Polar Coding for Noisy Write-Once Memories

Transcription

1 204 IEEE Iteratoal Symposum o Iformato Theory Polar Codg for Nosy Wrte-Oce Memores Eyal E Gad, Yue L,4, Joerg Klewer 2, Mchael Lagberg 3, Axao (Adrew) Jag 4 ad Jehoshua Bruck Calfora Isttute of Techology, Pasadea, CA New Jersey Isttute of Techology, Newark, NJ SUNY at Buffalo, Buffalo, NY Texas A&M Uversty, College Stato, TX {eegad, yl, bruck}@caltech.edu jklewer@jt.edu mkel@buffalo.edu ajag@cse.tamu.edu Abstract We cosder the osy wrte-oce memory (WOM) model to capture the behavor of data-storage devces such as flash memores. The osy WOM s a asymmetrc chael model wth o-causal state formato at the ecoder. We show that a estg of o-lear polar codes acheves the correspodg Gelfad-Psker boud wth polyomal complexty. I. INTRODUCTION Wrte-oce memory s a model for data-storage devces, where a set of bary cells are used to store data, ad the cell levels ca oly crease whe the data s rewrtte. Ths model was frst proposed [3], ad t s related to earler models for chael codg wth state formato [4], []. The WOM model was recetly detfed as a good model for flash memores, where rewrtg delays expesve block erasures ad by that leads to better preservato of cell qualty ad hgher performace [8], [5]. I ths paper we propose a codg scheme for WOM that corporates error-correcto capabltes to protect from ose the process of wrtg data to the memory. We cosder a stochastc model of wrtg ose due to Heegard [6], ad propose the frst codg scheme that acheves the capacty of the model wth polyomal complexty. The codg scheme s based o a estg of o-lear polar codes. The majorty of prevous work o WOM codes do ot cosder error correcto, e.g., [4] where a algebrac zero-error capacty-achevg codg scheme was preseted. A dfferet capacty-achevg scheme, based o polar codg, was proposed []. I cotrast, there s lttle work avalable whe errors are cosdered. For example, [3] a capactyachevg codg scheme was preseted whch corrects a eglgble fracto of errors. Further, prevous work by the authors [9] proposes a o-capacty-achevg soluto based o hgh-rate polar codes. Other o-capacty-achevg errorcorrectg WOM schemes were proposed [5], [6]. The ma cotrbuto of ths work s the dervato of a ested polar codg scheme for the osy WOM chael, wth a asymptotcally-optmal rate ad polyomal complexty. As typcally wth polar codg schemes, the algorthmc complexty of the proposed scheme s gve as O( log ) uder a Ths work was supported part by Itellectual Vetures, NSF grats 28005, , 44000, 44004, ad ad the US-Israel Batoal Scece Foudato (BSF) uder Grat No decodg error probablty of 2 Ω(/2 δ) for a block legth ad ay δ > 0. Comparg wth the capacty-achevg polar WOM codg scheme of [], we make two mportat cotrbutos. Frst, the scheme of ths paper poses a capablty to correct errors, whle the scheme [] dd ot have ths capablty. Secod, the scheme [] requres a large amout of radomess to be shared betwee the ecoder ad the decoder. The scheme that we propose ths paper requres a sgfcatly smaller amout of radomess to be shared. I fact, uder the reasoable approxmato of the cell states to be..d., the requremet for shared radomess s completely removed. The reducto of shared radomess s doe by the use of o-lear polar codes as proposed [7]. We ote that the codg scheme preseted ths paper s very smlar to the scheme preseted [5], for commucato broadcast chaels. Whle we were uaware of the result of [5] ad developed the scheme depedetly, ths paper also has two cotrbutos that were ot show [5]. Frst, we coect the scheme to the applcato of data storage ad flash memores, that was ot cosdered the prevous work. Secod, the aalyss [5] holds for chaels that exhbt a certa techcal codto of degradato. I ths paper we show that the model of WOM wth wrtg ose fact exhbt the requred degradato codto, ad by that show that the scheme acheves the capacty of the cosdered model. II. WOM CHANNEL MODEL A WOM s composed of cells, where each cell has a state from some fte alphabet. I ths paper we assume that the alphabet s {0, }. The reaso for ths s that polar codes are better uderstood wth bary codeword symbols. However, there s o other fudametal dffculty to exted the results of the paper to o-bary alphabets (see, e.g., []). The ma property of the WOM s that a cell at state 0 ca chage ts state to, but oce the cell state s, t caot be chaged aymore. The state of the cells s kow to a user that wshes to store formato o the memory. If some of the cells are state, a code s requred for the relable storage of formato, sce ot every sequece of bts ca be stored drectly. We cosder a stochastc..d. WOM model. Let S be a Beroull radom varable that correspods to the state of a /4/$ IEEE 638

2 204 IEEE Iteratoal Symposum o Iformato Theory cell. Sce the cells are..d., we defe the dstrbuto of a cell state by P(S ) β, () for some β [0, ]. We ote that ths model s ot completely accurate for the applcato of rewrtg flash memores. I rewrtg, the memory state s determed by the prevous wrte to the memory, whch the codeword dces are ot ecessarly dstrbuted depedetly. However, smulato results show that the..d. approxmato performs well practce. To make the cell states precsely..d., t was proposed modfcatos M3 ad M4 [, Secto IV] to share a radom permutato betwee the ecoder ad the decoder. However, t s ot clear f ths soluto could be mplemeted effcetly practce. The focus of ths work s o memores wth wrte errors, where the correspodg bt chael s show Fg.. Here X ad Y are Beroull radom varables that correspod to the put ad output of a sgle bt the memory. If the cell state S s, the output Y wll be as well, regardless of the value of the put X, correspodg to a stuck bt. Otherwse, f the cell state S s 0, we assume that the memory exhbts a symmetrc wrtg error wth crossover probablty α. Ths behavor s summarzed by P Y XS ( x, s) α f (x, s) (0, 0) α f (x, s) (, 0) f s. Fally, our model of study, we lmt the umber of cells that the ecoder attempts to chage from 0 to the memory. Ths lmtato leds tself aturally to the applcato of WOM to settgs whch multple wrtes are cosdered. We express ths lmtato by a parameter ϵ that bouds the expectato of the put X gve that S 0 as follows (2) E(X S 0) ϵ. (3) The capacty of the WOM model s gve the followg theorem. Theorem. [6, Theorem 4] The capacty of the memory descrbed by (), (2), ad (3) s C ( β)[h(ϵ α) h(α)], where ϵ α ϵ( α) + ( ϵ)α. The ested polar codg scheme of ths paper acheves the capacty of Theorem. The presetato of the scheme requres some otato from polar codg. III. POLAR CODING NOTATION For a postve teger, let [] {, 2,..., }. Let X (X, X 2,..., X ), Y (Y, Y 2,..., Y ) ad S (S, S 2,..., S ) be..d. copes of X, Y ad S, respectvely. For tegers < j, let X j represet the subvector (X, X +,..., X j ) ad for a set A [] let X A represet the subvector {X } A. Let be a power of 2. Defe a matrx G G log 2 ( ) 0 where G ad deotes the Kroecker power. A subset of the rows of G serves as a geerator matrx α α X X Y α Y α S 0 S Fg.. A bary WOM wth wrte errors. lear polar codg schemes. Let U be the product U X G X G. A subset of the coordates of the vector U wll cota the message to be stored the memory. Our aalyss uses the Bhattacharyya parameters. For a codtoal dstrbuto P Y X, where X s a Beroull radom varable, the Bhattacharyya parameter s defed by Z B (P Y X ) y P Y X (y 0)P Y X (y ). The Bhattacharyya parameter serves as a upper boud o the decodg error probablty polar codes for symmetrc settgs. Smlarly, defe the codtoal Bhattacharyya parameter as Z(X Y) 2 y PX,Y (0, y)p X,Y (, y). The codtoal Bhattacharyya parameter serves as a decodg error probablty boud polar codes for asymmetrc settgs. Note that the Bhattacharyya parameter ad the codtoal Bhattacharyya parameter are equal f X s dstrbuted uformly. IV. CODING SCHEME We start by presetg a rough overvew of our codg techque followed by a formal presetato. The achevablty of Theorem s show [4], [6] by radom codg. The dstrbuto by whch the codewords are draw ths achevablty proof s called the capacty-achevg dstrbuto, ad t s used our scheme. We deote ths dstrbuto by P X S. Accordg to the proof of [6, Theorem 4], we have ϵ( α) P X S ( 0) ϵ, P X S ( ) ϵ α. (4) Our costructo s based o a combato of two methods: o-lear polar codg [7] ad ested polar codg [0]. Nolear polar codes allow to acheve the capacty of asymmetrc chaels ad sources. I ested polar codg for Gelfad- Psker-type problems, the ecoder frst compresses the state usg a lossy source code, ad the trasmts the compressed state together wth the source message usg a chael code. The cosderato of the compressed state the choce of codeword allows the codtoal dstrbuto of the codeword gve the state to approxmate the capacty-achevg dstrbuto of Equato (4). Let s, u, x ad y be the realzatos of the radom varables S, U, X ad Y, respectvely. I a chael polar codg scheme, a vector u s used for represetg the source message ad a froze vector. The chael put s the codeword x u G. The codg scheme polarzes the codtoal etropes H(U U, Y ) for the dfferet coordates of the vector U. I a ested polar codg scheme, the vector U s also polarzed wth respect to the codtoal etropes H(U U, S ). I our case the codtoal etropes H(U U, S ) are defed accordg to the capacty-achevg codtoal dstrbuto of Equato (4). 639

3 204 IEEE Iteratoal Symposum o Iformato Theory I addto, o-lear polar codes, the vector U s also polarzed wth respect to the codtoal etropes H(U U ). We take advatage of these three dfferet polarzatos of the vector U our codg scheme. It s useful to defe the polarzed sets accordg to the codtoal Bhattacharyya parameter, as follows: A { : Z(U U, Y ) 2 /2 δ }, B { : Z(U U, S ) 2 /2 δ }, C { : Z(U U ) 2 /2 δ }, D { : Z(U U ) 2 /2 δ }, for some small δ > 0. The sets A, B, C ad D refer to sets of coordates of the vector u. Also defe the sets A c, B c, C c ad D c to be the complemets of A, B, C ad D wth respect to []. Itutvely, set A cotas the coordates of u that ca be decoded relably gve y. Set B cotas the coordates that have small relato to s. Set C s composed of coordates that do ot cota much formato about x, ad set D s composed of coordates that cota almost all the formato about x. Note that those sets are ot dsjot. The relato betwee the sets s depcted Fg. 2. The proposed codg scheme takes advatage of the polarzed sets as follows. Frst, the ecoder performs a lossy compresso of the state s by the method of [7]. The compresso s performed accordg to a formato set I S, defed to be the set of coordates that are ot the uo B C. The complemet of the formato set wth respect to [] s deoted as IS c B C. Each bt u for I S s set radomly to a value u wth probablty P U U,S (u u, s ). The coordates the set B are dstrbuted almost uformly gve (U, S ), ad therefore they do ot affect the jot dstrbuto (X, S ). For that reaso we ca place the source message those coordates. Sce we also eed the message to be decoded relably gve Y, we restrct t to the coordates the set A, whch are almost determstc gve (U, Y ). Takg both restrctos to accout, we set the coordates of u the tersecto A B to be equal to the source message. The rest of the coordates of u are set by the ecoder accordg to a set of Boolea fuctos, as [7]. Sce ths set of fuctos descrbes the code, t s kow to both the ecoder ad the decoder. For a postve teger, a Boolea fucto s deoted by λ : {0, } {0, }. Remember that the coordates the set I S are determed accordg to the state s, ad that the coordates A B are determed by the formato message. The rest of the coordates of u wll be determed accordg to a set of Boolea fuctos λ I c S \(A B) {λ } I c S \(A B). The ecoder sets a bt u accordg to the fucto λ (u ). Fally, remember that the set A cotas the relable coordates of u gve y. Therefore, the decoder ca guess the coordates A, ad wll estmate the rest of the coordates accordg to λ I c S \A. A coordate IS c ca be decoded ths way, sce f A, u ca be estmated relably, ad f IS c \ A, the u ca be recovered by λ. However, a D: almost radom gve U C: almost determstc gve U B I S C B: almost radom gve (U,S) B c : almost determstc gve (U,S) A c A B B c Fg. 2. Dfferet polarzatos of U X G. A c : almost radom gve (U,Y ) A: almost determstc gve (U,Y ) coordate I S ca be decoded relably oly f A, sce we do ot use Boolea fuctos for the set I S. Therefore, a relable decodg requres I S to be a subset of A. Whle we do ot kow f ths fact always holds, we ca show that the set dfferece I S \ A s very small. Therefore, our strategy s to store the vector u IS \A addtoal cells. Sce we show that the fracto I S \ A / approaches zero for large, ths strategy wll ot affect the asymptotc rate of the scheme or the expected weght of the codewords. The addtoal cells should be coded as well to esure ther relablty. I ths paper we assume that the addtoal cells are protected by a o-lear chael polar code, wthout utlzg the state formato at the ecoder. We do ot dscuss the detals of the codg of the addtoal cells. However, we ote that sce the block legth of the addtoal cells s much smaller tha the ma block legth, the probablty of decodg error at the addtoal cells s hgher. I partcular, the probablty of decodg error at the addtoal cells s 2 Ω(/2 δ) for ay δ > δ, whle the decodg error probablty the ma block of cells s O(2 /2 δ ), as stadard polar codes. The detals of ths ssue are dscussed the loger verso of ths paper [2]. We ow descrbe the codg scheme formally. Costructo 2. Ecodg Iput: a message m k {0, }k ad a state s {0, }. Output: a codeword x {0, }. ) Let u A B m k. The for from to, f I S, set { 0 wth probablty PU U,S (0 u u wth probablty P U U,S, s ) ( u, s ), ad f I c S \ (A B), set u λ (u ). 2) The vector u IS \A s stored addtoal cells. Fally, store the codeword x u G. Decodg Iput: a osy vector y. Output: a message estmato ˆm k. Estmate u by û (y, λ I c S \A ) as follows: ) Recover the vector u IS \A from the addtoal cells, ad assg û IS \A u IS \A. 2) For from to, set { arg maxu P û U U,Y (u û, y ) f A λ (û ) f IS c \ A. 3) Retur the estmated message ˆm k û A B. 640

4 204 IEEE Iteratoal Symposum o Iformato Theory Let be the umber of addtoal cells eeded to store the vector u IS \A. The ma theorem of ths paper addresses the propertes of Costructo 2. Theorem 3. Let the message m k be dstrbuted uformly. The for ay costats δ > δ > 0 ad ϵ > ϵ > 0, there exsts a set of Boolea fuctos λ IS \(A B) for whch Costructo 2 satsfes the followg: ) The rate of the scheme s asymptotcally optmal. Formally, lm k/( + ) C. 2) E[X S 0] ϵ. 3) The decodg error probablty s 2 Ω(/2 δ ). 4) The ecodg ad decodg complextes are O( log ). Due to space lmtatos, we skp the detaled proof of Theorem 3, ad refer the reader to the loger verso of ths paper [2]. I ths short verso we oly prove the rate optmalty of the codes (clam of Theorem 3). V. OPTIMALITY OF THE RATE To prove clam of Theorem 3, we combe two separate clams. Frst, we show Subsecto V-A that the fracto of addtoal cells s eglgble, ad ext, Subsecto V-B, we show that the rate of the ma block s asymptotcally optmal. Together the two results prove clam of Theorem 3. A. Fracto of Addtoal Cells s Neglgble The vector u IS \A s stored o addtoal cells. To esure a relable storage, a polar codg scheme requres oly a lear amout of redudacy I S \ A. Therefore, to show that the addtoal cells do ot affect the rate ad codeword weght of the scheme, t s eough to show that lm I S \ A / 0. To show ths, we use a dea from the proof of [0, Theorem 5]. Cosder the set Ī S { : Z(U U, S ) 2 /2 δ ad Z(U U ) 2 /2 δ }. By the defto of the sets above, Ī S I S. Therefore, by the dstrbutvty of tersecto over uo, we have I S \ A A c I S A c I S \ Ī S + A c Ī S. To show that lm A c I S / 0, we start by showg that Ī S A, whch mples that A c Ī S 0, ad therefore that A c I S A c (I S \ Ī S ) I S \ Ī S. From [7, Theorem ], we have lm I S \ Ī S / 0. Therefore, showg that Ī S A proves that lm A c I S / 0. To show that Ī S A, t s eough to show that Z(U U, Y ) Z(U U, S ) for all [], by deftos of the sets Ī S ad A. To show that Z(U U, Y ) Z(U U, S ) for all [], we use a sequece of reductos. Frst, we use the followg theorem: Theorem 4.[7, Theorem 2] Let Y be the alphabet of Y ad let Ỹ {0, } Y ad Ỹ ( X X, Y) where (X, Y) s depedet of X. Let Ũ X G ad defe W () Y, (ũ, ỹ ũ ) PŨ,Ỹ Ũ (ũ, ỹ ũ ). The Z(U U, Y ) Z B( W () Y, ). Theorem 4 mples that Z(U U, Y ) Z(U U, S ) f ad oly f Z B ( W () Y, ) Z B( W () S, ) for all []. To show the latter relato, we use the oto of stochastcally degraded chaels. A dscrete memory-less chael (DMC) W : {0, } Y s stochastcally degraded wth respect to a DMC W 2 : {0, } Y 2, deoted as W W 2, f there exsts a DMC W : Y 2 Y such that W (y x) y2 Y 2 W 2 (y 2 x)w(y y 2 ). We use the followg result o the Bhattacharyya parameters of degraded chaels: Lemma 5. [0, Lemma 2] (Degradato of W () ) Let W : {0, } Y W : {0, } Y 2 be two dscrete memoryless chaels (DMC) ad let W W. The for all, W () W () ad Z B (W () ) Z B (W () ). Lemma 5 reduces the proof of Z B ( W () Y, ) Z B( W () S, ) for all [] to showg that P S X P Ỹ X. The last degradato relato s prove the followg lemma, whch completes the proof that lm I S \ A / 0. Lemma 6. P S X P Ỹ X. Proof: We eed to show that there exsts a DMC W : {0, } 2 {0, } 2 such that P S X ( s x) PỸ X (ỹ x)w( s ỹ). (5) ỹ {0,} 2 To defe such chael W, we frst clam that P Y X,S ( x, 0)P X S (x 0) (ϵ α)p X S (x ). (6) Equato (6) follows drectly from Equato (4) sce P Y X,S ( 0, 0)P X S (0 0) P X S (0 ) P Y X,S (, 0)P X S ( 0) P X S ( ) Next, we clam that P X,S (x,) P X,Y (x,) α( ϵ) α( ϵ) ϵ α ( α)ϵ ( α)ϵ ϵ α β (ϵ α)( β)+β ϵ α, ϵ α. for ay x {0, }, ad therefore that P X,S(x,) [0, ]. Ths follows from P X,Y (x,) P X,S (x, ) (a) P X S (x )P S () P X,Y (x, ) P Y,X S (, x 0)P S (0) + P Y,X S (, x )P S () (b) P X S (x )β/[p Y X,S ( x, 0)P X S (x 0)( β) + P Y X,S ( x, )P X S (x )β] (c) P X S (x )β (ϵ α)p X S (x )( β) + P X S (x )β 64

5 204 IEEE Iteratoal Symposum o Iformato Theory β (ϵ α)( β) + β, where (a) follows from the law of total probablty, (b) follows from the defto of codtoal probablty, ad (c) follows from Equatos (2) ad (6). Deote the frst coordate of the radom varable Ỹ by Ỹ X X, ad the frst coordate of ỹ by y. The same otato s used also for S ad s. Sce P X,S(x,) s ot a P X,Y (x,) fucto of x ad s [0, ], we ca defe W as followg: f s ỹ ad (s, y) (0, 0) P X,S(x,) f s ỹ ad (s, y) (0, ) W( s ỹ) P X,Y (x,) P X,S (x,) P X,Y (x,) 0 otherwse. f s ỹ ad (s, y) (, ) We show ext that Eq. (5) holds for W defed above: PỸ X (ỹ x)w( s ỹ) ỹ {0,} 2 (d) ỹ {0,} 2 P X,Y X (ỹ x, y x)w( s ỹ) (e) ỹ {0,} 2 P X,Y (ỹ x, y)w( s ỹ) [ P X,Y ( s x, 0)W( s, 0 s, 0) ỹ {0,} 2 PỸ,Y X (ỹ, y x)w( s ỹ) + P X,Y ( s x, )W( s, 0 s, ) ] (s 0) + P X,Y ( s x, )W( s, s, )(s ) [ P X,Y ( s x, 0) ( + P X,Y ( s x, ) P ) ] X,S( s x, ) (s 0) P X,Y ( s x, ) + P X,Y ( s x, ) P X,S( s x, ) (s ) P X,Y ( s x, ) [P X,Y ( s x, 0) + P X,Y ( s x, ) P X,S ( s x, )](s 0) + P X,S ( s x, )(s ) (f) [P X ( s x) P X,S ( s x, )](s 0) + P X,S ( s x, )(s ) (g) P X,S ( s x, 0)(s 0) + P X,S ( s x, )(s ) P X,S ( s x, s) (h) P X,S X ( s x, s x) P S,S X ( s, s x) P S X ( s x), where (d) follows from the fact that Ỹ X X, (e) follows from the depedece of (X, Y) ad X, (f) ad (g) follow from the law of total probablty, ad (h) follows from the depedece of (X, S) ad X. So the chael W satsfes Equato (5) ad thus the lemma holds. B. The Rate of the Ma Block We eed to show that lm k/ ( β)[h(α ϵ) H(α)]. By [7, Equatos (38) ad (39)], we have lm B / H(X S), ad lm A c / H(X Y). So we get lm k/ lm A B / lm ( B Ac ) H(X S) H(X Y). Heegard showed [6] that H(X S) H(X Y) ( β)[h(α ϵ) H(α)]. Therefore the proof of clam Theorem 3 s completed. VI. DISCUSSION We preseted a capacty-achevg codg scheme for osy wrte-oce memores. The model of osy WOM ths paper dffers from the model [9] by the fact that here cells wth state s are ot proe to errors. For the error model [9], we were ot able to prove a chael degradato aalogous to Lemma 6 ad hece the results obtaed [9] are sub-optmal. A recet chag method descrbed [2] mples effcet capacty-achevg codg schemes wthout the degradato requremet. Thus, the chag method could be useful for the error model of [9]. However, the use of the chag method comes at the prce of a lower code rate the fte blocklegth regme. REFERENCES [] D. Burshte ad A. Strugatsk, Polar wrte oce memory codes, IEEE Tras. If. Theor., vol. 59, o. 8, pp , Aug [2] E. E Gad, Y. L, J. Klewer, M. Lagberg, A. Jag, ad J. Bruck, Polar codg for osy wrte-oce memores, Mauscrpt avalable at [3] A. Gabzo ad R. Shaltel, Ivertble zero-error dspersers ad defectve memory wth stuck-at errors, APPROX-RANDOM, 202, pp [4] S. Gel fad ad M. Psker, Codg for chael wth radom parameters, Problems of Cotrol Theory, vol. 9, o., pp. 9 3, 980. [5] N. Goela, E. Abbe, ad M. Gastpar, Polar codes for broadcast chaels, Proc. IEEE It. Symp. o If. Theor. (ISIT),, July 203, pp [6] C. Heegard, O the capacty of permaet memory, IEEE Tras. If. Theor., vol. 3, o., pp , Jauary 985. [7] J. Hoda ad H. Yamamoto, Polar codg wthout alphabet exteso for asymmetrc models, IEEE Tras. If. Theor., vol. 59, o. 2, pp , 203. [8] A. Jag, V. Bohossa, ad J. Bruck, Rewrtg codes for jot formato storage flash memores, IEEE Tras. If. Theor., vol. 56, o. 0, pp , 200. [9] A. Jag, Y. L, E. E Gad, M. Lagberg, ad J. Bruck, Jot rewrtg ad error correcto wrte-oce memores, Proc. IEEE It. Symp. If. Theor. (ISIT), 203, pp [0] S. Korada ad R. Urbake, Polar codes are optmal for lossy source codg, IEEE Tras. If. Theor., vol. 56, o. 4, pp , Aprl 200. [] A. Kusetsov ad B. S. Tsybakov, Codg a memory wth defectve cells traslated from Problemy Peredach Iformats, vol. 0, o. 2, pp , 974. [2] M. Modell, S. H. Hassa, I. Saso, ad R. Urbake, Achevg Marto s rego for broadcast chaels usg polar codes, Avalable at Jauary 204. [3] R. Rvest ad A. Shamr, How to reuse a wrte-oce memory, Iformato ad Cotrol, vol. 55, o. -3, pp. 9, 982. [4] A. Shplka, Capacty achevg multwrte WOM codes, IEEE Tras. If. Theor., vol. PP, o. 99, pp., 203. [5] E. Yaakob, P. Segel, A. Vardy, ad J. Wolf, Multple error-correctg wom-codes, IEEE Tras. If. Theor., vol. 58, o. 4, pp , Aprl 202. [6] G. Zemor ad G. D. Cohe, Error-correctg wom-codes, IEEE Tras. If. Theor., vol. 37, o. 3, pp , May