: Let c 1 be the iput to the ecoder E,p1 of the twowrite biary WOM code C 1 (p 1. For each m,p1 [M, p 1 ], we obtai E,p1 (m,p1, c = x = (x 1,..., x. For 1 i, if c 1,i = 0 the c,i = x i. Step : Let c 1 be the iput of the ecoder E,p hr of (w 1, w highweight twowrite biary WOM code C (p 1, p. For each m,p [M,p hr ], we obtai E,p hr (m,p, c 1 = y = (y 1,..., y []. We determie the vector y = (y 1,..., y [3] as follows. For all 1 i, y i = 0 if c 1,i = y i = 1, y i = if c 1,i = y i = 0, ad y i = 1 otherwise. For 1 i, if c 1,i = 1 the c,i = y i. We defie the decoder D : Im(E [M ] o the secod write as follows. For each c = (c,1,..., c, Im(E, we ca determie y = (y 1,..., y [] such that y i = 0 if ad oly if c,i =. Usig the decoder D,p hr of the (w 1, w  highweight twowrite biary WOM code C (p 1, p, we obtai D,p hr (y = m,p. Furthermore, we also ca determie x = (x 1,..., x [] such that x i = 0 if ad oly if c,i = 0. Usig the decoder D,p1 of the twowrite biary WOM code C 1 (p 1, we obtai D,p1 (x = m,p1. From m,p ad m,p1, we ca determie the uique m, ad decode by D (c = m. Third write: Let c = (c,1,..., c, be the cellstate vector after the secod write. We defie the biary vector c = (c,1, c,,..., c, [] such that c,i = 1 if ad oly if c,i =. O the third write, we determie the ecoder E 3 : [M 3 ] Im(E [3] as follows. Let p 1, = p 1 p. Let E,p1, be the ecoder o the secod write of the twowrite biary WOM code C 1 (p 1,. For each m 3 [M 3 ], we obtai E,p1, (m 3, c = z = (z 1,..., z. So E 3 (m 3, c = c 3 = (c 3,1,..., c 3, ca be defied such that c 3,i = 1 if z i = 0, c 3,i = 0 if c,i = 0 ad z i = 1, ad c 3,i = otherwise. The correspodig decoder D 3 : Im(E 3 [M 3 ] o the third write ca be defied as follows. For each c 3 Im(E 3, we determie z = (z 1,..., z [] such that z i = 0 if ad oly if c 3,i = 1. Usig the decoder D,p1, of the twowrite biary WOM code C 1 (p 1,, we obtai D 3 (c 3 = D,p1, (z = m 3. We observe that Costructio 7 uses twowrite biary WOM codes ad highweight twowrite biary WOM codes as importat compoet codes. A iterestig questio, which is addressed ext, is whether this costructio ca provide codes achievig the maximum sumrate. Theorem 8. If there exists a explicit costructio of (w 1, w  highweight twowrite biary WOM codes which achieve the ratetuple (R 1, R = (h(p 1, p 1 h(p for ay give w 1 = (1 p 1 ad w = (1 p 1 p, the there exists a explicit costructio of threewrite terary WlM codes which achieves the ratetuple (R 1, R, R 3 = (h(p 1, 1 p 1 + p 1 h(p, 1 p 1 p. I particular, there exists a explicit costructio of threewrite terary WlM codes for l = 1 which achieves the maximum sumrate R 3,3,1 = log 7. Proof: We ote that there exists a explicit costructio of twowrite biary WOM codes which achieves the ratetuple (h(p 1, 1 p 1 for ay give p 1 [0, 1/] [17]. Assume that there exists a explicit costructio of (w 1, w highweight twowrite biary WOM codes which achieves the ratetuple (h(p 1, p 1 h(p. I Costructio 7, the rate o the first write R 1 = h(p 1. O Step 1, Step of the secod write the rate is hr log M,p R,1 = log M,p 1 = 1 p 1, R, = = p 1 h(p, respectively. So, the rate o the secod write is R = R,1 + R, = 1 p 1 +p 1 h(p. O the third write, the rate is R 3 = 1 p 1 p. Hece, the costructed threewrite terary WlM codes achieve the ratetuple (R 1, R, R 3 = (h(p 1, 1 p 1 + p 1 h(p, 1 p 1 p, ad sumrate R sum = R 1 + R + R 3 = h(p p 1 + p 1 h(p + 1 p 1 p. For p 1 = 3/7 ad p = 1/3 we get the maximum sumrate R 3,3,1 = log 7. Lastly, we observe that i all three writes of the WlM codes from Costructio 7, every step is explicit ad is based o two compoet codes, twowrite biary WOM codes ad (w 1, w  highweight twowrite biary WOM codes. Therefore, the theorem is prove. It is ow possible to coclude with the followig corollary. Corollary 9. There exists a explicit costructio of threewrite terary WlM codes for l = 1 which achieves the maximum sumrate R 3,3,1 = log 7. The covergece rate of the codes achievig the maximum sumrate i Corollary 9 is expoetial. This follows sice they require the costructio of (w 1, w highweight twowrite biary WOM codes ad our best costructio is based o the techiques from [14], which also have expoetial covergece rate. I fact, it is possible to directly costruct WlM codes by the techiques from [14] with expoetial covergece rate. However, Costructio 7 is beeficial sice we believe that fidig highweight twowrite biary WOM codes with polyomial covergece rate will be a easier task, ad furthermore it eables us to preset practical WlM codes of short block legth but yet achieve high sumrate. This will be accomplished by explicit costructios of (w 1, w highweight twowrite biary WOM codes. To do so, we eed the followig defiitio from [6], [7]. Defiitio 10. For, t ad w with t + w, a (, t, w lowpower coolig (LPC code C of size M is defied as a collectio
5 of code sets {C 1, C,..., C M, where C 1, C,..., C M are disjoit subsets of {u [] : w(u w satisfyig the followig property: for ay set S [] of size S = t ad for i [M], there exists a vector u C i with supp(u S =. From this defiitio, we obtai the followig result. The proof is derived from the defiitios of lowpower coolig codes ad (w 1, w highweight twowrite biary WOM code. We omit the details of the proof due to the lack of space. Theorem 11. Give, w 1, w such that w 1 < w <, if there exists a (, w 1, w LPC code of size M, the there exists a [, ; M 1, M ] (w 1, w highweight twowrite biary WOM code such that M 1 = ( w 1 ad M = M. Recetly, LPC codes have bee ivestigated ad a few explicit costructios of LPC codes were preseted i [6], [7]. We use oe of these families of LPC codes to costruct a explicit highweight twowrite biary WOM code ad thus a explicit threewrite terary WlM code with l = 1. Followig is a example of asymptotically optimal LPC codes. Example 1. [6, Corollary 18] Fix τ ad ω = (1 τ/. The there exists a family of (, w 1, w  LPC codes C such that w 1 = τ, w = ω, ad lim (log C / = 1 τ. I other words, the rate of the codes coverges to 1 τ. From Theorem 11 ad Example 1, we obtai the followig. Corollary 1. For all 0 p , there exists a family of [,, M 1, M ] (w 1, w highweight twowrite biary WOM codes for w 1 = p 1 ad w = ( w 1 /, which achieves sumrate R = h(p p 1 with polyomial covergece rate. If we choose the above [,, M 1, M ] (w 1, w highweight twowrite biary WOM codes as a compoet code i Costructio 7 with p 1 = the the sumrate of our costructed threewrite terary WlM codes with l = 1 approaches h(p 1 + (1 p 1 + p (1 p 1 /.77. We coclude our result i the followig corollary. Corollary 13. There exists a explicit costructio of threewrite terary WlM codes of sumrate approachig.77 with polyomial covergece rate. As part of our future work, we will exted our techique i Costructio 7 to costruct qary twrite WlM code for all q, t, ad l. Furthermore, these ideas ca be leveraged i order to costruct the classical qary twrite WOM codes achievig maximum sumrate i geeral for all q ad t. Namely, we established the followig theorem. Theorem 14. If there exists a explicit costructio of costatweight twowrite biary WOM code which achieves the ratetuple (h(p 1, p 1 h(p for ay give parameter p 1, p [0, 1] the there exists a explicit costructio of qary twrite WOM code which achieves the maximum sumrate. Hece, the problem of fidig obiary WOM codes which achieve the maximum sumrate with polyomial covergece rate ca be reduced to the problem of fidig costatweight twowrite biary WOM codes with the same property. Due to the lack of space, these results will be discussed i the full versio of this work. IV. ACKNOWLEDGEMENT The research of Y. M. Chee, H. M. Kiah ad V. K. Vu is partially fuded by the Sigapore Miistry of Educatio uder grat MOE015T The research of E. 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