THe advent of high recording density enabling technologies,

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 An Iteratvely Decodable Tensor Product Code wth Applcaton to Data Storage Hakm Alhussen, Member, IEEE, and Jaekyun Moon, Fellow, IEEE arxv: v [cs.it] 29 Mar 200 Abstract The error pattern correctng code EPCC can be constructed to provde a syndrome decodng table targetng the domnant error events of an nter-symbol nterference channel at the output of the Vterb detector. For the sze of the syndrome table to be manageable and the lst of possble error events to be reasonable n sze, the codeword length of EPCC needs to be short enough. However, the rate of such a short length code wll be too low for hard drve applcatons. To accommodate the requred large redundancy, t s possble to record only a hghly compressed functon of the party bts of EPCC s tensor product wth a symbol correctng code. In ths paper, we show that the proposed tensor error-pattern correctng code T-EPCC s lnear tme encodable and also devse a low-complexty soft teratve decodng algorthm for EPCC s tensor product wth q-ary LDPC T-EPCC-qLDPC. Smulaton results show that T-EPCC-qLDPC acheves almost smlar performance to sngle-level qldpc wth a /2 KB sector at 50% reducton n decodng complexty. Moreover, KB T-EPCC-qLDPC surpasses the performance of /2 KB sngle-level qldpc at the same decoder complexty. Index Terms Tensor product codes, nter-symbol nterference, turbo equalzaton, error-pattern correcton, q-ary LDPC, mult-level log lkelhood rato, tensor symbol sgnatures, sgnature-correctng code, detecton postprocessng. I. INTRODUCTION THe advent of hgh recordng densty enablng technologes, poneered by gallopng mprovements n head and meda desgn and manufacturng processes, has pushed for smlar advances n read channel desgn and error correcton codng, drvng research efforts nto developng channelcapacty-approachng codng schemes based on soft teratve decodng that are also mplementaton frendly [], [2]. Soft decodable error correcton codes ECC, manly low densty party check LDPC codes, would eventually replace conventonal Reed-Solomon RS outer ECC, whch despte ts large mnmum dstance, possesses a dense party check matrx that does not lend tself easly to powerful belef propagaton BP decodng. There exsts vast lterature on the varous desgn aspects of LDPC coded systems for magnetc recordng applcatons. Ths ncludes code constructon [3] [6], effcent encodng [7], [8], decoder optmzaton [9] [], and performance evaluaton [2] [4]. In ths work, we propose an LDPC coded system optmzed for the magnetc recordng channel that spans contrbutons n most of these areas. Manuscrpt receved January 5, 2009; revsed August, Ths work was supported n part by the NSF Theoretcal Foundaton Grant Hakm Alhussen s wth Lnk-A-Meda Devces, Santa Clara, CA 9505, USA e-mal:hakma@lnk-a-meda.com. Jaekyun Moon s a Professor of Electrcal Engneerng at KAIST, Yuseonggu, Daeeon, , Republc of Korea e-mal:aemoon@ee.kast.ac.kr. The error-pattern correctng code EPCC s proposed n [5] [7] motvated by the well-known observaton that the error rate at the channel detector output of an ISI channel s domnated by a few specfc known error cluster patterns. Ths s due to the fact that the channel output energes assocated wth these error patterns are smaller than those of other patterns. A multparty cyclc EPCC was frst descrbed n [6], wth an RS outer ECC, possessng dstnct syndrome sets for all such domnant error patterns. To reduce the code rate penalty, whch s a severe SNR degradaton n recordng applcatons, a method to ncrease the code rate was ntroduced n [7] that also mproved EPCC s algebrac sngle and multple error-pattern correcton capablty. In ths method, the generator polynomal of the short base EPCC s multpled by a prmtve polynomal that s not already a factor of the generator polynomal. Also, the prmtve polynomal degree s chosen so as to acheve a certan desred codeword length. Moreover, [7] descrbes a Vterb detecton postprocessor that provdes error-event-relablty nformaton adng syndromemappng of EPCC to mprove ts correcton accuracy. However, mprovng the EPCC code rate by extendng ts codeword length ncreases the probablty of multple domnant error patterns wthn the codeword, and ths requres ncreasng the sze of the syndrome table consderably to mantan the same correcton power, whch eventually results n prohbtve decodng complexty. To mantan correcton power wth a manageable sze syndrome decodng table, [8] dscusses a more effcent method based on a lst decodng strategy that delvers satsfactory sector error rate SER gan wth an outer RS ECC. Later, ths lst decodng scheme was formulated as a soft-nput soft-output block n [9] and utlzed to enhance the performance of turbo equalzaton based on convolutonal codes CC. Nevertheless, the seral concatenaton scheme that proved successful wth RS hard decodng and CC-based turbo equalzaton does not work as well n seral concatenaton of long-epcc and LDPC. The reason s that when the LDPC decoder fals, especally n the water-fall regon, the sector contans a large number of multple error occurrences. When many such error events occur n a gven EPCC codeword, decodng by any reasonable sze lst decoder s formdable. Thus, an nner EPCC cannot n any capacty reduce the SER of a serally concatenated outer LDPC. On the other hand, f the EPCC codeword length s decreased substantally, then the number of errors per codeword s reasonable, as long as the overall code rate s somehow kept hgh. Here, the concept of tensor product constructon comes nto play. Tensor product party codes TPPC were frst proposed n [2] as the null-space of the party check matrx resultng

2 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 from the tensor product of two other party check matrces correspondng to a varety of code famles. As a result, based on the choce of the concatenated codes, TPPC would be classfed as an error correcton code f constructed from two ECCs, an error detecton code EDC f constructed from two EDCs, and an error locaton code ELC f constructed from an EDC and an ECC n a specal order. As a matter of fact, ELCs were ntroduced earler n [24] and ther algebrac propertes studed n detal, but later ncorporated n the unfed theme of TPPCs n [2]. Furthermore, a generalzed class of harddecodable ELCs was later suggested for applcaton n the magnetc recordng channel n [25]. In addton, TPPCs can be generalzed by combnng a number of codes on varous extenson felds wth shorter bnary codes. For ths more general case, a decodng algorthm was developed n [26]. An ECCtype TPPC was appled to longtudnal magnetc recordng n [22], and to perpendcular magnetc recordng n [23]. In [22], a hard decodable tensor product code based on a sngle party code and a BCH code s proposed as an nner code for RS. Ths code s sutable for low densty longtudnal recordng channels for whch domnant errors have odd weghts, such as {+} and {+,, +}. Also, [22] proposes that the hard decoder passes the corrected party bts to a Vterb detector reflectng channel states and party code states n order to compute the decoder output. Later, [23] presented two methods for combnng a tensor-product sngle party code wth a dstanceenhancng constraned code. Ths code combnaton acheved more satsfactory performance wth RS as an outer code n hgh densty perpendcular recordng channels. Our goal n ths work s to utlze the concept of tensor product concatenaton to construct hgh rate soft-decodable EPCCs on the symbol-level of the outer ECC. The EPCC target error lst s matched to the domnant error events normally observed n hgh densty perpendcular recordng channels. Snce domnant error events n perpendcular recordng are not only of odd weght [2], ths requres that our EPCC be a multparty code. However, n ths case, a Vterb detector matchng the channel and party wll have prohbtve complexty. In spte of ths, the performance of the optmal decoder of the baselne party-coded channel can be approached by the low complexty detecton postprocessng technque n [8]. We also present n detal a low complexty hghly parallel soft decoder for T-EPCC and show that t acheves a better performance-complexty tradeoff compared to conventonal teratve decodng schemes. A. Notatons and Defntons For a lnear code C : n,k,p, n denotes the codeword length, k denotes the user data length, and p n k denotes the number of code party bts. For a certan party check matrx H correspondng to a lnear code {C : Hc t 0, c C}, a syndrome s s the range of a perturbaton of a codeword Hc+e t s. A sgnature refers to the range under H for any bt block, not necessarly a codeword formed of data and party bts. The multlevel log-lkelhood rato mlllr of a random varable β GFq correspondng to the p.m.f. probablty mass functon p β Prβ, q 0 p β, can be defned as: γβ log pβ p 0β,γβ 0 0. [x] denotes a local segment [x,x +,...,x ] of the sequence x k. The perod of a generator polynomal on GF2 correspondng to a lnear code s equal to the order of that polynomal, as defned n [36]. Also, for a syndrome set {s } L 0 that corresponds to all the L possble startng postons of an error event, the perod P s defned as the smallest nteger such that s ρ+p s ρ [8]. Assume α L logα and β L logβ, then α + β L loge αl + e βl. Defne max αl β L α + β L maxα L,β L + log + e αl βl. Also, max {γ k } b b ka and max γ k are two dfferent representatons of the recursve mplementaton of max ka actng on the elements of the set {γ k } b ka. B. Acronyms TPPC: Tensor Product Party Code. qldpc: q-ary Low Densty Party Check code. RS: Reed Solomon code. BCJR: Bahl-Cocke-Jelnek-Ravv. T-EPCC: Tensor product Error Pattern Correcton Code. T-EPCC-qLDPC and T-EPCC-RS: Tensor product of EPCC and qldpc or RS, respectvely. LLR: Log-Lkelhood Rato. mlllr: mult-level Log-Lkelhood Rato. ML: Maxmum Lkelhood. MAP: Maxmum A Posteror. QC: Quas-Cyclc. SPA: Sum-Product Algorthm. II. REVIEW OF EPCC AND THE TENSOR PRODUCT CODING PARADIGM In ths secton we gve a bref revew on the concept of EPCC, ncludng the desgn of two example codes that wll be utlzed later n the smulaton study. Also, we revew the tensor product codng paradgm and present an encodng method that allows for EPCC-based lnear-tme-encodable TPPCs. A. EPCC Revew and Examples We revew constructng a cyclc code targetng the set of l max domnant error events {e k x,e2 k x,...,elmax k x} represented as polynomals on GF2 that can occur at any startng poston k n a codeword of length l T. A syndrome of error type e x at poston k s defned as s k x e k x mod gx, where gx s the generator polynomal of the code and mod s the polynomal modulus operaton. A syndrome set S for error type e x contans elements correspondng to all cyclc shfts of polynomal e x; elements of S are thus related by s k+ x s k mod gx.

3 ALHUSSIEN and MOON: AN ITERATIVELY DECODABLE TENSOR PRODUCT CODE WITH APPLICATION TO DATA STORAGE 3 Tensor symbol Tensor symbol 2 Tensor symbol n 2 - p 2 + Tensor symbol n 2 n bts p bts Fg.. TPPC C n,k,p C 2 n 2,k 2,p 2 codeword structure For unambguous decodng of e x and e x, {,}, we must have S S. Ths desgn requrement constransgx to have dstnct greatest common dvsors wth e x, for all targeted [6]. However, even f ths constrant s satsfed, an element n S can stll map to more than one error poston,.e., the perod of the syndrome set- and perod of gx- can be less than l T. Moreover, ths constrant s only suffcent but not necessary. As shown n [6], there may exst a lower degreegx that can yeld dstnct syndrome sets for the targeted error polynomals, resultng n a hgher rate EPCC. A search method to fnd ths gx s already dscussed n detal n [6] and [8]. We next gve two example EPCC constructons that wll be used throughout the paper. We target the domnant error events of the deal equalzed monc channel D n AWGN, whch s sutable as a partal response target n perpendcular magnetc recordng read channels. For ths channel, the domnant errors are gven by: e x, e 2 x + x, e 3 x + x + x 2, etc.,.e. they can be represented as polynomals on GF2 for whch all powers of x have nonzero coeffcents. The two EPCCs are: Example : Targetng error polynomals up to degree 4, we get the generator polynomal gx + x + x 3 + x 5 + x 6 of perod 2 va the search procedure of [6]. Choosng a codeword length of 2, 5 dstnct, non-overlappng syndrome sets are utlzed to dstngush the 5 target errors. Then, syndrome set S 3 wll have perod 6, whle all other sets have perod 2. A syndrome set of perod 6 means that each syndrome decodes to one of 2 possble error postons wthn the 2-bt codeword. Nonetheless, e 3 x can be decoded relably va channel relablty nformaton and the polarty of data support. The low code rate of 0.5 makes ths code unattractve as an nner code n a seral concatenaton setup for recordng channel applcatons. However, as we wll see later, a tensor code setup makes t practcal to use such powerful codes for recordng applcatons. Example 2: Targetng error polynomals up to degree 9, we have to record more redundancy. To accomplsh ths feat, a cyclc code wth 8 party bts, code rate 0.56, and a generator polynomal gx +x 2 +x 3 +x 5 +x 6 +x 8 of perod 8 s found by the search procedure n [6]. Then, syndrome sets S, S 3, S 5, and S 7 each have perod 8 and thus can be decoded wthout ambguty. Whle syndrome sets S 2, S 4, S 6, S 8, and S 0 each have perod 9, decodng to one of two postons. The worst s S 9 of perod2, whch would decode to one of 9 possble postons. Stll, the algebrac decoder can quckly shrnk ths number to few postons by checkng the data support, and then would choose the one poston wth hghest local relablty. B. Tensor Product Party Codes Constructon and Propertes of the TPPC Party Check Matrx: Consder a bnary lnear codec : n,k,p derved from the null space of party check matrxh c, and assumec corrects any error event that belongs to class ε. Also, consder a non-bnary lnear codec 2 : n 2,k 2,p 2 derved from the null space of party check matrx H c2 and defned over elements of GF2 p. Moreover, assume ths code corrects any symbol error type that belongs to class ε 2. As a prelmnary step, convert the bnary p n matrx H c, column by column, nto a strng of GF2 p elements of dmenson n. Then, construct the matrx H c3 H c H c2 as a p 2 n n 2 array of GF2 p elements. Fnally, convert the elements of H c3 nto p -bt columns, based on the same prmtve polynomal of degree p used all over n the constructon method. The null space of the p p 2 n n 2 bnary H c3 corresponds to a lnear bnary code C 3 : n n 2,k 3,p p 2. As shown n Fg., a C 3 codeword s composed of n 2 blocks termed tensor-symbols, each havng n bts. Also, t can be shown that C 3 can correct any collecton of tensor symbol errors belongng to class ε 2, provded that all errors wthn each tensor symbol belong to class ε [2]. Note that a tensor symbol s not an actual C codeword, and as such, usng the terms nner and outer codes would not be completely accurate. In addton, the tensor symbols are not codewords themselves, as can be seen n Fg., the frstk 2 tensor symbols are all data bts to start wth, and even the last p 2 tensor symbols, whch are composed of data and party bts, have non-zero syndromes under H c. Thus, a TPPC codeword does not correspond drectly to ether H c or H c2, and as a result, the component codebooks they descrbe are not recorded drectly on the channel. Another nterestng property of the resultng TPPC s that the symbol-mappng of the sequence of tensor-symbol sgnatures under H c forms a codeword of C 2, whch we refer to as the sgnature-correctng component code. 2 Encodng of Tensor Product Party Codes: The encodng of a TPPC can be performed usng ts bnary party check matrx, but the correspondng bnary generator matrx s not guaranteed to possess algebrac propertes that enable lnear tme encodablty. Thus, an mplementaton-frendly approach would be to utlze the encoders of the consttuent codes, whch can be chosen to be lnear tme encodable. Consder a bnary code C : n,k,p that s the null space of party check matrx H c, and a non-bnary code C 2 : n 2,k 2,p 2 defned on GF2 p, the tensor-product concatenaton s a bnary C 3 : n 3,k 3,p 3, where: n 3 n 2 n,k 3 n n 2 p p 2,p 3 p p 2

4 4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 Fg. 2. TPPC encoder of C n,k,p C 2 n 2,k 2,p 2 :a sgnatures calculated under H c, then the p -bt sgnatures are mapped to GF2 p symbols, b k 2 sgnatures encoded by generator matrx G c2 nto a C 2 codeword, then party symbols are mapped back to GF2, c p p 2 TPPC party bts are calculated by back substtuton. Assume that C s a cyclc code, and C 2 s any of the lnear tme encodable codes, where we choose a quas-cyclc QC component code for the purpose of ths study. Then, the encoders of C and C 2 communcate va the followng algorthm to generate a codeword of C 3, see Fg. 2: Receve a block of n k 2 +k n 2 k k 2 bts from the data source, call t maor block α. Dvde maor block α nto mnor block β of n k 2 bts, and mnor block γ of k n 2 k k 2 bts.e. k p 2 bts. Dvde blockβ nto k 2 columns each of n bts. Then, for each column, calculate the ntermedate p -bt sgnature under the party check matrx of C. Usng a feedback shft regster FSR to calculate the sgnatures, the computatonal cost s n operatons per sgnature, and n k 2 for ths entre step. v Convert ntermedate sgnatures from p -bt strngs nto GF2 p symbols. v Encode the k 2 non-bnary sgnatures nto a C 2 codeword of length n 2. Usng FSRs to encode the quas-cyclc C 2, the computatonal complexty of ths step s n 2. v Convert computed sgnatures back nto p -bt strngs. v Dvde block γ nto p 2 columns each of k bts. Add p blanks n each column to be flled wth the party bts of C 3. Then, algn each column wth the p 2 sgnatures computed n the prevous step, leavng p blanks n each column. v Fll blanks n the prevous step such that the sgnature of data plus party blanks underc equals the correspondng algned sgnature from step v. The party can be calculated usng the systematc H c and the method of back substtuton whch requres a computatonal complexty n per column. The total computatonal complexty of ths encodng algorthm s n k 2 +n 2 +n p 2,.e. t s n n 2 n 3, whch s the TPPC codeword length. Thus, we have shownwth some constrants- that f C and C 2 are lnear tme encodable, then C 3 C C 2 s lnear tme encodable. III. T-EPCC-RS CODES To demonstrate the algebrac propertes of TPPC codes, we present an example code sutable for recordng applcatons wth /2 KB sector sze. Consder two component codes: A bnary cyclc 8,0 EPCC of example 2 above wth rate 0.556, 8 party bts, and party check matrx n GF2 8 : [ α α 2 α 3 α 4 α 5... H epcc α 6 α 7 α 33 α 34 α 96 α α 82 α 236 α 234 α 27 α 92 α 93 ] 8. A 255,95 RS over GF2 8, of rate 0.765, t 30, and 60 party symbols. The resultng TPPC s a bnary 4590,40 code, of rate 0.896, and redundancy of 480 party bts. For ths code, a codeword s made of bt tensor symbols, of whch, any combnaton of 30 or less tensor symbol errors are correctable, provded that each 8-bt tensor symbol has a sngle or multple occurrence of a domnant error that s correctable by EPCC, those beng combnatons of error polynomals up to degree 9. Furthermore, although the EPCC consttuent code has a very low rate of 0.556, the resultng T-EPCC has a hgh rate of Notably, n the vew of the 8-bt EPCC, ths 6% reducton n recorded redundancy corresponds to an SNR mprovement of 2 db n a channel wth rate penalty 0log 0 /R, and 4. db n a channel wth rate penalty 0log 0 /R 2. A. Hard Decodng of T-EPCC-RS Codes Hard decodng of T-EPCC-RS drectly reflects the code s algebrac propertes, and thus, serves to further clarfy the concept of tensor product codes. Hence, we dscuss the hard decodng approach before gong nto the desgn of soft decodng of T-EPCC codes. The decodng algorthm s summarzed by the followng procedure, see Fg. 3: After hard slcng the output of the Vterb channel detector, the sgnature of each tensor symbol s calculated under H epcc. Each sgnature s then mapped nto a Galos feld symbol, where the sequence of non-bnary sgnatures

5 ALHUSSIEN and MOON: AN ITERATIVELY DECODABLE TENSOR PRODUCT CODE WITH APPLICATION TO DATA STORAGE 5 8 bts Tensor symbol Tensor symbol 2 Tensor symbol 3 Tensor symbol 255 Compute EPCC bnary Sgnatures and convert to GF2 8 EPCC Sgnature EPCC Sgnature 2 EPCC Sgnature 3 EPCC Sgnature bts or GF2 8 symbol RS hard decodng n GF2 8 or any lst soft decodng algorthm. RS symbol RS symbol 2 RS symbol 3 RS symbol 255 Convert back to bnary EPCC error syndromes. EPCC Error Syndrome 8 bts Lkely domnant Error 8 bts EPCC Error Syndrome 2 Lkely domnant Error 2 EPCC Error Syndrome 3 Lkely domnant Error 3 Fnd most lkely sngle and double errors. Add to ML word EPCC Error Syndrome 255 Lkely domnant Error 255 Fg. 3. Hard decoder of 8,0,8 EPCC 255,95,2t RS of t 30 over GF2 8. consttute an RS codeword - that s f the channel detector dd not suffer any errors. Any hard-nput RS decoder, such as the Berlekamp- Massey decoder, acts to fnd a legtmate RS codeword based on the observed sgnature-sequence. If the number of sgnature-symbols n error s larger than the RS correcton power, RS decodng fals and the tensor product decoder halts. v Otherwse, f RS decodng s deemed successful, the corrected sgnature-symbol sequence s added to the orgnal observed sgnature-symbol sequence to generate the error syndrome-symbol sequence. v Each error syndrome-symbol s mapped nto an EPCC bt-syndrome of the correspondng tensor symbol. v Fnally, EPCC decodes each tensor symbol to satsfy the error-syndrome generated by the component RS, n whch t faces two scenaros: A zero error-syndrome at the output of RS decodng ndcates ether no error occurred or a multple error occurrence that has a zero EPCC-syndrome, whch goes undetected. In ths case, the EPCC decoder s turned off to save power. A non zero error-syndrome wll turn EPCC correcton on. If the error-syndrome ndcates a sngle error occurrence n the target set, then, the EPCC sngle error algebrac decoder s turned on. On the other hand, f the error-syndrome s not recognzed, then EPCC lst decodng s turned on wth a reasonable-sze lst of test words. Note that although the number of EPCC codewords tensor symbols s huge, the decoder complexty s reasonable snce EPCC decodng s turned on only for nonzero error-syndromes. IV. T-EPCC-qLDPC CODES We learned from the desgn of T-EPCC-RS that the component sgnature-correctng codeword length can be substantally shorter than the competng sngle level code. Although the mnmum dstance s bound to be hurt f the ncreased redundancy does not compensate for the shorter codeword length, employng teratve soft decodng of the component sgnaturecorrectng code can recover performance f desgned properly. Whle LDPC codes have strctly lower mnmum dstances compared to comparable rate and code length RS codes, the sparsty of ts party check matrx allows for effectve belef propagaton BP decodng. BP decodng of LDPC codes consstently performs better than the best known soft decodng algorthm for RS codes. Snce the TPPC expanson enables the use of 2 to 4 tmes shorter component LDPC compared to a competng sngle level LDPC, a class of LDPC codes effcent at such short lengths are crtcal. LDPC codes on hgh order felds represent such good canddates. In that respect, [29] showed that the performance of bnary LDPC codes n AWGN can be sgnfcantly enhanced by a move to felds of hgher orders extensons of GF2 beng an example. Moreover, [29] establshed that for a monotonc mprovement n waterfall performance wth feld order, the party check matrx for very short blocks has to be very sparse. Specfcally, column weght 3 codes over GFq exhbt worse bt-error-rate BER as q ncreases, whereas column weght 2 codes over GFq exhbt monotoncally lower BER as q ncreases. These results were later confrmed n [30], where they also showed through a densty evoluton study of large q codes that optmum degree sequences favor a regular graph of degree 2 n all symbol nodes. On the other hand, for satsfactory error floor performance, we found that usng a column weght hgher than 2 was necessary. Ths becomes more mportant as the mnmum dstance decreases for lower q. For nstance, we found that a column weght of 3 mproved the error floor behavor of GF2 6 -LDPC at the expense of performance degradaton n the waterfall regon.

6 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 A. Desgn and Constructon of qldpc The low rate and relatvely low column weght desgn of qldpc n a TPPC results n a very sparse party check matrx, allowng the usage of hgh grth component qldpc codes. To optmze the grth for a gven rate, we employ the progressve edge growth PEG algorthm [30] n qldpc code desgn. PEG optmzes the placement of a new edge connectng a partcular symbol node to a check node on the Tanner graph, such that the largest possble local grth s acheved. Furthermore, PEG constructon s very flexble, allowng arbtrary code rates, Galos feld szes, and column weghts. In addton, modfed PEG-constructon wth lneartme encodng can be acheved wthout notceable performance degradaton, facltatng the desgn of lnear tme encodable tensor product codes. Of the two approaches to acheve lnear tme encodablty, namely, the upper trangular party check matrx constructon [30] and PEG constructon wth a QC constrant [3], we choose the latter approach, for whch the desgned codes have better error floor behavor. T-EPCCqLDPC lends tself to teratve soft decodng qute naturally. Next, we present a low complexty soft decoder utlzng ths mportant feature. B. Soft Decodng of T-EPCC-qLDPC To fully utlze the power of the component codes n T- EPCC-qLDPC, we need to develop a soft teratve verson of the hard decoder of T-EPCC-RS. To lmt the complexty of the proposed soft decoder, sub-optmal detecton post-processng s adopted nstead of the maxmum a posteror MAP detector to evaluate tensor symbol sgnature relabltes. The complexty of the optmal MAP detector matched to both the channel of memory length L and H epcc of row length p s exponental n p + L. We present a practcal soft detecton scheme that separates soft channel detecton from tensor symbol sgnature detecton, though, through a component sgnature-correctng LDPC n a TPPC setup, approaches the ont MAP performance through channel teratons. The man stages of the decoder are, see Fg. 4: Detecton postprocessng: Utlzng a pror nformaton from the prevous decodng teraton, bnary Vterb generates the hard ML word based on channel observatons, for whch the error sequence s calculated and passed to the correlator bank. A bank of local correlators estmates the probablty of domnant error type/locaton pars for all postons nsde each tensor symbol. 2 Sgnature p.m.f. calculaton: For each tensor symbol, the lst of most lkely error patterns s constructed. Ths lst ncludes sngle occurrences and a predetermned set of ther combnatons. The lst s then dvded nto sublsts, each under the sgnature value t satsfes. For each tensor symbol, usng each sgnature value s error lkelhood lst, we fnd the sgnature p.m.f. of that symbol. 3 q-ary LDPC decodng: Usng the observed sequence of sgnature p.m.f. s, we decode the component q-ary LDPC va FFT-based SPA. For each tensor symbol, the LDPC-corrected sgnature p.m.f. s convolved wth the observed sgnature p.m.f. at ts nput to generate the error-syndrome p.m.f.. 4 EPCC decodng: For each tensor symbol, we fnd the lst of most probable error-syndromes and generate a lst of test error words to satsfy each syndrome n the lst. A bank of parallel EPCC sngle-error correctng decoders generates a lst of most probable codewords along wth ther relabltes. 5 Bt-LLR feedback: Usng the codeword relabltes we generate bt-level relabltes that are fed back to the Vterb detector and the detecton postprocessng stage. Those bt-level relabltes, servng as a pror nformaton, favor paths whch satsfy both the ISI and party constrants. We explan each of these steps n the followng sectons, but we replace any occurrence n the text of syndrome sgnature p.m.f. by syndrome sgnature mult-level log-lkelhood ratos mlllr, as decodng wll be entrely n the log doman for reasons explaned below. r k λ k Hard-decson bnary Vterb c c ˆ + [ e ] h k + c : ML word ˆk q k { Ce } l T 2 { Ce } l k t l T l + l - e C E E Convert from local error pattern relablty to sgnature p.m.f. mlllr ch γ Sg t C ɶ + h + l + l k [ e ] * k + k [ s ] + k q h w w 0 T 0 l { max Ce } lt 0 k t + l T Fg. 5. Bank of parallel error-matched correlators to fnd error pattern type/poston relabltes. Detecton Postprocessng: At ths decoder stage we prepare a relablty matrx CE for error type/poston pars - captured n a tensor symbol of lengthl T - that s usable by the next stage to calculate the tensor symbol s sgnature mlllr: CE 2. l max 0 l T Ce Ce 0 Ce Ce 2 0 Ce2 l T l T Ce Ce lmax 0 Ce lmax Ce lmax l T wherece k s the error pattern type/ postonk relablty measure computed by the maxmum a posteror MAP- based error-pattern correlator shown n Fg. 5. The bank of local correlators dscussed here was also employed n [8]

7 ALHUSSIEN and MOON: AN ITERATIVELY DECODABLE TENSOR PRODUCT CODE WITH APPLICATION TO DATA STORAGE 7 LDPC teraton Map to bt-level a pror nfo EPCC lst decoder e γ Syn GF 6 2 Convolve 6 GF2 GF2 6 -LDPC Log-FFT-SPA r k λ k Bnary Vterb RS decoder t 6 c ˆk Global teraton h k Correlatore q k Correlatore 2 + Correlatore lmax 380 Tensor symbol : : 0 α 62 Syndrome Lst of lkely errors and relabltes ch γ Sg GF α 62 bˆk : : Fg. 4. T-EPCC-qLDPC soft decoder of 2,6,6 EPCC 380, 323 GF2 6 -LDPC. for AWGN channels, and n [9] for data-dependent nose envronments. We now dscuss how to generate these local metrcs. Let r k be the channel detector nput sequence r k c k h k + w k, where c k s the bpolar representaton of the recorded codeword sequence, h k s the partal response channel of length l h, and w k s zero-mean AWGN nose wth varance σ 2. Also, let q k r k ĉ k h k c k ĉ k h k +w k be the channel detector s output error sequence. If a target error pattern sequence e k occurs at postons from k to k +l, then q k can be wrtten as q k [c ĉ ] +l h k +w k [e ] +l h k +w k [s ] +lh +w k where s k s the channel response of the error sequence, and s gven by s k e k h k, and l h l +l h 2. Note that we defne the start of the tensor symbol at 0. So, f < 0, then the error pattern startng poston s n a precedng tensor symbol. The relablty for each error pattern wth startng poston,, can be computed by the local a posteror probabltes gnorng tensor symbol boundares for now: Pr Pr [e ] +l [s ] +lh [r] +lh,[ĉ] +lh l h + [q] +lh,[ĉ] +lh l h The most lkely assumed error type/poston par n a tensor symbol maxmzes the a posteror probablty rato of ts relablty to the relablty of the most probable error event the competng event n ths case would be the ML word tself, wth no error occurrence assumed at the output of Vterb detecton. Hence, utlzng 3 and Bayes rule, the rato to maxmze becomes Pr e ĉ,[q] +lh Pr [ML word] +l ĉ,[q] +lh Pr [q] +lh [ĉ] +lh Pr Pr [q] +lh l h +,[s ] +lh [ĉ] +lh l h +,[ s ] +lh Pr [s ] +lh [ s ] +lh where [ s ] +lh s the ML word s noseless channel response. Gven the nose model, [q] +lh s a sequence of ndependent Gaussan random varables wth varance σ 2. Therefore, maxmzng 4 can be shown to be equvalent to maxmzng the log-lkelhood local measure [8]: h +l Ce k 2σ 2 qk 2 q k s where the a pror bas n 5 s evaluated as: log Pr[ s ] +l h Pr[s ] +lh +l h k,ĉ k + k 2 log Pr[ s ] +l h λ k +l h k,ĉ k 4 Pr[s ] +lh 5 λ k 6 where λ k s the a pror LLR of the error-event bt at poston k as receved from the outer soft decoder, and we are assumng here that error event sequences do not nclude 0 bts,.e., the ML sequence and error sequence do not agree for the entre duraton of the error event. Equaton 5 represents the local error-pattern correlator output n the sense that t essentally descrbes the correlator operaton between q k and the channel output verson of the domnant error pattern e wthn the local regon [, + l h ]. However, equaton 5 gnores that errors can span tensor symbol boundares when < 0 or + l > l T. For nstance, an error n the frst bt of the tensor symbol can result from a sngle error event n that

8 8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 bt, a double error event n the last bt of the precedng tensor symbol, a trple error event occurrng two bts nto the prevous symbol, and so on. Hence, the probablty of an error n the frst bt s the sum of all these parent error event probabltes. Moreover, ths can be easly generalzed to boundary errors extendng beyond the frst bt. In a smlar manner, an error n the last bt of a tensor symbol can result from a sngle error event n that bt, a double error event startng n that bt and contnung nto the next tensor symbol, a trple error startng at the last bt and contnung nto the next tensor symbol, and so on. Agan, the probablty of an error event n that bt s the sum of the probabltes of all these parent events. Moreover, we have to nullfy the probablty of the parent error events n the modfed relablty matrx snce they are already accounted for n the last bt s relablty calculaton. Furthermore, ths can also be generalzed to error events startng earler than the last bt and extendng nto the next tensor symbol. In summary, to calculate a modfed metrc relevant to the current tensor symbol, we utlze the followng procedure: at 0, modfy Ce 0 Ce 0 max lmax k Ce k k+, ndependently for each, where l max s the maxmum length of a targeted error pattern. Startng at and l T, do: Ce max lmax Ce k. k k k >, set Ce. Set +,. v If < l max go back to. We assume here that domnant error events span only two tensor symbols at a tme and that they do not nclude error free gaps, whch s certanly true for the case study of ths paper. Followng ths procedure we obtan the modfed relablty matrx CE. 2 Sgnature mlllr Calculaton: For each tensor symbol, utlzng CE, we need to fnd the p.m.f. or the log doman mlllr of ts sgnature Sg GF2 pepcc, for EPCC wth p epcc party bts. To lmt the computatonal complexty of ths calculaton, we construct a sgnature only from the domnant errors and a subset of ther multple occurrences. Denote PrSg ˆ α as the runnng estmate of the p.m.f. at α, and ˆγSg α log PrSg ˆ α as the runnng estmate of mlllr. Denote a one dmensonal ndex of CE as p rc p c l max + p r correspondng to the p r -th row and p c -th column of CE and error Ep rc. We choose the domnant lst as the L patterns wth the largest correspondng elements of CE havng ndexes {p rc }L. Based on ths lst, we developed the followng procedure to compute ˆγSg α : Step Sngle occurrences: ˆγSg α prc L max kp rc Ck, k : G f q epcc H epcc [ĉ + l T l T Ek] t α 7 where q epcc 2 pepcc, and G f. s an operator that maps p epcc -bt vectors nto GFq epcc symbols. Step 2 Double occurrences: ˆγSg α ˆγ Sg α max } p rc L { Ck+ Cm,prc L kp rc,mprc, {k,m} : D k m Ek,Em > E free, G f H epcc [ĉ + l T l T Ek Em] t α q epcc 8 where D s the error free dstance between the two errors, E free l h s the error free dstance of the channel beyond whch the errors are ndependent.... Step M M occurrences: ˆγSg α max { M ξ Cq ξ {q,q 2,...,q M } : D s,t,s t ξ ˆγ Sg α } p rc L q ξ p rc,ξ,...,m, Eq s,eq t > E free, ξm G f H epcc [ Eq ξ ĉ + lt l T ] t α q epcc Step M + ML-sgnature relablty; computed so that the resultng sgnature p.m.f. sums to : γ Sg α βml max 0 q epcc max ML+ˆγ Sg α, β ˆγ Sg α βml max γ Sg α βml ˆγ Sg α βml 9 0 ˆγSg α ˆγSg α + ˆγSg α βml, Step M +2 Normalzaton:,,...,q epcc ; β ML +. γsg α ˆγSg α ˆγSg α. 3 In steps through M, to calculate the log-lkelhood of sgnature assumng value α, we sum the probabltes of all presumed sngle and multple errors n the ML word whose sgnatures equal α. Ths s equvalent to performng the max operaton n the log doman on error relabltes dctated by CE. However, to lmt the complexty of ths stage, we only use a truncated set of possble error combnatons, n all steps from to M. Also, for sgnature values that do not correspond to any of the combnatons, we set ther relablty 2

9 ALHUSSIEN and MOON: AN ITERATIVELY DECODABLE TENSOR PRODUCT CODE WITH APPLICATION TO DATA STORAGE 9 to, or more precsely, a reasonably large negatve value n practcal decoder mplementaton. Snce there are many such sgnature values, the correspondng constructed p.m.f. wll be sparse. In step M +, the lkelhood of the ML sgnature value s computed so that the p.m.f. of the tensor symbol sgnature sums to. In ths step, the max operaton n s a reflecton of the fact that n prevous steps, through M, some multple error occurrences have the same sgnature as the ML tensor symbol value. So, we have to account for such error nstances n the runnng estmate of the ML sgnature relablty. These events correspond to cases where error events are not detectable by H epcc,.e., they belong to the null space of H epcc. In step M + 2, the mlllr of the tensor symbol s centered around ˆγSg 0 to prevent the qldpc SPA messages from saturatng after a few BP teratons. 3 q-ary LDPC Decodng: Now, the sequence of sgnature mlllrs s passed as mult-level channel observatons to the qldpc decoder. We choose to mplement the log-doman q-ary fast Fourer transform-based SPA FFT-SPA decoder n [35] for ths purpose. The choce of log-doman decodng s essental, snce f we use the sgnature p.m.f. as nput, the SPA would run nto numercal nstablty resultng from the sparse p.m.f. generated by the precedng stage. The LDPC output posteror mlllrs correspond to the sgnatures of tensor symbols, rather than the syndromes of errors expected by EPCC decodng. Smlar to the decoder of T-EPCC-RS, error-syndrome Syn e s the fnte feld sum of the LDPC s nput channel observaton of sgnature, Sg ch, and output posteror sgnature relablty, Sg p. Moreover, the addton of hard sgnatures corresponds to the convoluton of ther p.m.f. s, and ths convoluton n probablty doman corresponds to the followng operaton n log-doman: ˆγSyn e αβe max ˆγSyn e αβe γsg ch α β ch +γsg p, αβp {β ch,β p } : α βe α β ch GFq epcc α βp, β ch,0,...,q epcc 2; β p,0,...,q epcc 2. 4 The error-syndrome mlllr s later normalzed, smlar to LDPC BP mlllr message normalzaton, accordng to: γsyn e αβe ˆγSyn e αβe ˆγSyn e α, β e,0,...,q epcc EPCC Decodng: An error-syndrome wll decode to many possble error events due to the low mnmum dstance of sngle-error correctng EPCC. However, EPCC reles on local channel sde nformaton to mplement a lst-decodng-lke procedure that enhances ts multple error correcton capablty. Moreover, the short codeword length of EPCC reduces the probablty of such multple error occurrences consderably. To mnmze power consumpton, EPCC s turned on for a tensor symbol only f the most lkely value of the error-syndrome mlllr s nonzero,.e., arg max γsyn e αβ α, α β GFq epcc ndcatng that a resolvable error has occurred. After ths, a few syndrome values, 3 n our case, most lkely accordng to the mlllr, are decoded n parallel. For each of these syndromes, the lst decodng algorthm goes as [8], [9]: A test error word lst s generated by nsertng the most probable combnaton of local error patterns nto the ML tensor symbol. An array of parallel EPCC sngle-pattern correctng decoders decodes the test words to produce a lst of vald codewords that satsfy the current error-syndrome. The probablty of a canddate codeword s computed as the sum of lkelhoods of ts parent test-word and the error pattern separatng the two. Each canddate codeword probablty s based by the lkelhood of the error-syndrome t s supposed to satsfy. In addton, when generatng test words, we only combne ndependent error patterns that are separated by the error free dstance of the ISI channel. 5 Soft Bt-level Feedback LLR Calculaton: The lst of canddate codewords and probabltes are used to generate bt level-probabltes n a smlar manner to [9], [27]. The converson of word-level relablty nto bt-level relablty for a gven bt poston can be done by groupng the canddate codewords nto two groups, accordng to the bnary value of the hard decson bt n that bt poston, and then performng group-wse summng of the word-level probabltes. Three scenaros are possble for ths calculaton: The canddate codewords do not all agree on the bt decson for locaton k; then, gven the lst of codewords and ther accompanyng a posteror probabltes, the relablty λ k of the coded bt c k s evaluated as λ k log c S + k c S k Prc ĉ,r Prc ĉ,r 6 where S + k s the set of canddate codewords where c k +, and S k s the set of canddate codewords where c k Although rare for such short codeword lengths, n the event that all codewords do agree on the decson for c k, a method nspred by [27] s adopted for generatng soft nformaton as follows λ k β ter λ max ˆd k 7 where ˆd k s the bpolar representaton of the agreedupon decson, λ max s a preset value for the maxmum relablty at convergence of turbo performance, and the multpler β ter < s a scalng factor. β ter n the frst global teratons and s ncreased to as more global teratons are performed and the confdence n bt decsons mproved. Thus, ths back-off control process reduces the rsk of error propagaton. The heurstc scalng n 7 s agan useful when EPCC s turned off for a tensor symbol, n case the most lkely error-syndrome beng 0. Then, the base hard value of the tensor symbol corresponds to the most lkely error event

10 0 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 found as a sde product n stage 2 of the T-EPCC-qLDPC decoder. C. Stoppng Crteron for T-EPCC-qLDPC and RS Erasure Decodng Due to the ambguty n mappng tensor symbols to sgnatures and syndromes to errors n stages 2 and 4 of the decoder, respectvely, the possblty of non-targeted error patterns, or errors that have zero error-syndromes that are transparent to H epcc, a second lne of defense s essental to take care of undetected errors. Therefore, an outer RS code of small correcton power t out s concatenated to T-EPCC-qLDPC to take care of the mperfectons of the component EPCC. Several concurrent functons are offered by ths code, ncludng: Stoppng Flag: If the RS syndrome s zero, then, global teratons are halted and decsons are released. Outer ECC: Attempt to correct resdual errors at the output of EPCC after each global teraton. Erasure Decodng: If the RS syndrome s nonzero, then, for those tensor symbols that EPCC was turned on, declare ther bts as erasures. Next, fnd the correspondng RS symbol erasures, and attempt RS erasure decodng whch s capable of correctng up to2 t out such erasures. In ths case, T-EPCC acts as an error locatng code. V. SIMULATION RESULTS AND DISCUSSION We compare three codng systems based on LDPC: conventonal bnary LDPC, q-ary LDPC, and T-EPCC-qLDPC, where all the component LDPC codes are regular and constructed by PEG wth a QC constrant. We study ther sector error rate SER performance on the deal equalzed partal response target +0.85D corrupted by AWGN, and wth codng rate penalty 0log 0 /R. The nomnal systems run at a codng rate of 0.9. The mnmum SNR requred to acheve relable recordng at ths rate s 3.9 db, estmated by followng the same approach as n [28]. A. Sngle-level BLDPC & qldpc Smulaton Results In Fg. 6, we compare SER of the followng LDPC codes, each constructed by PEG wth a QC constrant: A 4550,4095 GF2-LDPC, of column weght 5, and crculant sze 9 bts. The channel detector s a 2 state bnary BCJR. A 570,50 GF2 8 -LDPC, of codeword length 4560 bts, column weght 2, and crculant sze of 5 symbols. The channel detector s a symbol-bcjr wth 256 branches emanatng from each of 2 states. A 760,684 GF2 6 -LDPC, of codeword length 4560 bts, column weght 2, and crculant sze of 9 symbols. The channel detector s a symbol-bcjr wth 64 branches emanatng from each of 2 states. A 775,700 GF2 6 -LDPC, of codeword length 4650 bts, column weght 3, and crculant sze of 25 symbols. The channel detector s a symbol-bcjr wth 64 branches emanatng from each of 2 states. For the bnary LDPC turbo equalzer, we run a maxmum of 0 50 teratons, 0 global, and 50 LDPC BP teratons. For the q-ary turbo equalzers, on the other hand, we run a maxmum of 3 50 teratons. A column weght of 2 gves the best waterfall performance of q-ary LDPC. However, GF2 6 -LDPC exhbts an error floor as early as at SER 6 0 4, whereas a hgher order feld of GF2 8 does not show such a tendency down to 0 5. Nevertheless, the prohbtve complexty of GF2 8 symbol-bcjr makes GF2 6 -LDPC a more attractve choce. Stll, we need to sacrfcegf2 6 -LDPC s waterfall performance gans to guarantee a lower error floor. For that purpose, we move to a column weght 3 GF2 6 -LDPC that s.37 db away at 0 5 from the ndependent unformly-dstrbuted capacty C I.U.D. of the channel [28], and 0.37 db away from GF2 8 - LDPC a the same SER. In ths smulaton study, we have observed that whle bnary LDPC can gan up to 0.4 db through 0 channel teratons before gan saturates, GF2 8 - LDPC and GF2 6 -LDPC acheve very lttle teratve gan by gong back to the channel, between 0.09 to 0.2 db through 3 channel teratons. One way to explan ths phenomenon, s that symbol-level LDPC decodng dvdes the bt stream nto LDPC symbols that capture the error events ntroduced by the channel detector, renderng the bnary nter-symbol nterference lmted channel nto a memoryless mult-level AWGN lmted channel. Nonetheless, error events spannng symbol boundares rentroduce correlatons between LDPC symbols that are broken only by gong back to the channel. In other words, f t was not due to such boundary effects, a q-ary LDPC equalzer would not exhbt any teratve turbo gan whatsoever. Nonetheless, full-blown symbol BCJR s stll too complex to ustfy salvagng the small teratve gan by performng extra channel teratons [33]. Ths s where error event matched decodng comes nto the pcture, whch leads us to the results of the next secton. B. T-EPCC-qLDPC Smulaton Results We frst construct two T-EPCC-qLDPC codes of rate 0.9, the same rate as the competng sngle-level qldpc. These TPPC s are based on EPCC 2,6 of example. The codes constructed are: TPPC-A: A /2 KB sector, bnary 4680,422 TPPC, of rate 0.9, and 468 party bts, based on a component 390,32 PEG-optmzed QC GF2 6 -LDPC, of rate 0.8, column weght 3, and crculant sze 26. TPPC-B: A KB sector, bnary 9360,8424 TPPC, of rate 0.9, and 936 party bts, based on a component 780,624 PEG-optmzed QC GF2 6 -LDPC, of rate 0.8, column weght 3, and crculant sze 52. Frst, we study the SER of T-EPCC-qLDPC ust up to the component GF2 6 LDPC decoder, and only at the frst channel pass. Ths SER s functon of the Vterb symbol error rate, and the accuracy of generatng sgnature mlllrs, n addton to the component LDPC employed. Ths SER represents the best that the TPPC code can do, under the assumpton of perfect component EPCC,.e., as long as qldpc generates a clean codeword of sgnature-symbols, then EPCC generates

11 ALHUSSIEN and MOON: AN ITERATIVELY DECODABLE TENSOR PRODUCT CODE WITH APPLICATION TO DATA STORAGE Sector Error Rate C I.U.D Bnary LDPC, 0 50, Col wt 5. GF2 6 LDPC, 3 50, Col wt 3. GF2 6 LDPC, 3 50, Col wt 2. GF2 8 LDPC, 3 50, Col wt E / N db b o Fg. 6. Comparng SER of: 0 50 teratons of bnary LDPC, 3 50 teratons of GF2 8 -LDPC of column weght 2, and 3 50 teratons of GF2 6 -LDPC of column weghts 2 and 3. Mnmum SNR to acheve relable recordng at codng rate 0.9 s 3.9 db for +0.85D. a clean codeword of data-symbols. Fg. 7 shows the deal SER of these two TPPC codes, assumng perfect EPCC, compared to sngle-level GF2 6 -LDPC and GF2 8 -LDPC. Ideal /2 KB TPPC has about the same SER as sngle level GF2 6 - LDPC at SER. In /2 KB TPPC, the component GF2 6 -LDPC has half the codeword length of the sngle level counterpart, savng 50% of the decoder complexty, whle delverng smlar SER performance. The TPPC component qldpc faces a harsher channel than sngle-level qldpc, because the symbol error probablty of 6-bt data symbols s strctly less than the symbol error probablty of 6-bt sgnature symbols, where sgnature symbols are compressed down from 2-bt data symbols. Also, the shorter codeword length of component qldpc hurts ts mnmum dstance. Stll, these mparments are effectvely compensated for by an.4% ncrease n the redundancy of the TPPC component LDPC. On the other hand, f we match the codeword length of TPPC s component LDPC to sngle-level LDPC, as part of constructng KB TPPC, then, KB TPPC wll have smlar decoder complexty to /2 KB sngle-level LDPC wth about 0.2 db SNR advantage for KB deal TPPC at SER. Due to the mperfectons of EPCC desgn, ncludng mscorrecton due to one-to-many syndrome to error poston mappng, and undetected errors due to EPCC s small mnmum dstance, achevng the deal performance n Fg. 7 s not possble n one channel pass. In addton, an outer code s necessary to protect aganst undetected errors and provde a stoppng flag for the teratve decoder. Hence, one can thnk of an mplementaton of the full T-EPCC-GF2 6 LDPC decoder that ncludes an outer t 6 42,409 RS for the /2 KB case, and an outer t 2 842,88 RS for the KB case, so as to protect aganst EPCC resdual errors. These outer RS codes are defned on GF2 0 and have rate However, ths concatenaton setup wll run at a lower code rate of 0.875, whch can ncur an SNR degradaton larger than 0.25 db for a nose envronment characterzed by the rate penalty 0log 0 /R δ, δ 2. In a more thoughtful approach, one can preserve the nomnal code rate of 0.9 and redstrbute the redundancy between the nner TPPC and outer RS to acheve an mproved tradeoff between mscorrecton probablty and the nner TPPC s component LDPC code strength. In that sprt, we construct the followng concatenated codes: TPPC-C: A /2 KB sector, bnary 4560,428 TPPC, of rate 0.925, and 342 party bts, based on a component 380,323 PEG-optmzed QC GF2 6 -LDPC, of rate 0.85, column weght 3, and crculant sze 9. An outer t 6 422, 40 RS code of rate s ncluded, resultng n a total system rate of 0.9. TPPC-D: A KB sector, bnary 920,8436 TPPC, of rate 0.925, and 684 party bts, based on a component 760,646 PEG-optmzed QC GF2 6 -LDPC, of rate 0.85, column weght 3, and crculant sze 38. An outer t 2 844,820 RS code of rate s ncluded, resultng n a total system rate of 0.9. The control mechansm of teratve decodng for these codes s as follows: f EPCC results n less than 6 RS symbol errors for the /2 KB desgn or less than 2 for the KB desgn, or f EPCC generates more errors than ths, but declares less than 2 erasures for /2 KB or 24 erasures for KB, then, decodng halts and decsons are released. Otherwse, one more channel teraton s done by passng EPCC soft bt-level LLR s to Vterb detecton and the bank of error-matched correlators. Smulaton results n Fg. 8, for a nose envronment of rate penalty 0log 0 /R, demonstrate that after 3 channel teratons, the deal and practcal performances of the new TPPC codes almost lock, whle ncurrng mnmal SNR degradaton. Also, /2 KB TPPC saves 50% of decoder complexty whle achevng the same SER performance as sngle level LDPC for an addtonal SNR cost of 0.04 db at SER 0 5. Hence, TPPC-C represents a tradeoff between the lower complexty of GF2-LDPC and performance advantage of GF2 6 -LDPC, whereas KB TPPC has the same decodng complexty as sngle-level LDPC whle furnshng 0.8 db gan at SER. In terms of channel detector mplementaton complexty, the complexty and latency of GF2 6 -BCJR n the sngle level code far exceeds the overall complexty of the non-ldpc parts of two level T-EPCC-GF2 6 LDPC ncludng Vterb detecton. At the same tme, sgnature mlllr generaton, EPCC decodng, and bt-llr generaton are all mplemented tensor-symbol by tensor-symbol, achevng full parallelsm on the tensor-symbol level. Furthermore, t s only when qldpc fnds a syndrome error that EPCC decodng s turned on for each tensor symbol. To elmnate redundant computatons n the teratve decoder, branch metrc computaton n Vterb and 5 s only requred at the frst pass. For all subsequent teratons, however, only the a pror bas s updated n the second term of 5, and the branch update of Vterb [34]. One very mportant feature of the TPPC setup, that snglelevel LDPC lacks, s ts robustness to boundary error events.

12 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 28, NO. 2, FEBRUARY 200 Sector Error Rate Bnary LDPC, 0 50, Col wt 5. GF2 6 LDPC, 3 50, Col wt 3. GF2 8 LDPC, 3 50, Col wt 2. TPPC-A: Ideal /2 KB, TPPC-B: Ideal KB, Sector Error Rate Bnary LDPC, 0 50, Col wt 5. GF2 6 LDPC, 3 50, Col wt 3. GF2 8 LDPC, 3 50, Col wt 2. TPPC-C: Ideal /2 KB, TPPC-C: Real /2 KB, TPPC-D: Ideal KB, TPPC-D: Real KB, E b / N o db E b / N o db Fg. 7. Comparng SER of: 0 50 teratons of bnary LDPC, 3 50 teratons ofgf2 8 -LDPC of column weght2,3 50 teratons ofgf2 6 - LDPC of column weght 3, and 0 50 teratons of deal /2 KB and KB T-EPCC-GF2 6 LDPC based on column weght 3 LDPC. The presence of a syndrome-constrant means that errors spannng boundares are broken by EPCC when attemptng to ndependently satsfy the adacent tensor symbol syndromes, then, n the next turbo teraton, adacent tensor-symbols are decorrelated. Ths mechansm enables TPPC to recover from these errors by teratve decodng. However, for errors wth a zero error-syndrome whch go undetected by EPCC, outer RS protecton becomes handy. Based on the fact that TPPC enables an ncrease n the redundancy of ts component LDPC, n addton to smulaton results demonstratng the utlty of such lowered rate n combatng the harsher compressed channel, we conecture that as the sector length of both TPPC and sngle-level LDPC s drven to nfnty, TPPC wll acheve strct error rate SNR gans. Ths s manly because of ts surplus of redundancy compared to the sngle level code at the same rate penalty, whereas channel condtons and EPCC correcton power do not change wth replcaton of tensor symbols, and the error rate performance of LDPC asymptotcally approaches the nose threshold n the lmt of nfnte codeword length. Therefore, wthn a channel-capacty achevng argument, n the lmt of nfnte codeword length, we take the vew that TPPC wll brdge the gap to capacty further than any sngle level system could. Moreover, the advantage of TPPC for larger sector szes s more tmely than ever as the ndustry moves to the larger 4 KB sector format [33]. VI. CONCLUSIONS In a tensor product setup, codes of short codeword length and low rate can be combned nto hgh rate codes of nce algebrac propertes. We showed that encodng of tensor product codes s lnear tme f the component codes are lnear Fg. 8. Comparng SER n envronment of rate penalty0log 0 /R: 0 50 teratons of bnary LDPC, 3 50 teratons of GF2 8 -LDPC of column weght 2, 3 50 teratons of GF2 6 -LDPC of column weght 3, and 3 50 teratons of practcal /2 KB T-EPCC-GF2 6 LDPC+RSt 6, and KB T-EPCC-GF2 6 LDPC+RSt 2, both based on column weght 3 LDPC. tme encodable. We also demonstrated how the codeword length and rate of channel matched EPCC can be substantally ncreased by combnng wth a strong RS or LDPC of short codeword length. We also ncorporated an outer RS code of low correcton power to clean out the resdual errors of T-EPCC-RS or T-EPCC-LDPC TPPCs. In concluson, ths work establshed T-EPCC-qLDPC as a reasonable complexty approach to ntroducng non-bnary LDPC to the perpendcular recordng read channel archtecture, pavng the way to relable hgher recordng denstes. ACKNOWLEDGMENT The authors would lke to thank the anonymous revewers for ther constructve comments that helped enhance the techncal qualty and presentaton of ths paper. REFERENCES [] Hao Zhong, We Xu, Nngde Xe, and Tong Zhang, Area-effcent mnsum decoder desgn for hgh-rate quas-cyclc low-densty party-check codes n magnetc recordng, IEEE Transactons on Magnetcs, vol. 43, no. 2, pp , Dec [2] Hao Zhong, Tong Zhong, and Erch F. Haratsch, Quas-cyclc LDPC codes for the magnetc recordng channel: Code desgn and VLSI mplementaton, IEEE Transactons on Magnetcs, vol. 43, no. 3, pp. 8-23, Mar [3] N. Varnca, and A. Kavcc, Optmzed low-densty party-check codes for partal response channels, IEEE Communcatons Letters, vol. 7, no. 4, pp , Apr [4] S. Sankaranarayanan, B. Vasc, and E. M.Kurtas, Irregular low-densty party-check codes: Constructon and performance on perpendcular magnetc recordng channels, IEEE Transactons on Magnetcs, vol. 39, no. 5, pp , Sept [5] B. Vasc, and O. Mlenkovc, Combnatoral constructons of lowdensty party-check codes for teratve decodng, IEEE Transactons on Informaton Theory, vol. 50, no. 6, pp , June 2004.

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Ryan, LDPC decoder strateges for achevng low error floors, Informaton Theory and Applcatons Workshop, 2008, pp , Jan Feb [2] R. D. Cdecyan, E. Eleftherou, and T. Mttelholzer, Perpendcular and longtudnal recordng: A sgnal-processng and codng perspectve, IEEE Trans. Magn., vol. 38, no. 4, pp , Jul [3] X. Hu, and B. V. K. V. Kumar, Evaluaton of low-densty party-check codes on perpendcular magnetc recordng model, IEEE Transactons on Magnetcs, vol. 43, no. 2, pp , Feb [4] Hongxn Song, R. M. Todd, and J. R. Cruz, Applcatons of low-densty party-check codes to magnetc recordng channels, IEEE Journal on Selected Areas n Communcatons, vol. 9, no. 5, pp , May 200. [5] J. Moon, and J. Park, Detecton of prescrbed error events: Applcaton to perpendcular recordng, n Proc. IEEE ICC, vol. 3, pp , May [6] J. Park, and J. Moon, Hgh-rate error correcton codes targetng domnant error patterns, IEEE Trans. Magn., vol. 42, no. 0, pp , Oct [7] J. Park, and J. 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Cruz, Reduced-complexty decodng of Q-ary LDPC codes for magnetc recordng, IEEE Trans. Magn., vol. 39, no. 2, pp , Mar [36] R. Ldl, and H. Nederreter, Introducton to fnte felds and ther applcatons, 2nd ed. New York: Cambrdge Unversty Press, 994. Hakm Alhussen receved the B.S. and M.S. degrees n electrcal engneerng wth hgh honors from Jordan Unversty of Scence and Technology JUST, Irbd, n 200 and 2003, respectvely, and the MSEE and Ph.D. degrees n electrcal engneerng from the Unversty of Mnnesota Twn- Ctes, Mnneapols, n 2008 and 2009, respectvely. From 2003 to 2004, he was an Instructor wth the Department of Electrcal Engneerng at the Unversty of Yarmouk, Jordan. From 2004 to 2008, he held Research and Teachng Assstant postons wth the Department of Electrcal and Computer Engneerng, the Unversty of Mnnesota. Snce September 2008, he has been a Systems Desgn Engneer wth Lnk-A-Meda Devces, Santa Clara. Hs man research nterest are n the applcatons of codng, sgnal processng, and nformaton theory to data storage systems. Jaekyun Moon s a Professor of Electrcal Engneerng at KAIST. Prof. Moon receved a BSEE degree wth hgh honor from SUNY Stony Brook and then M.S. and Ph.D. degrees n Electrcal and Computer Engneerng at Carnege Mellon Unversty. From 990 through early 2009, he was wth the faculty of the Department of Electrcal and Computer Engneerng at the Unversty of Mnnesota, Twn Ctes. Prof. Moon s research nterests are n the area of channel characterzaton, sgnal processng and codng for data storage and dgtal communcaton. He receved the McKnght Land-Grant Professorshp from the Unversty of Mnnesota. He also receved the IBM Faculty Development Awards as well as the IBM Partnershp Awards. He was awarded the Natonal Storage Industry Consortum NSIC Techncal Achevement Award for the nventon of the maxmum transton run MTR code, a wdely-used errorcontrol/modulaton code n commercal storage systems. He served as Program Char for the 997 IEEE Magnetc Recordng Conference. He s also Past Char of the Sgnal Processng for Storage Techncal Commttee of the IEEE Communcatons Socety. In 200, he co-founded Berma, Inc., a fabless semconductor start-up, and served as foundng Presdent and CTO. He served as a guest Edtor for the 200 IEEE J-SAC ssue on Sgnal Processng for Hgh Densty Recordng. He also served as an Edtor for IEEE Transactons on Magnetcs n the area of sgnal processng and codng for He worked as consultng Chef Scentst at DSPG, Inc. from 2004 to He also worked as Chef Technology Offcer at Lnk-A-Meda Devces Corp. n He s an IEEE Fellow.