Sum-Capacity of Multiple-Write Noisy Memory

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1 2011 IEEE Iteratioal Symposium o Iformatio Theory Proceedigs Sum-Capacity of Multiple-Write Noisy Memory Lele Wag ad Youg-Ha Kim Departmet of Electrical ad Computer Egieerig Uiversity of Califoria, Sa Diego La Jolla, CA USA {lew001, yhk}@ucsdedu Abstract Motivated by the emergig iterests i o-volatile solid-state computer memories such as flash memories, this paper studies the problem of repeatedly storig iformatio o memory cells with oise ad state The goal is to reliably covey t messages by writig Xj o a -cell oisy memory py j x j,y j 1), which stores Yj at the j-th write We model this problem as a chael with state ad itroduce the multiple-write oisy memory model, which icludes the write-oce memory ad flash memory models as special cases The t-write sum-capacity for the multiple-write oisy memory is established as C sumt) max [ IX 1;Y 1)+ t IU ] j;y j) IU j;y j 1)), j2 where the maximum is over all pmfs px 1) t j2 puj yj 1) ad fuctios x ju j,y j 1),j 2,,t We derive three outer bouds o the capacity regio ad discuss their extesio to other classes of memory models These results exted Wolf, Wyer, Ziv, ad Körer s work o the biary write-oce memory ad Fu ad Vick s work o the geeralized writeoce memory to oisy memories I INTRODUCTION AND MAIN RESULT A write-oce memory WOM) [12] is a biary iformatio storage medium, where each cell of the memory is iitialized at 0-state ad is allowed to chage from 0-state to 1-state, but ot vice versa Suppose we wish to reliably store t idepedet messages by writig t times o a -cell WOM The goal is to maximize the total umber of bits stored per cell Oe way to deal with this problem is to desig codewords such that i ay two cosecutive writes, there is o trasitio from 1 to 0 i the same cell positio The total umber of codewords satisfyig such property characterizes the limit o storig iformatio o the WOM A uderlyig assumptio i this combiatorial approach is that the memory cell always faithfully store the writte codewords A alterative probabilistic) way to attack this problem is through chael codig for commuicatio over a DMC with state Codewords are assumed to be arbitrary without ay costrait, while the memory cell may ot always record the iput faithfully Specifically i the WOM, whe a cell is at 0-state, it records what is writte Whe a cell is at 1-state, it always stores 1 regardless of what the ecoder writes see Figure 1) Clearly, the kowledge of the cell state iformatio help to make the commuicatio more reliable How much do the ecoder ad the decoder kow about the cell state? The cell state i the WOM is the output of the previous write ad before each write, the ecoder has access to the whole output sequece, while the decoder does ot Hece, the problem is readily recogized as a chael with state where the state iformatio is available ocausally oly at the ecoder X S 0 Y X S 1 Y Fig 1 Biary write-oce memory May attempts have bee made to determie the theoretical limit of maximum umber of bits stored per cell Several code costructios for the biary WOM were proposed by Rivest, Shamir [12] ad Cohe, Godlewski, ad Merkx [3] Wolf, Wyer, Ziv, ad Körer [13] established the zero-error) capacity regio of the biary WOM Fiat ad Shamir [5] geeralized the biary WOM by allowig q-ary alphabet ad extedig the legal state trasitios to ay directed acyclic graph This model has bee suggested as a good model for flash memories [9] Practical code costructios based o the geeralized WOM have bee proposed by Zémor, Cohe [15], Jiag, Bruck [9], Yaakobi, Siegel, Vardy, ad Wolf [14] amog others Fu ad Vick [6] established the zero-error) capacity regio of the geeralized WOM Further geeralizig the determiistic models for memories, Heegard [8] cosidered oisy memories ad foud the capacity regio for the WOM with symmetric oise The mai goal of this paper is to establish the capacity regio of oisy memories Toward this goal, we focus o the memories of the form t py j x j,y j 1 ), j1 where y 0 by covetio This model is simple, yet still captures the major challege i updatig iformatio o memory cells such as the issue of spatial depedece of the memory state It is geeral eough to iclude all the aforemetioed models of WOM, geeralized WOM, oisy WOM as special cases More specifically, we cosider the multiple-write oisy memory MWNM) model X 1 X 2 X t, py 1 x 1 )py 2 x 2,y 1 ) py t x t,y t 1 ), Y 1 Y 2 Y t ), as show i Figure 2 A 2 R1,,2 Rt,) code for the MWNM cosists of t message sets [1:2 R1 ],,[1:2 Rt ], t ecoders, where ecoder1 assigs a codewordx 1 m 1) to each message m 1 [1:2 R1 ] at the first write ad US Govermet work ot protected by US copyright 2494

2 M 1 M 2 M t X1 Y1 Ecoder 1 py 1 x 1 ) Decoder 1 X2 Y2 Ecoder 2 py 2 x 2,y 1 ) Decoder 2 Ecoder t X t Fig 2 Y 1 Y t 1 py t x t,y t 1 ) Y t multiple-write oisy memory Decoder t ecoder j [2 : t] assigs a codeword x j m j,yj 1 ) to each message m j [1:2 Rj ] with side iformatio yj 1 at the j-th write, ad t decoders, where decoder j [1: t] assigs a estimate ˆm j or a error e to each received sequece yj at the j-th write The average probability of error is defied as P ) e P{ ˆM 1,, ˆM t ) M 1,,M t )} A rate tuple R 1,,R t ) is said to be achievable for the MWNM if there exists a sequece of 2 R1,,2 Rt,) codes such that lim P e ) 0 The capacity regio C is the closure of all achievable rate tuples R 1,,R t ) The t- write sum-capacity C sum t) of the MWNM is the maximum achievable sum-rate t j1 R j As the mai result of this paper, we will prove the followig theorem i Sectio III Theorem 1: The t-write sum-capacity of the MWNM is C sum t) max [ t IX 1 ;Y 1 )+ IU j ;Y j ) IU j ;Y j 1 )) ], j2 where the maximum is over all pmfs px 1 ) t j2 pu j y j 1 ) ad fuctios x j u j,y j 1 ) for j [2:t] The achievability proof follows directly by usig the Gelfad Pisker codig scheme [7] The mai challeges i establishig the sum-capacity lie i the proof of the coverse Note that here the chael state sequece is the output of the previous write ad hece may ot be iid over space Such spatial correlatio i the state sequece makes this problem differet from the stadard setup of chaels with discrete memoryless state However, i our proof of the coverse, by carefully combiig the stadard techiques such as Fao s iequality, Csiszár sum idetity [4], ad auxiliary radom variables idetificatio, we ca cacel out such correlatio cumulatively I additio to the sum-capacity, three outer bouds o the capacity regio are established i Sectios II ad III We discuss their optimality i special cases The ier ad outer bouds recover all existig capacity results Heegard [8] cosidered a more geeral memory model t j1 py j,s j x j,s j 1 ), where s j, x j, ad y j deote the chael state, iput, ad output at the j-th write, respectively ˆM 1 ˆM 2 ˆM t However, as a cost of geerality, it is extremely hard to characterize the capacity regio or eve the sum-capacity The capacity regio of Heegard s model remais ope after more tha three decades Recetly, the phase-chage memory PCM) has received cosiderable attetio as the ext geeratio o-volatile solid-state memory techology A PCM cell stores data usig its amorphous ad crystallie states The cell chages betwee these two states through high temperature However, due to the sesitivity of the cells to temperature, it is importat to balace the heat i both time ad space whe programmig the cells Several models have bee proposed to study the PCM Jiag, Bruck, ad Li [10] itroduced the α, β, p) space-time costraied codes for the PCM, where each segmet of β cells ca be programmed at most p times i ay α cosecutive writes, which ca be viewed as a special case of Heegard s model As a alterative, Ahlswede ad Zhag s model of write-efficiet memory [2] has bee suggested as a good model for the PCM [11] I their model, cost fuctios are itroduced based o the assumptio that trasitios from 0 to 1 or 1 to 0 are costly, while writig the same symbol is cheap These two models differ from our MWNM model i that the ecoder may have oly partial access to the state iformatio ad that cost fuctios are itroduced Noetheless, we will briefly discuss how our results o the MWNM ca be applied to the models for phase-chage memories i Sectio IV II TWO-WRITE MEMORY We establish three outer bouds o the capacity regio The sum-capacity of the MWNM will be established alog the way For brevity, we first cosider the 2-write case i this sectio ad provide its extesio to t writes i Sectio III We first recall the followig Propositio 21 Special case of [8, Theorem 2]): A rate pair R 1,R 2 ) is achievable for the 2-write MWNM py 1 x 1 )py 2 x 2,y 1 ) if R 1 IX 1 ;Y 1 ), R 2 IU 2 ;Y 2 ) IU 2 ;Y 1 ) for some pmf px 1 )pu 2 y 1 ) ad fuctio x 2 u 2,y 1 ) Remark 21: This ier boud does ot become larger if we evaluate it for geeral pmf px 1 )pu 2,x 2 y 1,x 1 ) To see this, we first ote that the ier boud does ot become larger if evaluated over px 1 )pu 2,x 2 y 1 ), which follows similar argumets as i the Gelfad Pisker codig scheme The, it suffices to show for ay rate pair IX 1 ;Y 1 ),IU 2 ;Y 2 ) IU 2 ;Y 1 )) attaied by p x 1 )p u 2,x 2 y 1,x 1 ), there exist some pmf px 1 ) pu 2,x 2 y 1 ) attaiig the same rate pair For ay p x 1 )p u 2,x 2 y 1,x 1 ), let px 1 ) p x 1 ) ad pu 2,x 2 y 1 ) x 1 p u 2,x 2 y 1,x 1 )px 1 y 1 ) p py 1 x 1 )p x 1 ) u 2,x 2 y 1,x 1 ) x 1 x 1 py 1 x 1 )p x 1 ) 2495

3 The, p x 1,y 1 ) px 1,y 1 ),p u 2,y 2 ) pu 2,y 2 ), ad p u 2,y 1 ) pu 2,y 1 ), which fully determie the achievable rate pair IX 1 ;Y 1 ),IU 2 ;Y 2 ) IU 2 ;Y 1 )) We establish three outer bouds o the capacity regio Propositio 22: If a rate pair R 1,R 2 ) is achievable for the 2-write MWNMpy 1 x 1 )py 2 x 2,y 1 ), the it must satisfy R 1 IX 1 ;Y 1 ), R 1 +R 2 IX 1 ;Y 1 )+IU 2 ;Y 2 ) IU 2 ;Y 1 ) for some pmf px 1 )pu 2,x 2 y 1,x 1 ) Proof: To establish the first iequality, cosider R 1 HM 1 ) IM 1 ;Y1 )+ǫ IM 1 ;Y 1i Y1,i+1)+ǫ a) b) IM1,Y 1,i+1 ;Y 1i) IY 1,i+1 ;Y 1i) ) +ǫ IM1,Y 1,i+1,X 1i;Y 1i ) IY 1,i+1 ;Y 1i) ) +ǫ IX1i ;Y 1i ) IY 1,i+1;Y 1i ) ) +ǫ 1) IX 1i ;Y 1i )+ǫ, where a) follows sice X 1i is a fuctio of M 1 ad b) follows sice M 1,Y1,i+1 ) X 1i Y 1i form a Markov chai For the secod iequality, the key is to idetify U 2i such that U 2i X 2i,Y 1i ) Y 2i form a Markov chai for each i [1:] Let U 2i M 2,Y2 i 1,Y1,i+1 ) The R 2 IM 2 ;Y 2i Y2 i 1 )+ǫ a) IM 2,Y i 1 2 ;Y 2i )+ǫ IM2,Y i 1 2,Y 1,i+1;Y 2i ) IY1,i+1 ;Y 2i M 2,Y2 i 1 ) ) +ǫ IM2,Y2 i 1,Y1,i+1 ;Y 2i) IY 1i ;Y2 i 1 M 2,Y1,i+1) ) +ǫ IM2,Y2 i 1,Y1,i+1;Y 2i ) IY 1i ;Y2 i 1,M 2,Y1,i+1)+IY 1i ;Y1,i+1) ) +ǫ IU2i ;Y 2i ) IU 2i ;Y 1i ) +IY 1i ;Y 1,i+1) ) +ǫ, 2) where a) follows from the Csiszár sum idetity Now combiig 1) ad 2), we have the boud for sum-rate R 1 +R 2 ) IX1i ;Y 1i )+IU 2i ;Y 2i ) IU 2i ;Y 1i ) ) +ǫ Usig the stadard time sharig radom variable ad takig completes the proof of Propositio 22 Remark 22: The sum-rate outer boud i Propositio 2 together with the ier boud i Propositio 21 establishes the 2-write sum-capacity C sum 2) max IX 1 ;Y 1 )+IU 2 ;Y 2 ) IU 2 ;Y 1 ) ), where the maximum is over all pmfs px 1 )pu 2 y 1 ) ad fuctio x 2 u 2,y 1 ) Remark 23: The outer boud i Propositio 22 is tight for ay sum-rate R 1 +λr 2 with λ [0,1] ad maxr 1 +λr 2 ) max [ IX 1 ;Y 1 )+λiu 2 ;Y 2 ) IU 2 ;Y 1 )) ], where the maximum is over all pmfs px 1 )pu 2 y 1 ) ad fuctios x 2 u 2,y 1 ) We ow preset the secod outer boud Propositio 23: If a rate pair R 1,R 2 ) is achievable for the 2-write MWNMpy 1 x 1 )py 2 x 2,y 1 ), the it must satisfy R 1 IX 1 ;Y 1 ), ad R 2 IU 2 ;Y 2 Y 1 ) for some px 1 )pu 2,x 2 y 1,x 1 ) Proof: The first iequality is exactly the same as i Propositio 22 To establish the secod iequality, choose U 2i M 2,Y2 i 1,Y1,i+1 Y 2i, i [1:], as desired Cosider,Y i 1 1 ) AgaiU 2i X 2i,Y 1i ) R 2 IM 2 ;Y2 Y 1 )+ǫ IM 2 ;Y 2i Y2 i 1,Y1 )+ǫ IM 2,Y i 1 2,Y 1,i+1,Y i 1 1 ;Y 2i Y 1i )+ǫ IU 2i ;Y 2i Y 1i )+ǫ Usig stadard time sharig radom variable ad takig completes the proof Remark 24: The auxiliary radom variable U 2 idetified i Propositio 23 is differet from the oe i Propositio 22 As a result, we do ot have a otrivial boud o the sum-rate i Propositio 23 Remark 25: The outer boud i Propositio 23 is tight whe py j x j,y j 1 ) is determiistic, ie, the geeralized WOM Takig U 2 Y 2, we have R 1 IX 1 ;Y 1 ) HY 1 ) ad R 2 IU 2 ;Y 2 Y 1 ) HY 2 Y 1 ) for some px 1 )px 2 y 1 ), which recovers the capacity regio of the geeralized write-oce memories established by Fu ad Vick [6] This outer boud follows essetially from the fact that the capacity whe side iformatio is available oly at the ecoder is upper bouded by the capacity whe side iformatio is available at both the ecoder ad the decoder I the determiistic case, these two capacities coicide We ow preset the third outer boud 2496

4 Propositio 24: If a rate pair R 1,R 2 ) is achievable for the 2-write MWNMpy 1 x 1 )py 2 x 2,y 1 ), the it must satisfy R 1 IU 1 ;Y 1 ), R 2 IU 2 ;Y 2 Y 1,U 1 ), R 1 +R 2 IU 1 ;Y 1 )+IU 2 ;Y 2 ) IU 2 ;Y 1 ) for some pu 1,x 1 )pu 2,x 2 y 1,u 1,x 1 ) Proof: I this outer boud, we itroduce a ew auxiliary radom variable U 1 To establish the first iequality, we expady1 differetly way from the previous proofs Idetify U 1i M 1,Y1 i 1 ) The R 1 a) IM 1 ;Y 1i Y i 1 1 )+ǫ IM1,Y i 1 1 ;Y 1i ) IY i 1 1 ;Y 1i ) ) +ǫ IU1i ;Y 1i ) IY 1,i+1;Y 1i ) ) +ǫ, 3) where a) follows from the fact that i 1 IY1 ;Y 1i ) HY 1i) HY1 ) IY 1,i+1 ;Y 1i) To establish the secod iequality, idetify U 2i M 2,Y2 i 1,Y1,i+1 ) The R 2 HM 2 Y1,M 1 ) IM 2 ;Y2 Y 1,M 1)+ǫ IM 2 ;Y 2i Y i 1 2,Y 1,M 1)+ǫ IM 2,Y i 1 2,Y 1,i+1;Y 2i Y 1i,M 1,Y i 1 1 )+ǫ IU 2i,Y 2i Y 1i,U 1i )+ǫ Sice we use the same U 2i as i Propositio 22, the iequality i 2) cotiues to hold Combiig 3) ad 2) establishes the third iequality R 1 +R 2 ) IU1i ;Y 1i )+IU 2i ;Y 2i ) IU 2i ;Y 1i ) ) Usig the stadard time sharig radom variable completes the proof of Propositio 24 Remark 26: The outer boud i Propositio 24 is tight both for ay sum-rate R 1 + λr 2, where λ [0,1], ad for the determiistic case Takig U 1 X 1, this outer boud recovers the outer boud i Propositio 22 I the determiistic case, this outer boud recovers the outer boud i Propositio 23 However, it is ot clear whether this is true i geeral III EXTENSION TO MORE THAN TWO WRITES The ier boud ad outer bouds for the 2-write MWNM exted aturally to the geeral t-write MWNM Propositio 31 Special case of [8, Theorem 2]): A rate tuple R 1,,R t ) is achievable for the t-write MWNM py 1 x 1 ) t j2 py j x j,y j 1 ) if R 1 IX 1 ;Y 1 ), R 2 IU 2 ;Y 2 ) IU 2 ;Y 1 ), R t IU t ;Y t ) IU t ;Y t 1 ) for some pmf px 1 ) t j2 pu j y j 1 ) ad fuctios x j u j,y j 1 ) for j [2:t] Remark 31: For a similar reaso as i Remark 21, this ier boud does ot become larger whe evaluated for geeral pmfs pu 1,x 1 ) t j2 pu j,x j y j 1,u j 1,x j 1 ) We ow exted Propositio 22 to the t-write MWNM Propositio 32: If a rate tuple R 1,,R t ) is achievable for the t-write MWM py 1 x 1 ) t j2 py j x j,y j 1 ), the it must satisfy the coditios R 1 IU 1 ;Y 1 ), R 1 +R 2 IU 1 ;Y 1 )+IU 2 ;Y 2 ) IU 2 ;Y 1 ), t R 1 + +R t IU 1 ;Y 1 )+ IUj ;Y j ) IU j ;Y j 1 ) ) j2 for some pu 1,x 1 ) t j2 pu j,x j y j 1,u j 1,x j 1 ) Proof: To boud the sum-rate, the key is still to cacel the oegative terms i 1 IYj 1 ;Y j 1,i) We play a similar trick as i the proof of Propositio 24 to cacel them cumulatively For each j [2 : t], idetify U ji M j,y i 1 j,yj 1,i+1 ) ad assume Y0 We have R j IM j ;Y ji Yj i 1 )+ǫ IMj,Y i 1 j,yj 1,i+1;Y ji ) IY i 1 j ;Y ji ) IYj 1,i+1;Y ji M j,y i 1 j ) ) +ǫ IMj,Y i 1 j,yj 1,i+1;Y ji ) IY i 1 j ;Y ji ) IY j 1,i ;Y i 1 j M j,yj 1,i+1) ) +ǫ IMj,Yj i 1,Yj 1,i+1 ;Y ji)+iyj 1,i+1 ;Y j 1,i) IY j 1,i ;Y i 1 j,m j,y IUji ;Y ji ) IU ji ;Y j 1,i ) a) j 1,i+1) IY i 1 j ;Y ji ) ) +ǫ +IYj 1,i+1 ;Y j 1,i) IYj i 1 ;Y ji ) ) +ǫ IUji ;Y ji ) IU ji ;Y j 1,i ) +IY i 1 j 1 ;Y j 1,i) IY i 1 j ;Y ji ) ) +ǫ, where a) follows from the fact IY j 1,i+1 ;Y j 1,i) HY j 1,i) HYj 1 ) i 1 IYj 1 ;Y j 1,i) It ca be readily observed that the oegative terms i 1 IYj 1 ;Y j 1,i),j 2,,t, telescope ad thus will be caceled i the sum-rate Usig the stadard time sharig radom variable ad takig completes the proof 2497

5 Remark 32: The ier boud i Propositio 31 ad the outer boud i Propositio 32 are tight for sum-rate, which is the mai result stated i Theorem 1 We ow exted Propositio 23 to the t-write MWNM Propositio 33: If a rate tuple R 1,,R t ) is achievable for the t-write MWNM py 1 x 1 ) t j2 py j x j,y j 1 ), the it must satisfy the iequalities R 1 IU 1 ;Y 1 ), R 2 IU 2 ;Y 2 Y 1 ), R t IU t ;Y t Y t 1 ) for some pu 1,x 1 ) t j2 pu j,x j y j 1,u j 1,x j 1 ) Proof sketch: The proof follows similar argumets as i Propositio 23 with U ji M j,yj i 1,Yj 1,i+1 i 1,Yj 1 ) for j [1:t] ad Y0 Remark 33: This boud is tight for the determiistic case We ow exted Propositio 24 to the t-write MWNM Propositio 34: If a rate tuple R 1,,R t ) is achievable for the t-write MWNM py 1 x 1 ) t j2 py j x j,y j 1 ), the it must satisfy the iequalities R 1 IU 1 ;Y 1 ), R 2 IU 2 ;Y 2 Y 1,U 1 ), R t IU t ;Y t Y t 1,U t 1 ), R 1 +R 2 IU 1 ;Y 1 )+IU 2 ;Y 2 ) IU 2 ;Y 1 ), t R 1 + +R t IU 1 ;Y 1 )+ IUj ;Y j ) IU j ;Y j 1 ) ) j2 for some pu 1,x 1 ) t j2 pu j,x j y j 1,u j 1,x j 1 ) Proof sketch: The proof follows similar argumets as i Propositio 24 with U ji M j,y i 1 j,yj 1,i+1 ) for j [1 : t] ad Y0 Remark 34: This boud is tight for the determiistic chaels ad for the sum-rate of arbitrary chaels IV EXTENSION TO PHASE-CHANGE MEMORY I [10], Jiag, Bruck, ad Li itroduced the α, β, p) space-time costraied codes for the PCM Ulike i the MWNM, there is a hidde memory state that is ot available at the ecoder Thus, to fid the sum-capacity for rewritig o such memories, we look ito Heegard s model t j1 py j,s j+1 x j,s j ), where S 1 for covetio Throughout our proof of the coverse, the techiques we used are Fao s iequality, Csiszár sum idetity, ad the chai rule for mutual iformatio, which are valid for Heegard s model as well The oly differece that eeds to be checked is whether our idetificatio of auxiliary radom variables still guaratees the right Markov chai coditio U ji X ji,s ji ) Y ji,s j+1,i ) for i [1:],j [2:t] This ca be verified for all our three idetificatios Thus Propositio ca be readily exteded to Heegard s geeral model I particular, the sum-capacity is upper bouded as C sum t) max [ t IX 1 ;Y 1 )+ IU j ;Y j ) IU j ;Y j 1 )) ], j2 where the maximum is over px 1 ) t j2 pu j,x j s j ), ad lower bouded usig the same expressio as C sum t) max [ t IX 1 ;Y 1 )+ IU j ;Y j ) IU j ;Y j 1 )) ], j2 except that the maximum is over px 1 ) t j2 pu j,x j y j 1 ) These two bouds ca be differet i geeral, however I [2], Ahlswede ad Zhag proposed the write-efficiet memory They itroduced a cost fuctio to characterize whether there is a chage i a memory cell Motivated by this result, we ca icorporate a cost fuctio bx,s) i the MWNM model By the mootoicity, cocavity, ad cotiuity of the capacity-cost fuctio C sum t,b) i B ad the typicality average lemma [1], our results o the MWNM model ca be exteded to the sum-capacity with cost costrait B as C sum t,b) max [ IX 1 ;Y 1 )+ t IU j ;Y j ) IU j ;Y j 1 )) ], j2 where the maximum is over px 1 ) t j2 pu j,x j y j 1 ) such that 1/t) E [ t j1 bx j,y j 1 ) ] B REFERENCES [1] A El Gamal ad Y-H Kim 2010) Lecture otes o etwork iformatio theory [Olie] Available: [2] R Ahlswede ad Z Zhag, Codig for write-efficiet memory, Iformatio ad computatio, vol 83, pp 80 97, 1989 [3] G D Cohe, P Godlewski, ad F Merkx, Liear biary code for write-oce memories, IEEE Tras Iform Theory, vol 32, o 5, pp , 1986 [4] I Csiszár ad J Körer, Broadcast chaels with cofidetial messages, IEEE Tras Iform Theory, vol 24, pp , 1978 [5] A Fiat ad A Shamir, Geeralized write-oce memories, IEEE Tras Iform Theory, vol 30, o 1, pp , 1984 [6] F W Fu ad A J H Vick, O the capaicty of geeralized writeoce memory with state trasitios described by a arbitrary directed acyclic graph, IEEE Tras Iform Theory, vol 45, o 1, pp , 1999 [7] S I Gelfad ad M S Pisker, Codig for chaels with radom parameters, Problem of Cotrol ad Iformatio Theory, vol 9, o 1, pp 19 31, 1980 [8] C Heegard, O the capaicty of 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