O the Use of Hard-Decisio LDPC Decoders o MLC NAND Flash Memory Khoa Le ad Fakhreddie Ghaffari ETIS, UMR-8051, Uiversité Paris Seie, Uiversité de Cergy-Potoise, ENSEA, CNRS, Frace. {khoa.letrug, fakhreddie.ghaffari}@esea.fr Abstract The soft-decisio Low-Desity Parity-Check (LDPC) decoders are widely used i the multi-level per cell (MLC) NAND flash memory to esure the correctess of stored data. However, to obtai the good error correctio capability, these soft-decisio decoders require the soft-iformatio, which results a log o-chip memory sesig ad dramatically exteds the memory read latecy. I this paper, we explore the possibility of usig some recet itroduced Bit Flippig (BF) hard decisio decoders, i replacig the soft-decisio decoders, to correct the retetio errors appeared i the MLC NAND flash memory. The applied BF decoders require oly the hard iformatio read from that memory to decode ad thus, by avoidig soft-iformatio sesig, the memory read latecy is improved. Furthermore, by usig the characteristic of the retetio error where some iformatio bits ca be determied as error-free, we propose the adaptatio of the BF decoders to be better adapted to MLC NAND flash memory. We show that, the adapted BF decoders ca correct more erroeous bits thaks to these error-free values. The simulatio results show that, the adapted BF decoders ruig o the flash memory provide a good error correctio performace, beig close to that of the soft-decisio decoders, with oly the hard iformatio of the memory. I additio, it is show that, the BF decoders provide a higher decodig speed compared to the soft-decisio decoder. I. INTRODUCTION NAND flash memory have bee used i practical applicatios thaks to its favorable features such as the high data throughput, low-power cosumptio, compactess [1]. I order to icrease the storage capacity of NAND flash memory, multi-level cell techology have bee developed to store more tha 1 bit per cell, e.g. 2 bits per cell i [2], 3 bits per cell i [3] or 4 bits per cell i [4]. This icreasig storage desity results to the fact that, the stored data is more proe to error due to various uderlyig physical factors [5], appearig at the type of retetio error, cellto-cell iterferece (CCI) error etc. This MLC techology sigificatly icreases the bit error rate ad a powerful error correctio code with soft-iformatio has to be employed to yield a good error correctio performace. There are several works o the use of the soft-decisio LDPC codes for error correctio o MLC NAND flash memory [2][6][7]. There are also several works o efficiet hardware implemetatio of LDPC decodig algorithm o that type of memory [8]. All of these researches have demostrated that, the soft-decisio LDPC decoders ca achieve a oticeable error correctio gai over the covetioal bose-chaudhuri-hocqueghem (BCH) code. However, sice the MLC NAND flash memory read latecy is proportioal to the memory sesig precisio, the use of soft-iformatio memory sesig causes sigificatly icrease of memory read latecy. Low-Desity Parity-Check (LDPC) are the well-kow error correctio code (ECC) applied i several commuicatio stadards such as IEEE 802.11, 802.3a, 802.16a etc. [9][10][11] ad storage applicatios as i [12][13][14]. LDPC ca be effectively decoded by the iterative decodig algorithms which are the iterative process where messages are iteratively passed betwee the computig uits: Variable Nodes (VNs) ad Check Nodes (CNs). A LDPC decoder is either the soft-iformatio message passig decoder, i.e. Mi Sum (MS), Sum Product (SP) [15], or hard-decisio decoder, i.e. Bit Flippig (BF), Gallager-A,B [16]. The soft-decisio decoders require the soft-iformatio from chael ad durig the decodig process, the soft-iformatio are passed betwee the computig uits. These soft-decisio decoders provide the outstadig error correctio, approachig the capacity. However, the decoder complexity is dramatically icreased while the decodig speed is relatively low, due to the itesive computatio. O the cotrary, the hard decisio decoders have bee recetly attracted much attetio due to the high decodig throughput, low-complexity implemetatio ad the progressive improvemet i error correctio, approachig or eve surpassig the soft-decisio decoders [17]. Recetly, several BF decoders such as Gradiet Descet Bit Flippig (GDBF) [18], Probabilistic Gradiet Descet Bit Flippig (PGDBF) [19], Probabilistic Parallel Bit Flippig (PPBF) [20] decoders, have bee itroduced, i which the error correctio performace is sigificatly improved, beig very close or eve comparable to the performace of softdecisio decoders. These decoders are show to be very lowcomplexity, which comes from the fact that, oly biary messages are iteratively passed betwee the computig uits simplifyig the coectio etworks. Furthermore, the computig uits which process these biary messages are very simple ad therefore, ehace the decodig throughput. More iterestigly, these decoders require oly the hard iformatio from the chael to start the decodig process. I this paper, we ivestigate o the use of BF decoders, i replacig the soft-decisio decoders, to correct the retetio errors i the MLC NAND flash memory. We focus o the GDBF ad PGDBF o MLC NAND flash memory i terms of error correctio as well as the hardware efficiecy such as the decodig throughput ad decoder complexity, by compariso
with the soft-decisio decoders. We take beefit from the fact that, the BF decoders require oly hard iformatio from chael to avoid the log memory sesig, ad from that, the memory read latecy is obviously reduced. Moreover, by usig the characterizatio of the retetio errors i MLC NAND flash memory, we ca determie certai bits which are reliable. We the adapt the BF decoders such that the reliable VNs are kept uchaged i the VN processig uits (VNUs) while decodig oly o other ureliable VNs. We show that, with the additioal iformatio i determiig the reliable VNs i MLC NAND flash memory, the modified BF decoders ca correct more retetio errors tha the covetioal couterparts. The rest of the paper is orgaized as followig. I Sectio II, we recall the characteristic of the retetio oise i MLC NAND flash memory ad characterize the priciple to determie the reliable bits whe the retetio errors occur. The LDPC decodig priciple ad the deployed BF decoders (the GDBF ad PGDBF decoders) are also itroduced. I Sectio III, we preset the adapted BF decoders (deoted as A-GDBF ad A-PGDBF) dedicated for MLC NAND flash memory. I Sectio IV, we show for the test case that, the BF decoders provide a comparable error correctio to that of the soft-decisio decoders (the MS decoder) while the average umber of decodig clock cycles is oly 1/4. Although the advatages of avoidig soft-iformatio sesig is ot quatified, the fact that the adapted BF decoders use oly the hard-decisio iformatio reduces obviously the read latecy of MLC NAND flash memory. Sectio V cocludes the paper. II. THE MLC NAND FLASH MEMORY RETENTION ERROR CHARACTERIZATION AND BF LDPC DECODERS RECALL A. The characterizatio of retetio errors i MLC NAND flash memory MLC NAND flash memory allows storig more iformatio i a sigle cell which is realized by the umber of charges trapped i the floatig gate ad the umerous voltage thresholds to idetify the storage states [2]. I each MLC NAND flash memory cell, the voltage levels betwee two adjacet storage states become closer, makig it beig more proe to errors whe beig read. Fig. 1 shows the relatio betwee the stored data ad its correspodig threshold voltage distributio for a 2-bits per cell NAND flash memory. There are 4 states of the stored data which are usually represeted by the Gray code to reduce the raw bit error probability. While the data is stored i the memory cell, the programmed cell gradually loses its charge i the floatig gate (Fig. 1b). This leads evetually to a chage i logic states, causig the error kow as the retetio error. The appearace of retetio error depeds o several factors e.g. the umber of Program/Erase (P/E), the storig time. There are also other types of errors i MLC NAND flash memory such as the cell-to-cell iterferece (CCI) or programmig error (which appears whe the data is beig writte). However, several works i literature showed that, the retetio error is the mai source of error appearig i MLC NAND flash memory [5]. We assume i this work that oly Retetio directio Number of charges 0 01 00 10 11 Logic values v th v ref3 v ref2 v ref1 Desired distributio Distributio after losig charges a. b. Figure 1. The retetio behavior i MLC NAND flash memory. retetio errors appear i the read data from MLC NAND flash memory. Also, we limit the illustratios i this work to oly 2-bits per cell MLC NAND flash memory. Whe readig data from a MLC NAND flash memory cell, the retur data is i the form as (most-sigificat-bit, least-sigificat-bit) or (MSB,LSB). Further iformatio o the MLC NAND flash memory such as the orgaizatio, program/read process ca be foud i [2][6][7]. The charge trapped i the floatig gate is gradually dimiished formig the fact that, the retetio error is always i the type of oe-directio state trasitios. Ideed, it ca be see i Fig. 1 that, the logic state 01 is realized at the highest charge trapped. The loss of charge chages oly that logic state to the lower charge level states such as states 00 or 10 ad there is o logic state that ca form the state 01 whe the retetio errors happe. This implies the fact that, the 01 read from MLC NAND flash memory cell is the error-free (reliable) iformatio. This observatio have also itroduced i [21]. The work of [21] also showed more cases where we ca determie the reliability of the read data. Ideed, the MSB whe readig 00 from the cell is reliable because if the retetio error occurs, it chages from the state 01 to 00 ad oly LSB is chaged. With the same priciple, the LSB of 10 ad MSB of 11 are also the reliable data. Note that, we assume i this work that, oly sigle step retetio error (where the logic state is oly chaged to its closest adjacet state) appears i MLC NAND flash memory. It is a acceptable assumptio sice the multiple state jump is extremely rare thaks to the observatio i [5][22]. By usig this characterizatio of retetio error, we propose to have a pre-processig o the read data from the MLC NAND flash memory to determie the reliable iformatio before lauchig the LDPC decodig process. This extra iformatio will help the BF correct more v th
erroeous bits i the decodig process. B. GDBF ad PGDBF decoders A LDPC code is defied by a sparse parity-check matrix H (M, N), N > M. Each row represets a parity check fuctio, computed by a CN, o the VNs represeted by the colums. The VN v (1 N) is checked by the CN c m (1 m M) if the etry H(m, ) = 1 ad they are called eighbors. LDPC code ca also be represeted by a bipartite graph called Taer graph composig two groups of odes, the CNs c m, (1 m M) ad the VNs v, (1 N). The VN v (1 N) coects to the CN c m (0 m M) by a edge i the Taer graph if the etry H(m, ) = 1. We deote the set of CNs coected to the VN v as N (v ), (1 N), ad N (v ) is called VN degree. Similarly, N (c m ), (1 m M) deotes all VNs coected CN m ad N (c m ) is the CN degree. This paper works o the regular LDPC code, i.e. N (v ) = d v ad N (c m ) = d c, m. Whe LDPC ECC is applied i the MLC NAND flash memory, the user data is ecoded ad stored i form of a codeword. A vector x = {x } = {0, 1} N, (1 N) is called a codeword if ad oly if Hx T = 0. We deote y = {y }, (1 N) as the read versio of x from the memory. The retetio error i this work is modeled as a crossover behavior such that each bit of x is flipped with a probability α, called raw Bit Error Rate (BER), i.e., Pr(y = x ) = 1 α ad Pr(y = 1 x ) = α, 1 N. We use the superscript (k) alog with the ode (VN ad CN) otatios to idicate that ode value at the k iteratio. The decodig process of BF decoders is a iterative process where the biary messages are iteratively passed betwee the VNs ad CNs. The CN operatio is the parity check operatio which is formulated i Equ. 1 where is the Exclusive-OR (XOR) operatio. The decodig process will be termiated if all the CN are satisfied i.e. c (k) m = 0, m, otherwise the decodig process cotiues with the VN operatios. E (k) c (k) m = v (k) (1) v N (c m) = v (k) y + c m N (v ) c (k) m (2) I each VN operatio, first of all, a so-called eergy value is computed usig Equ. 2 basig o the all eighbor CN values, the VN curret value (v (k) ) ad its value received from the chael (y ). The maximum eergy value, E max, (k) is computed by a Maximum Fider (MF). I GDBF decoder, the value of a VN will be flipped if ad oly if its eergy value is equal to the maximum eergy value. I PGDBF, a biary radom geerator is implemeted geeratig for each VN a radom sigal R (k) where Pr(R (k) = 1) = p ad Pr(R (k) = 0) = 1 p ad p is a pre-defied sigal. The value of each VN i PGDBF is flipped if ad oly if its eergy equals to the maximum eergy ad R (k) = 1. The GDBF ad PGDBF provide the good error correctio performace while requirig oly the hard-iformatio as show i [19][23]. We preset i the ext MLC NAND flash memory x ^ Raw data HD read ad pre-processig I BF LDPC Decoders y Decoded codeword Figure 2. The utilizatio of BF decoders o the MLC NAND flash memory system. sectio the adaptatio of these decoders to work o the MLC NAND flash memory. III. ADAPTED BF (A-BF) DECODERS FOR MLC NAND FLASH MEMORY By uderstadig the retetio error characteristic, we preset the adaptatio of the BF decoders to MLC NAND flash memory system. The MLC NAND flash memory system is described i the Fig. 2. The adaptatio icludes the additioal pre-processig uit which provides the reliability sigal for each data bit read from memory. The secod part of the adaptio is the modificatio i the decodig process basig o the reliability iformatio from the pre-processig module. A. The pre-processig module The target of desigig the pre-processig block is to idetify the reliability of each bit i the data read from MLC NAND flash memory ad to use this iformatio improvig the performace of the BF decoders. As preseted above, 2 bits (MSB ad LSB) are stored i the same cell ad basig o their values, the reliability ca be determied. We deote the reliability of the MSB as I MSB ad that of the LSB as I LSB. The value of these variables represets the reliability of the bits (reliable bit is deoted by 1, otherwise, it is marked by 0). We show the values of I MSB ad I LSB as the fuctios of MSB ad LSB i table I. It ca be see that, whe the read data is 01, both of MSB ad LSB are reliable. I the other cases, oly oe bit (either MSB or LSB) is reliable. The pre-processig module is desiged to idetify the reliability of a stored codeword ad to produce a biary vector I = (I 1, I 2,..., I N ). Each elemet of I represets the reliability of the correspodig bits (or VNs) i the read data. I the MLC NAND flash memory layout, two bits (MSB ad LSB) i a cell may belog to the same codeword [22] or they are i two differet codewords. I the latter case, pre-processig module eeds the read of two cosecutive codewords (i N cells) i order to compute I for a codeword, while i the former case, it eeds the iformatio i N/2 cells. A computig block is desiged to produce the values I MSB
MSB LSB XOR LSB I MSB I Figure 3. The computig uit to idetify the reliability of the read data from oe MLC NAND flash memory. ad I LSB o the value read (MSB ad LSB). The detailed circuit of this computatio block is described i Fig. 3. I geeral, the pre-processig module may be implemeted with oly 1 computig block ad used serially or may blocks are implemeted operatig i parallel for acceleratig the readig speed. It ca be see that, the pre-processig uit is a simple combiatory circuit ad it would require a egligible hardware overhead. Table I THE RELIABILITY DETERMINATION BASED ON THE DATA READ FROM A MLC NAND FLASH CELL. MSB LSB I MSB I LSB 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 0 B. The adapted BF decoders for MLC NAND flash memory The pre-processig uit provides the reliability iformatio of each bit read from the MLC NAND flash memory. We adapt the BF decoders to be used i this type of memory. We have applied our strategy for the GDBF ad PGDBF which are amed as A-GDBF ad A-PGDBF, respectively. The A- PGDBF decodig algorithm is described i Algorithm 1. It ca be see that, the A-PGDBF follows similarly the decodig step of the origial PGDBF ad the modificatio is o the VN operatio where a verificatio step is added. I the VN processig of A-PGDBF, the flippig step is proceeded oly if that VN value is idicated as ureliable (I = 0). This priciple is also applied for the A-GDBF. It ca be see that, the reliability iformatio of each VN is used i the adapted decoders, chagig the behaviors of VNs, compared to the origial decoders. Furthermore, this iformatio helps the adapted BF decoders correct more errors i the read data tha their origial couterparts. We illustrate a example where the origial decoder (GDBF) fails to correct the errors while the adapted decoder (the A-GDBF) ca correct i Fig. 4. It is show that, the GDBF fails with oly 3 error VNs located o a trappig set (5,3) (i Fig. 4a) [23] (the white circles (squares) represet the correct VNs (satisfied CNs) while the black circles (squares) represet the erroeous VNs (usatisfied CNs), with the assumptio that all zero codeword is set). GDBF fails to fid the correct codeword because whe flippig the erroeous VNs ( ad ), it flips also the correct VNs ( ad ) ad oscillates betwee the two states as i Fig. 4a ad Fig. 4b. O the cotrary, i A-GDBF decoder, the erroeous VNs will be certaily flipped while the correct VNs will be flipped oly whe they are ureliable. A-GDBF ca correct all the errors i Fig. 4a after 2 iteratios (the error cofiguratio as i Fig. 4a Fig. 4c Fig. 4d) if the correct VNs ad are reliable. It ca also correct all error VNs i 3 iteratios by goig Fig. 4a Fig. 4e Fig. 4c Fig. 4d if is reliable while is ureliable. A-GDBF fails to correct the errors whe both ad are ureliable. Therefore, while GDBF always fails, A-GDBF ca correct 75% whe this error patter appears. The advatage i error correctio of the adapted BF decoders is show further i the simulatio results i the ext sectio. Algorithm 1 The Adapted-PGDBF decodig algorithm for MLC NAND Flash memory 1: Iitializatio k = 0, v (0) = y 2: while k It max do 3: for 1 m M do CN processig 4: Compute c (k) m, usig Equ. (1). 5: ed for 6: if c (k) = 0 the Check sydrome 7: exit while 8: ed if 9: for 1 N do VN processig 10: Compute E (k), usig Equ. (2). 11: ed for 12: Sort E max (k) = max (E (k) ) 13: for 1 N do 14: if I = 0 the Added verificatio 15: Geerate R (k) 16: if E (k) from B(p). = 1 the = v (k) 1 = E max (k) ad R (k) 17: v (k+1) 18: ed if 19: ed if 20: ed for 21: k = k + 1 22: ed while 23: Output: v (k) IV. ERROR CORRECTION PERFORMANCE AND ANALYSIS I this sectio, we preset the simulated error correctio performace of adapted BF decoders o the MLC NAND flash memory with the retetio errors. Although the practical LDPC code i the MLC NAND flash memory is usually with very high code rate ad the log codeword [8], we simulated our BF decoders o a moderate code rate (R = 0.75), ad relative short (N = 1296) code to save the simulatio time. We assume that, the MSB ad LSB i a MLC NAND flash memory cell belog to 2 differet codewords ad we decode oly the codeword formed by the LSB bits. As show above,
10 4 10 5 A-GDBF baselie GDBF A-PGDBF baselie PGDBF MS a. b. Bit Error Rate (BER) 10 6 10 7 10 8 10 9 c. d. 10 10 0.01 Raw BER, α 0.001 Figure 5. The decodig performace of the BF decoders o MLC NAND flash memory. e. Figure 4. The error patter with which the GDBF fails to correct the error while the A-GDBF ca. the LSB codeword has higher probability to be erroeous tha the MSB codeword. We produce, therefore, the worst-case decodig performace of our BF decoders. We also show the decodig performace of the quatized MS decoder with 4 bits for LLR ad 6 bits for the AP-LLR iformatio [24]. The MS decoder is implemeted with the layered schedulig i which a decodig iteratio is composed by 4 layers. It ca be see i the Fig. 5 i geeral that, the BF decoders provide the good error protectio ability for MLC NAND flash memory. Ideed, the A-PGDBF provides a BER at 8.10 10 ad A-GDBF is with 4.10 8 whe the raw BER at 2.10 3. The A-PGDBF is much better tha the A-GDBF decoder i term of error correctio at the cost of hardware overhead whe beig implemeted, as observed i may works [23][25]. The reliability iformatio of each bit helps the BF decoders correct more retetio errors. I fact, we itroduce also i the Fig. 5 the decodig performace of the BF decoders whe the bit reliability iformatio are ot used ad deote them as baselie. It is show that, the A-GDBF loses aroud 5dB i BER while A-PGDBF loses 2dB at raw BER 2.10 3. Also, the A-GDBF seems to be affected o all values of α while the A-PGDBF is affected o the error floor regio oly. I geeral, the MS is better tha all BF decoders i error correctio at the cost of hardware complexity [23] ad the decodig throughput. We show i Fig. 6 the compariso o the average umber of clock cycles eeded to decode a codeword, as a fuctio of α. Due to the lack of space, we do ot show the detail of the BF ad MS hardware architectures (which ca be foud i [23] ad [24]). The MS decoders requires 2 clock cycles to decode 1 layer (it does 4 layers i a iteratio) while Average umber of clock cycles 16 14 12 10 8 6 4 2 0.01 Raw BER, α A-GDBF baselie GDBF A-PGDBF baselie PGDBF MS 0.001 Figure 6. The average clock cycles required to decode 1 codeword. the BF decoders requires 1 clock cycle to fiish 1 iteratio. The differece i the average umber of clock cycles ca be iterpreted to the differece i decodig throughput if the decoders operate at the same frequecy. It ca be see i the Fig. 6 that, the BF decoders provide a much higher decoder speed tha that of the MS decoder. At raw BER 2.10 3, the BF decoders is 4 times faster tha the MS decoder. Note further that, the BF decoders require oly the hard iformatio from MLC NAND flash memory while the soft-decisio decoders eed the soft-iformatio. Although we do ot quatify i detail this advatage, it will obviously improve the readig latecy whe the BF decoders are used i MLC NAND flash memory. V. CONCLUSION We propose i this paper a ivestigatio i usig the recet itroduced BF hard-decisio LDPC decoders (the GDBF ad PGDBF decoders), i replacig the soft-decisio, to correct the retetio error i MLC NAND flash memory. The motivatio
comes from the fact that, BF decoders require oly the hardiformatio from the chael to proceed the decodig process, ad from that, the log soft-iformatio sesig ca be omitted. Furthermore, by usig the characteristic of retetio error, we have adapted these BF decoders i such a way that they ca correct more errors i MLC NAND flash memory. The adapted BF decoders show a admirable error correctio improvemet, beig very close or eve comparable to soft-decisio decoders. The advatage of usig BF decoders are also cofirmed by the very small umber of clock cycles eeded for decodig a codeword which is iterpreted to the high decodig throughput. The promisig results i terms of error correctio capability, decodig throughput ad complexity lead the BF decoders to become the good ECC cadidates i the ew geeratio MLC NAND flash memory. ACKNOWLEDGMENT This work was fuded by the frech ANR uder grat umber ANR-15-CE25-0006-01 (NAND project). REFERENCES [1] R. S. Liu, M. Y. Chuag, C. L. Yag, C. H. Li, K. C. 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