Coding for Write l-step-up Memories

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1 Codig for Write l-step-up 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 Duisburg-Esse, 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 o-biary rewritig codes, called write l-step-up memories (WlM codes. From a iformatio-theoretic poit of view, this codig scheme is a geeralizatio of o-biary write-oce 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 over-shootig problem. We first provide a exact formula for the capacity regio ad the maximum sum-rate of WlM codes. Lastly, we preset several explicit costructios of high-rate WlM codes with efficiet ecodig/decodig algorithms. I. INTRODUCTION Write-oce 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 write-oce memory which is kow i literature as WOM codes. Later, Fiat ad Shamir [] studied a codig scheme for geeralized write-oce memory where each cell has q levels that ca oly be icreased. This codig scheme is called o-biary WOM codes. I the 1980 s ad 1990 s, a few differet models of WOM codes were studied due to its beauty from a iformatio-theoretical poit of view ad its applicatios i puch cards ad optical disks [] [6]. Furthermore, there are several versios of WOM codes, icludig write ui-directioal 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 o-volatile 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 floatig-gate (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 over-shootig. 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 error-correctig codes [19] ad rak-modulatio codes [0]. The goal of this paper is to iitiate the study of a codig scheme which combats both limited-edurace ad overshootig i flash memory. Oe approach to meet this challege is by usig error-correctig codes together with WOM codes as was studied i [1], [1], []. However, i order to directly reduce the over-shootig problem, this work proposes to use a geeralizatio of o-biary 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 l-step-up 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: ɛ-error ad zero-error. I this paper, we focus o the more practical model where the ecoder is iformed ad the decoder is uiformed for the zero-error case. Although this paper is the first to study WlM codes, there are several closely related models such as codes for edurace-limited memories (ELM [3] ad write-costraied memories (WCM codes [4]. Results o the exact capacity regio ad the maximum sum-rate of WlM codes i some cases ca be accomplished usig kow techiques. However, we are ot aware of ay explicit costructio of high-rate 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 cell-state 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 o-biary WOM codes. Defiitio 1. A [, t; M 1,..., M t ] q q-ary t-write WOM code is a codig scheme comprisig of q-ary 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, q-ary t-write WlM codes are defied. Defiitio. A [, t, l; M 1,..., M t ] q q-ary t-write WlM code is a codig scheme comprisig of q-ary 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 j-th write is the ratio betwee the umber of writte bits ad the umber of cells, that is, log Mj R j =. The sum-rate is the sum of all rates o t writes, that is, R sum = t j=1 R j. I q-ary t-writes 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 q-ary t-write WlM code log Mj such that R j ɛ for all 1 j t. The capacity regio of q-ary t-writes WlM is the set of all achievable rate tuples ad is deoted by C q,t,l, ad the maximum sum-rate 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 o-biary 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 cout-vector 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 cout-vector is icreased by oe. Hece, this cout-vector is similar to a word i WlM code for l = 1. Yet, these codes are still ot idetical ad furthermore the cout-vector 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 1-chage t-write 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 q-ary t-write 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 q-ary t-write 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 write-costraied memories (WCM which is a WOM code with costs o the state-trasitios. I this codig scheme, each cell-state 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 capacity-achievig codes still remais a iterestig challege which is addressed i the paper. II. THE CAPACITY OF q-ary WlM I this sectio, we preset the capacity regio ad the maximum sum-rate 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 q-ary t-write 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 legth-t vectors i [l+1] t such that their Lee-weight 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 q-ary t-write 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 sum-rate 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 sum-rate 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. q-ary WOM codes could be viewed as a special case of q-ary 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 q-ary t-write WlM codes is R q,t,1 (q 1 log(t + 1 whe t q. Comparig to the maximum sum-rate of biary t- write WOM codes, which is log(t + 1, R q,t,1 is (q 1 times larger. Furthermore, the maximum sum-rate of q-ary t-write 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 q-ary t-write 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 ɛ-close to a rate tuple (R 1,..., R t or a sum-rate R. More specifically, it is said that a costructio approaches the rate tuple or sum-rate with polyomial, expoetial rate if (ɛ is polyomial, expoetial i 1/ɛ, respectively. Next, a special family of two-write biary WOM code, called high-weight two-write biary WOM code, is preseted ad will be used i our costructio. Defiitio 6. A [, ; M 1, M ] (w 1, w -high-weight 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 -costat-weight two-write 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 two-write 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 costat-weight two-write biary WOM codes with the same covergece rate. High-weight two-write 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 ] two-write 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 two-write 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 three-write 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 three-write terary WlM

4 code is the same as the pair of ecoder/decoder o the first write of two-write 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 cell-state 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 two-write 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 -high-weight two-write 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 - high-weight two-write 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 two-write 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 cell-state 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 two-write 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 two-write biary WOM code C 1 (p 1,, we obtai D 3 (c 3 = D,p1, (z = m 3. We observe that Costructio 7 uses two-write biary WOM codes ad high-weight two-write biary WOM codes as importat compoet codes. A iterestig questio, which is addressed ext, is whether this costructio ca provide codes achievig the maximum sum-rate. Theorem 8. If there exists a explicit costructio of (w 1, w - high-weight two-write biary WOM codes which achieve the rate-tuple (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 three-write terary WlM codes which achieves the rate-tuple (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 three-write terary WlM codes for l = 1 which achieves the maximum sum-rate R 3,3,1 = log 7. Proof: We ote that there exists a explicit costructio of two-write biary WOM codes which achieves the rate-tuple (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 -high-weight two-write biary WOM codes which achieves the rate-tuple (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 three-write terary WlM codes achieve the rate-tuple (R 1, R, R 3 = (h(p 1, 1 p 1 + p 1 h(p, 1 p 1 p, ad sum-rate 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 sum-rate 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, two-write biary WOM codes ad (w 1, w - high-weight two-write 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 three-write terary WlM codes for l = 1 which achieves the maximum sum-rate R 3,3,1 = log 7. The covergece rate of the codes achievig the maximum sum-rate i Corollary 9 is expoetial. This follows sice they require the costructio of (w 1, w -high-weight two-write 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 high-weight two-write 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 sum-rate. This will be accomplished by explicit costructios of (w 1, w -high-weight two-write 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 low-power coolig codes ad (w 1, w -high-weight two-write 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 -high-weight two-write 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 high-weight two-write biary WOM code ad thus a explicit three-write 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 -high-weight two-write biary WOM codes for w 1 = p 1 ad w = ( w 1 /, which achieves sum-rate R = h(p p 1 with polyomial covergece rate. If we choose the above [,, M 1, M ] (w 1, w -high-weight two-write biary WOM codes as a compoet code i Costructio 7 with p 1 = the the sum-rate of our costructed three-write 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 sum-rate approachig.77 with polyomial covergece rate. As part of our future work, we will exted our techique i Costructio 7 to costruct q-ary t-write WlM code for all q, t, ad l. Furthermore, these ideas ca be leveraged i order to costruct the classical q-ary t-write WOM codes achievig maximum sum-rate i geeral for all q ad t. Namely, we established the followig theorem. Theorem 14. If there exists a explicit costructio of costatweight two-write 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 q-ary t-write WOM code which achieves the maximum sum-rate. Hece, the problem of fidig o-biary WOM codes which achieve the maximum sum-rate with polyomial covergece rate ca be reduced to the problem of fidig costat-weight two-write 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. 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