Efficient Box and Match Algorithm for Reliability-Based Soft-Decision Decoding of Linear Block Codes

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1 Effiient Box and Math Algorithm for Reliability-Based Soft-Deision Deoding of Linear Blok Codes Wenyi Jin LSI Logi 765 Syamore Drive Milpitas, CA Mar Fossorier EE Department University of Hawaii at Manoa Honolulu, HI Abstrat In this paper, effiient methods to improve the box and mathing algorithm (BMA) are presented. Firstly, an effiient approah is introdued to terminate the deoding if a loal optimal andidate satisfies a probabilisti suffiient ondition. The false alarm probability assoiated with the use of the probabilisti suffiient ondition is also derived. Seondly, by onstruting a ontrol band whih is assumed error free, the mathing apability of the BMA is enhaned. More preisely, the performane of BMA of order is nearly ahieved with a small inrease in omplexity and no inrease in memory with respet to the BMA of order. A tight performane analysis is derived based on the theory of order statistis. An error floor assoiated either with false alarms or with errors in the ontrol band is introdued, but this error floor an be ontrolled using the analysis in both ases. Simulation results show that the performane of the enhaned BMA for the deoding of the RS(255,239) ode with BPSK signaling over an AWGN hannel is about 0.1 db away from that of maximum likelihood deoding at the word error rate (WER). I. INTRDUCTIN The box and mathing algorithm (BMA)[3] is an effiient most reliable basis (MRB) based soft deision deoding algorithm. The BMA roughly redues the omputational ost of the ordered statisti deoding (SD) algorithm [1] by its squared root at the expense of memory. In addition to onsidering all odewords assoiated with error patterns of Hamming weight at most on the MRB, the BMA with order also onsiders all odewords assoiated with error patterns of Hamming weight at most on the most reliable positions (MRPs), with, where is the dimension of the ode. This algorithm is referred to as BMA( ) and the values outside the MRB as the ontrol band (CB). In order to redue the average omplexity, it is desired to know before the omputation of a andidate whether it an not be optimal or has a very low possibility to be optimal. This is the ase when it does not satisfy some deterministi neessary ondition (DNC) or probabilisti neessary ondition (PNC) for optimality. It is also desired to know prematurely that a andidate is optimal or has a high probability to be optimal. This is the ase when it satisfies some deterministi suffiient ondition (DSC) or probabilisti suffiient ondition (PSC) for optimality. DNCs and DSCs for MRB list deoding have been studied in [1][5][6][7][8][9] based on a priniple first introdued in [4]. A PNC was proposed in [10] and its effetiveness was shown via simulations. DNCs and the DSCs an redue the average omplexity without degrading the error performane. However, the existing ones are not very effiient for long odes at pratial signal to noise ratio (SNR) as based on the ode minimum distane. PNCs an redue the omputation omplexity at the prie of performane degradation, whih may be not negligible, but need to be properly analyzed [11]. To ahieve better performane, BMA( ) with larger order is desired. However, not only the number of proessed andidates but also the memory size inrease exponentially with. Hene it is desired to inrease the deoding apability of BMA( ) by onsidering an additional set of promising andidates, instead of resorting to BMA(! #"$%& ). In this paper, we first introdue a PSC whih greatly redues the average omplexity in all SNR regions. This PSC is espeially effiient for long odes. We also derive an upper bound for the false alarm probability assoiated with this PSC. Then we investigate a new method to effiiently improve the performane of BMA('( ), that an approah or even outperform BMA() *"+ ) without inreasing the memory size. We denote this type of algorithm as enhaned BMA(,- ). The PSC is used in the enhaned BMA(.- ) to redue the average omplexity. II. PRELIMINARY Let / be a binary $687 linear blok ode of length 1 and dimension with minimum Hamming distane 596 defined by its generator matrix :. Suppose BPSK signaling is used for transmission over an AWGN hannel with variane ;<>=. Assume that eah signal has unit energy. Let? = 7 be a odeword in /. This odeword is mapped onto a sequene of BPSK signals L = 0NM <M D> EHEGE MFIKJ!D 7, where MPRS0CT"U7CVXW. This ode sequene L is transmitted over the AWGN hannel. Let Y = 0AZ < XZ D FEGEHE Z IJ!D 7 be the reeived vetor. In this ase, the reliability of the hard deision [B0AZ 7]\_^a` assoiated with the

2 Œ œ š È ¹ Ê ¾ Ç Î Ò ½ Ä Ð É Í Æ ³ Ã Ô Ó Ñ Ï Ì «¼ Å» Â Ë ª º l è reeived signal Z \'b is simply proportional to its amplitude Z. A. Most Reliable Basis Finding the MRB is the first step to be performed in MRB reproessing type algorithms. The basi proedure is the following: 1) rder the reeived symbols based on their reliability values in dereasing order. This order of reeived symbols defines a permutation d D. 2) Permute the olumns of generator matrix : based on d4d, whih defines a permuted matrix :Te. Gaussian elimination is then performed to put :Te in redued ehelon form in order to determine the most reliable independent positions (MRIPs). A seond permutation df` may be neessary to make this redued ehelon form matrix into a matrix :-g in systemati form. The sequene Y is permuted aordingly to form the vetor h defined as follows: hijd 2k d 1k YUlHl E (1) Define hnm as the vetor orresponding to the MRB, and define hpo as the vetor orresponding to the least reliable q ptu q basis q ptu (LRB). q Hene msr msrv for w'xyzj{ z}, and o!r H~ o4r vx~ for w'xye z#{9e z+1. Let $eƒ k $em X eo be the hard deision of h k h m Xh o l. Let be the parity hek matrix defined by the MRB. form the identity matrix The most right 1_ olumns of and orrespond to the LRB. The left olumns orrespond to the MRB. In general, these olumns form a dense matrix, defined as ˆ. III. PRBABILISTIC SUFFICIENT CNDITIN (PSC) Let Õm Š Ž ž Ÿ Fig. 1. µ ± ² À Á Preliminaries be the error pattern orresponding to h within the MRB. Define vetor $e < e k em Ö Õm eo l. Define <# e <!Ø e Ù. It is readily seen that F< onsists of the linear ombination of the olumns orresponding to error positions within the LRB. The Hamming weight of < is the number of error bits within the LRB. A olumn of the dense matrix ˆ an be assumed to follow a binomial distribution and with high probability has a weight lose to 0A1'&.7X=. In a MRB-reproessing algorithm, a list ÚÜÛ2ÝÞÛ of MRB error patterns ÕK is used to produe andidates. We assume that the patterns ÕK are proessed in a predefined order and that a ost funtion has to be minimized. A pattern Õ9 whih minimizes this ost funtion over all patterns proessed before ÕK is referred to as a loal optimum of Ú Û2ÝÞÛ. For eah ÕK, the vetor e e k $em Ö Õ X eo l is formed, and the related syndrome ß $e e Ø Ù is omputed. It follows that v Ö < (2) vá.â W ã äuå æ â ç ã ä Hene is the summation of < and the olumns where Õ and Õ m are different. Sine olumns in the dense matrix ˆ are assumed to follow a binomial distribution, then with high probability U has a muh larger Hamming weight than < if ÕKêé ßÕm. In this paper, we introdue a new PSC for MRBreproessing algorithms as follows. Suppose a loal optimal andidate orresponding to the MRB error pattern Õ is found by a MRB-reproessing algorithm during reproessing. If the weight of is smaller than a threshold ë, this loal optimal andidate is delared to be optimal, and deoding an be terminated. The advantages of this method are : (1) a miss event does not lead to a deoding error; (2) the PSC an greatly redue the average omplexity; (3) the miss probability and false alarm probability an be derived based on the theory of order statistis. The fat that the syndrome orresponding to a vetor with no error in the MRB has in general relatively small Hamming weight was also used in [12][13]. A DSC used for reliabilitybased syndrome deoding and a PSC used for minimum weight syndrome deoding were derived in [12]. However both this DSC and this PSC still depend on the minimum Hamming distane of the ode. In [13], the syndrome weight of the original reeived vetor was ompared with a threshold before deoding to determine whether the MRB is error free or not. If the MRB was determined to be error free, then error positions in the LRB were flipped aording to the syndrome. therwise, a method based on the SD onept was used to find the error positions within the MRB by proessing a list of andidates determined by the olumns of the parity hek matrix. The effiieny of this deoding algorithm highly depends on the threshold as not only a false alarm event but also a miss event an lead to a deoding error. Furthermore, all the andidates in a sublist are proessed by that algorithm, and no performane analysis is provided. IV. PERFRMANCE ANALYSIS A PSC an effiiently redue the average omputation omplexity. There are two types of events, however, whih are not desirable when a PSC is used. ne is the miss event, and the other is the false alarm event.

3 ì õ è ý è To analyze these two events, let us first define the PSC heking list Ú whih ontains only the MRB error patterns orresponding to all the loal optimal andidates in ÚÜÛ2ÝÞÛ. Hene Ú is a random set, whih depends on the reeived. vetor. Define the number of MRB error patterns in Ú as Ú In the following analysis, we assume Õmí\îÚ Û2ÝCÛ, whih implies Õmï\ðÚ sine only the degradation due to the PSC is onsidered (see [11] to relate this estimate to the performane of a MRB-reproessing algorithm). Assume BMA(Üñ ) is used. All the events defined below are onditioned on a given SNR value. A. Miss Event Define the Hamming weight of a vetor? as ò]6 k? l. Then the miss event ó%ô is defined as ÕK3\'Ú öøò 6 k U l x ë (3) The miss event implies that although the optimal andidate is within the searh list, it an not be delared optimal when it is proessed. Furthermore, there exists no MRB error pattern Õ in Ú, Õ Õ é m, whih satisfies the PSC. As a result all the andidates in the list have to be proessed. Hene the miss event does not degrade the error performane. We provide a simple upperbound of the missing probability: Z9úUóûô-ü-x Z9ú ò]6 k 0 l ëtü (4) Reall that ò 6 k < l is the number of errors in the LRB, whih dereases as the SNR inreases. With a proper hoie of the threshold ë, Z9úUó ô ü an beome very small. Sine the miss event does not degrade the performane, we mainly fous on the false alarm event, whih degrades the error performane. B. False Alarm Event Define the set of MRB error patterns ý as follows ý}ú>õ9ƒö$õ9 \_Úû%Õ9]é ßÕm %ò 6 k l xëtü (5) The set ý ontains all MRB error patterns in Ú whih are not the true MRB error pattern but satisfy the PSC. Define the event óûþ>ÿ r D as Kt ý " and ò 6 k < l ë (6) For óûþ>ÿ r D, although the optimal andidate is in Ú, the related syndrome does not satisfy the PSC, and there exists at least one MRB error pattern Õ in Ú, Õ Õ é m, whih satisfies the PSC. Hene the related odeword is erroneously delared to be optimal. Define the set of MRB error patterns as follows +ú>õ9ƒö ÕK3\'ý-Õ9 is proessed before Õm ü (7) Define the event óþuÿ r ` as,t " and ò 6 k F< l x ë (8) For ó þuÿ r `, although the optimal andidate is in Ú and the related syndrome satisfies the PSC, there exists at least one MRB error pattern Õ in Ú, Õ Õ é m, whih satisfies the PSC and is proessed before Õ m. In this ase too, a deoding error ours although the optimal andidate is in Ú and satisfies the PSC. Define the false alarm event óþuÿ as óûþ>ÿ8ßóþuÿ r D óþuÿ r ` (9) Sine events ó þ>ÿ r D and ó þ>ÿ r ` are disjoint, it follows that Z9úUó þ>ÿ ü Z9úUó þ>ÿ r D>üê Z9úUó þuÿ r ` ü (10) Z$ú>óûþ>ÿ$ü is not easy to be evaluated beause the list of andidates onsidered by BMA(3ð ) is a random variable, whih is not as strutured as that of SD( ). However the list of andidates onsidered by BMA(Ü ) is a subset of that onsidered by SD( ), we an therefore set an upperbound of Z9úUó þ>ÿ ü for BMA( ' ) by deriving an upperbound of (10) for SD( ), whih is developed as follows. Define as the number of errors in the MRB. Using total probability and based on the assumption that SD( ) is used, (10) an be written as Z9úUó þuÿ ü v æ < From (6), we obtain Z$ú>ó þ>ÿ r D Z9ú>óûþ>ÿ r ` ø {.ü ï {.ü Ø Z9úUó þ>ÿ r D ø {,ü Kt Z9ú " and ò 6 k F< l ë From (8), we obtain Z9ú ï {.ü (11) ø{.ü (12) Z9úUóûþ>ÿ r ` ø {,ü Kt Z9ú " and ò 6 k F< l x ë ø{.ü (13) For a MRB error pattern Õ \'Ú%ûÕ Õ é m, define þ>ÿk0h{k7ü Z9úUò 6 k l i x ë ò]6 k < l ë% ï {.ü Z$úUò]6 k Ö < l x ë ò]6 k < l ë% ø {.ü (14) where is defined as vá.â W ã ä æ å â ç ã ä Define the binomial distribution as 021 ;ðf7ü ; 1 v (15) I 0Þ"]f7 J4I (16) The è olumn weight of v in ˆ an be well approximated by 0Aò]6 k v l 1ð4Xw.E $7 if 1_ is large enough. This was verified by simulation. Then ò]6 k l an also be approximated by 0Aò]6 k l 1ð 4w,E7 regardless of how many olumns in are involved in the summation of (15). Note that the weight of a olumn in an not be smaller than 5 6 i", where 5 6 is the minimum Hamming distane of the ode. In fat the weight of should range from 5 6 to 1 if olumns are involved in the summation of (15). In the following, these boundary effets are negleted as of minor influene and allowing olumns of weight smaller than 596# " even inreases the probability of a false alarm.

4 x x ý " v " " " x x x J L I K N D M H P From simulations, we observe that ò]6 k < l is also well approximated by a binomial distribution 02ò]6 k < l 1_447, where the parameter depends on the SNR, the number of errors in the MRB, and is onditioned on the event ú ò]6 k < l ëtü. However, to obtain the distribution of ò]6 k Ö < l, we do not need to know the value of as shown in the following. Let Ö < 02ò%DCò ` FEGEHEGCò]IKJ 7, where ò]r Ö <Fr. It follows that k l k l ò 6 k Ö < l k l k < l k < l Z9ú S"ü (17) Z9úU <Fr S"ü (18) Z9ú ˆR "ü (19) Hene, sine ò]6 is distributed as 0Aò]6 1+4Xw.E $7, has the same distribution as ò]6 if ò]6 is distributed as 0Aò]6 1!f7, regardless of the value of. Then based on (2) and (15), (14) an be simplified as þ>ÿ8 þ>ÿ0g{k7 Z9ú ò]6 k For order reproessing, define so that Ú Û2ÝCÛ 0G{K7 æ < l x ëtü (20) {s#w, " EHEGEHX (21) t Z9ú ò]6 k < l ë% ï {.ü xy"] 0C"] þ>ÿ7 áâ W o! #"$ S"] k "] þ>ÿ l&% o' ("$)H#* % JpD % o' ("$)H#* % (22) The inequality in (22) follows from the fat that we onsider all the MRB error patters in Ú Û2ÝCÛ instead of just those in Ú. Then (12) an be upperbounded by Z9úUóþUÿ r D ï {.ü % o ("$ )G+* % Ø Z9ú ò]6 k < l ë ø{.ü (23) Tt To alulate Z9ú ò 6 k F< l x ë%, {.ü, we use similar approximations. Note that when w, Õ m is proessed first and no other MRB error pattern an be proessed before Õ m. In this ase w. therwise, we still onsider the worst ase, in whih all the MRB error patterns proessed from phase(w ) to phase({ ) are in Ú. Following the analysis methods used above, we obtain Z9úUóþUÿ r ` ï {.ü % o' ("$-) v.* % Ø Z9ú ò]6 k < l xë ø {.ü (24) Then (11) an be upperbounded as Z9úUó þuÿ ü % o' ("$-)G+* % Ø Z9ú ò 6 k F< l ë ø#w,ü Ø Z9ú øwü % o' ("-)H#* % Ø Z9ú ò 6 k < l ë ø {.ü v æ D % o ("$ ) v.* % Ø Z$úUò]6 k < l xë ø {.ü Ø Z9ú ø {,ü9 (25) Z$ú/ ø{.ü is omputed from ordered statistis [14][15]. for example, we readily obtain 021 Z9úUóþUÿ 0N7 ü % o #"$ ) ` * % Ø Z9ú ò]6 k < l ë ø#w,ü Ø Z9ú øwü % o ("$ ) ` * % Ø Z9ú ò]6 k < l ë øs"ü % o ("$ ) D-* % Ø Z9ú ò]6 k < l x ë ø}"ü Ø Z9ú ø "ü % o #"$ ) ` * % Ø Z9ú ïü (26) It follows that (26) an be used as an upperbound of Z9úUó þ>ÿ ü for BMA("]& ). The error probability 43 of a MRB reproessing type algorithm-ý with PSC an be upper bounded by the union bound as where mso'5 3 mso'5 g Ûn Z9úUóûþ>ÿ$ü9 (27) is the probability of an MLD error and g Û is the probability that the transmitted odeword is not in the list onsidered by algorithm-ý. It is desired that Define 8>ô Z$ú>ó þ>ÿ ü76 m o!5 V. ENHANCED BMA gû (28) as the number of errors in the MRB, and define 8:9 as the number of errors in the ontrol band. Fig.2 depits the onept of BMA( ). ; < = E F G A B C Fig. 2. Conept of BMA(R$STVUXW ). In this setion, we develop the enhaned BMA(ƒ' ). To desribe the proedure learly, we assume Tø. We apply the PSC to all the simulations below. The threshold an be seleted based on (27) and (28) suh that a ontrolled error floor is allowed. In the following simulations, we selet the threshold suh that no false alarm event is observed.

5 µ º ¹ ^ ] ` _ b a y f w q k v p j u oi t nh s mg r l e x Ÿ ž œ š d ƒ} Œ «Ž ± à  Á Å Ä É È Z Ñ Í Ï Ð Î Ì è ` ` Fig. 3. ~ z { ˆ Š ª Y ² ³ ½ ¾ À» ¼ [ \ Conept of ÒÔÓÖÕØ (Ù/Ú&ÛÝÜXÞ:ß with enhaned mathing. A. BMA(âáäãÝåçæ ) with Enhaned Mathing è EBMA('áéãêåëæ )ì 1) Algorithm: Figure 3 depits an enhaned íxîðïòñ$âáäãóå æ'ô è$õ íxîðïxñâáäã7åöæ'ô ì. Define the region of the first ã2å æ positions in the ordered sequene ø: as í. If ãúåûæ remains small, with high probability, í is error free, espeially for long odes. We first find the MRB of ø: without inluding any position in í. Then we onstrut a vetor üêý by plaing this MRB in the first æ positions of ü ý, moving region í to positions from æòþûÿ to ã of ü ý (as the ontrol band), and plaing the remaining positions of ø: in the positions from ã þ ÿ to of ü ý. Denote the orresponding generator matrix as ý and the hard deision of ü ý as ý. It follows that with high probability, the ontrol band of ü ý is error free (i.e ). Based on ü, õ íxîðïxñ'áéã å æ'ô is performed in two stages. Define as the number of errors in the MRB, and define as the number of errors in the ontrol band. In stage 1, íxîðïòñ$âáäãóå æ'ô is performed to orret all the error patterns with or. In this stage, all the MRB error patterns with Hamming weight 2 are stored in the orresponding boxes following the proedures desribed in [3]. These boxes are aessed in stage 2, where an enhaned mathing is used to orret most of the error patterns with or. Define ÿ and "! as two parameters used for enhaned mathing, where #$%& æ å(' and )*! æëå+. Define, % and,! as the position index of the MRB error pattern that has been stored in a box in stage 1. The enhaned mathing onsists of two steps. Define -.%/á.-/! á0-21 as the error positions of a MRB error pattern with Hamming weight 3. In the first step, we generate the set ï of all the MRB error patterns with Hamming weight 3 suh Ç Æ Ë Ê that %43 - %53 -! æ. For eah ï, there may be a box : initialized in stage 1, suh that for any MRB error pattern 7; stored in :, the positions from æ to ã of ý=<> ñ?7 1@< 7;êôBA ý.c are all zero. We denote this type of mathing as 0-mathing. We then uniquely proess eah of the MRB error pattern 7 1D< 7;. It is readily observed that if E and F, then ü is orretable provided that,hg ã1dij%, where,hg ã1 is the position of the third MRB error of üêý. Define -0%/á0-/!:á.-/1 á0-2k as the error positions of a MRB error pattern with Hamming weight 4. In the seond step, we generate all the MRB error patterns with Hamming weight 4 suh thatl! 3-0% 3 -/! 3 -/1 3-2K 3 æ. We perform the same and 0-mathing as desribed above. It follows that if M N, then ü ý is orretable provided,hg ã 1 I$!. Note that in stage 2 of the enhaned mathing, we try to approah the deoding apability of BMA(3) with muh less andidates proessed than for BMA(3). Furthermore, the memory used for the enhaned mathing is the same as that of BMA(2). It is readily seen that both the deoding apability and the omputation omplexity of the seond stage depend on % and!. The smaller % and! are, the better the performane is. However, the omputation omplexity of stage 2 inreases rapidly as % and! derease. These values are seleted so that the omplexity of stage 2 remains lose to that of stage 1. 2) Computation Complexity: Define the maximum number of boxes visited in stage 1 as ý. %. Define the maximum number of boxes visited in the first step and seond step of stage 2 as ý.!p% and ý.!0!, respetively. Define the maximum number of boxes visited in EBMA(âáäã åûæ ) as ý RSRUTWVYX[Z\!P ]_^a`b. It follows that ý. RUTWVYX[Z\!d ]^a`b ý. % þ ý.!p% þ ý.!! á (29) where ý. % % ìøþ è! ì, ý!p% è `^feg 1 ì, and ý!! è `^feph K ì. As expeted, ý.!p% and ý.!! inrease exponentially with ævåi % and æxåj!. 3) Performane Analysis: In the following analysis, we use definitions losely following those of [14], [15]. Define kl ñnmpo á0mq% ádrsr\r#á.mat^u%äô as the hard deision of the reeived vetor ø, where maus ðÿ if vulw and mau[ otherwise. The reliability of this hard deision is taken as xsuy zy v{u_y. In the reeived sequene ø, assume transmission errors have ourred and the orresponding reliability values are reordered in dereasing order. For ÿw~}, let ñ? ô represent the } -th ordered reliability value among hard deision errors in a reeived sequene of length, so that % ñ? ôy ƒ! ñn ôy r\rsr H ñn ô. The remaining å5 reliability values orresponding to the orret hard deisions are also reordered in dereasing order. For ÿ åj, let ˆñ å ô represent the -th ordered reliability value among the remaining åš orret hard deisions in the reeived sequene of length. It follows that H%:ñ å ô5! ñ å ôn r\rsr t^ ñ å ô. The density funtions of ñn ô and ˆ ñ å ô have been expressed in [14] and allow to evaluate probabilities of the form ëñn ñ? ô ˆ ñ å+ ôô.

6 ó ó ó q t Define the event ó&œž as g Û g ó@œ2 %+ú region ontains at least one errorü$ (30) as ÛÜ+ú optimal andidate is not in the list onsidered by EBMA(, )ü$ (31) and the event ó and the event ó@ as ó@ }ú EBMA( & ) failsü (32) Based on the union bound and total probability, we obtain Z$ú>ó@ ütx mso'5 It follows that (see Fig.3) Z9úUó Z9úUó Z$ú>ó g ÛP ó Œ2 üê Z9ú>ó g ÛP l ó ŒŽ ü (33) D ü gû ó@œ2 Uü Z9ú/8>ô tw 8:9 "ü (34) gû l ó@œ2 Uü Z9ú/8>ô t 8:9Üwü Z9ú 8Uô+,8 9)ßw.öp T Z9ú/8>ôy, 8:9Üw,ûH T$ $`ü9 (35) From (34)-(35), the union bound in (33) an be omputed from the joint ordered statistis of!e B. Biased-MRB-EBMA()&4š ) and ne v [14], [15]. 1) Algorithm: For the seleted values D and `, the biasing method of hapter 4 an be used to further improve performane while the omputation omplexity inreases linearly with the number of biasing iterations. œ Figure 4 depits the onept of biased MRB ó ý 0N,4,7, whih basially onsists of two steps. In the first step, œ ý 0N,7 defined in Figure 3 is performed. In the seond step, we bias the MRB of h and repeat stage 2 of œ ý 0N],7 iteratively as follows. Define the bias as the following binary random variable: ž Ÿšf }"B= (36) šf }"B= where š is a positive real value. Define Œ as the systemati parity hek matrix generated Y in step 1 of biased ó ý0 ]ñ,7. The first olumns of #Œ defines the MRB of huœ. The last 1i# olumns of #Œ represent the identity matrix. Define = 02ò < Xò D FEGEGEHXò] J!D 7 with ò. The elements of are permuted aording to the reliability values ò, whih defines a permutation d D. The first olumns of the matrix #Œ are then permuted aording to d D, whih defines a new matrix e Œ. Note that the permuted first olumns of e Œ still defines the MRB. The orresponding generator matrix : eœ an be diretly obtained from e Œ without Gaussian elimination. Using the same permutation as that defined by e Œ and Œ, we obtain hneœ from h Œ. There is the same number of errors in the MRB of h Œ and hre Œ, but in different positions due to permutation. The error pattern ú/8 ô 8 9 w H z$ ` ü Y or ú/8 ô 8 9 w p z KDBü an not be orreted by ó ý0n,s},7. However, after permutation, it is possible that p of hneœ is hanged suh Fig. 4. Conept of ÒW /Û. ª Ü Ó4«4ÒÖÜY ÔÒØÓÖÕÔ #Ù/Ú&ÛÝÜXÞ Ú? ß. that üêý beomes orretable. Sine we only bias the MRB, we all this algorithm í -/ âã åöî± úí å õ íxîðïòñ$âáäã å æ á0 âô è or íxî å õ íxîðïxñâáäã å æ á0 âô ì. 2) Performane Analysis: The bias amplitude and the number of iterations determine the deoding apability of BM- EBMA(âáéã2åûæ á ). We derive a lower bound Sˆ\²0³[ T[VYR of BM-EBMA(âáäã å æ á0 ) by assuming that all the error bloks with ± or ±0 ± are orretable. This is the best performane that BM-EBMA(âáéã å æ á0 ) an ahieve with any and a large enough number of iterations. It follows that ˆ\²0³[ T[V ^arurutwvyx[z\!d ]^a`b DvL µ'0 ÿ þ DvL Ļ (37) As examples of EBMA('áéã åúæ ) and BM-EBMA('áéã åúæ á0 ), we onsider the binary image of the (255,239) Reed Solomon (RS) ode so that ¹" and æ6 ÿ{º!ÿ/. Let ã7åæ», % æxå and! æòåöÿ{ ". The maximum number of boxes visited by BMA(âáä ) is ý. % F¼ æ ÿ ½ þ¾¼ æ "½J!.o{À Á.o (38)

7 Â Ä ó 3 3 and the maximum number of boxes visited in the seond stage of EBMA( ) is 1UŒ r ` D1UŒ r `` w KD ` <{Å 0 ` DÅ ÆC< w ` à `` Å ` (39) We observe that 1 Œ r ` D%1 Œ r `X` Ä 1 Œ r D, so that the omplexity of EBMA( ) remains of the same order as that of BMA( ) Ç note that 1UŒ r D Ä <då D ` for BMA(3,22)È. In Fig.5, we plot (33) for EBMA( ) and (37) for BM-EBMA($š ). We observe that the performane of EBMA(, ) is between that of BMA( ) and BMA( ), while the lower bound of BM-EBMA(,,0š ) an approah that of BMA( $ ). 0 log 10 (WER) of BMA (2,22) of BMA (3,22) Upperbound of EBMA(2,22), k J 1 =100 P low of BM EBMA(2,22) Simulated EBMA(2,22), k J 1 =100 Simulated BM EBMA(2,22,0.15), k J 1 =100, 20 it SNR 5 Fig. 6. Simulated error performane of EBMA(É/SËÉÉ ) and BM- EBMA(É/SËÉÉ/SŽÌÍÏÎ.Ð ) for RS(255,239). log 10 (WER) of BMA(2,22) of BMA(3,22) P upper of EBMA(2,22), k J 1 =100 P low of BM EBMA(2,22) TSB MLD Upperbound SNR Fig. 5. Error performane analysis of EBMA(É/S2ÉÉ ) and BM-EBMA(É/S2ÉÉ/SŽÊ ) for RS (255,239). In Fig.6, we plot the orresponding simulation results and (33) for EBMA(, ). We also plot the simulation results of BM-EBMA($w,EG"/ ) with 20 iterations. We observe that the upperbound of EBMA(, ) is tight. Simulation results show that EBMA($ ) beomes muh better than BMA(2,22) with enhaned mathing. The performane of BM-EBMA(w,EG"/ ) with 20 iterations approahes the lower bound of BM-EBMA(,0š ), and is very lose to the performane of BMA(3). Note that not only the number of andidates onsidered by BM-EBMA(,$Xw.EH" ) with 20 iterations is muh smaller than that of BMA(3), but the memory used by BM-EBMA(,,Xw.EH" ) is just a small fration of that used by BMA(3). C. Biased-Blok-EBMA(,Ü&!0š ) 1) Algorithm: In this setion, enhaned mathing is performed with the biasing method. At eah iteration, all the positions of h Œ are biased exept the CB. The biased symbols of h Œ outside the CB are reordered in dereasing reliability values, whih defines a permutation d D. The olumns of : Œ are permuted based on d D whih defines a permuted matrix :TeŒ. Gaussian elimination is then performed to put :TeŒ in redued ehelon form in order to determine the MRIPs. A seond permutation df` may be neessary to make this redued ehelon form matrix into a matrix :-g in systemati form. The sequene h Œ is permuted aordingly to form the vetor hneœ defined as follows: h eœ ßd 2k d 1k huœ lhl E (40) Then œ ý 0N$7 with 0-mathing and enhaned 0-mathing an be performed on úu:-g>xhneœ Xd D Cdf`Bü. We denote this algorithm as œ ò Y /š/8u5 }[? BMFñ}ó ý 0N, S!0š7 Ç ý0n,%&4š7 È. Note that in both BM-EBMA(,y 4š ) and BB- EBMA(3_4š ), eah iteration is independent of the others so that it an be performed in parallel, offering a tradeoff with respet to lateny. The potential improvement brought by eah iteration depends on the magnitude š of, KD and `. 2) Performane Analysis: We derive the list error probability g Û r Ñ[Ñ JaÒUÒSÑam ) ` r g J r ÿ * of BB-EBMA(,s}40š ) with the biasing magnitude š, assuming enough iterations are performed. BB-EBMA(T40š ) ontains two stages. In the first stage, BMA(,,8 ) is performed on. In the seond stage, BMA(2) with 0-mathing and enhaned 0-mathing are performed on the biased sequene hne iteratively. Define ó ÑSÑ3D as ó ÑSÑ3D Sú optimal andidate is not in the list onsidered by Stage 1 of BB-EBMA0N,%&4š7ü (41) and ó Ñ[Ñ ` as ó@ñsñ ` Sú optimal andidate is not in the list onsidered by Stage 2 of BB-EBMA0N,%&4š7ü (42) It follows that gû r ÑSÑÜJpÒSÒUуm ) ` r g J r ÿ * Z$ú>ó Ñ[уD and ó Ñ[Ñ `ü (43)

8 3 3 Ý Ý ó ó ` ó Define ÑSÑ3DD and Ñ[уD as ÑSÑ3DXD +úuó ÑSÑ3D and ó&œž Bü (44) ó Ñ[уD `SúUó Ñ[Ñ3D and ó@œ2 Uü9 (45) where ó@œ2 was defined in (30). It follows from total probability that g Û r ) ` r g J r ÿ * ó ` ó Ñ[Ñ JaÒUÒUуm Z9úUó ÑSÑ3DXD and Ñ[Ñ `Bü Z9úUó Ñ[Ñ3D and Ñ[Ñ `ü (46) Assume { transmission errors have ourred in. It is readily derived that Z9úUó Ñ[уD `¾š91p5 ó Ñ[Ñ `Bü Z9ú{ H $0H{K7 w J `021''{K7 and HÓ$0H{K7 w g JpÔ 021_ð{K7 and g J 0A1'ð{K7êw D 0H{K7 and HÕ0G{K7ƒ+š j,g JaÖ 0A1'ð{K7n šü9 (47) where ú 0G{K7 J ` 0A1ñ{97 and Ó 0G{K7 g JHÔK0A1ñ&{K7 ü defines the event that BMA( ) is in error, ú{ g J 021Üê{K7ê 4DB0G{K7ü defines the event that first p most reliable positions are error free and ú Õ 0G{K7 jšñ± g JpÖ$0A1i {973šü indiates that at eah iteration, the first positions always ontain more than 7 errors when the bias amplitude is š. As a result, this error blok an not be orreted by BB-EBMA( 40š ). When region ontains errors, enhaned 0-mathing in stage 2 always fails. Hene an error blok is orretable only if BMA(2) of stage 1 sueeds, or BMA(2) with 0-mathing of stage 2 sueeds. Define ó ÑÜr Û as the event that there are Ø errors in the region, "8xµØ)x#]. It follows that Z9úUó Ñ[уDD and ó Ñ[Ñ `ü g J Z$ú>ó Ñ[уD and ó ÑÜr Û and ó Ñ[Ñ `ü Û æ D Z9úUó Ñ[уD and ó ÑÜr D and ó ÑSÑ `ü Z9úœ H $0H{K7êw J `$021 ð{k7 and HÓ0G{K7êj,g JHÔ 0A1_ð{97 and D 0H{K7êj,g J 0A1'ð{97êµ `$0H{K7 and Ô 0H{K7ƒ(š,g J $021 ð{k7n š ü (48) In (48), we use the fat that óÿñür D is the dominant event when region ontains errors, sine is the most reliable region with small width %&. ú D 0G{K7êj,g J,0A1 _{97êµ `$0H{K7ü is the dominant event that the region ontains errors; ú Ô 0G{K7nÙš w g J 021 {K7! (šü indiates that BMA(2) with 0-mathing fails at eah iteration. Fig.7 depits g Û r Ñ[Ñ JaÒUуm ) ` r g J r ÿ * for the binary image of RS(255,239), with š} w,e w,xw.eh"$xw,eg"/, respetively, and 9ˆß. We observe that the larger the bias amplitude is, the smaller the list error probability is. However, the onvergene of the iterative approah with a large bias is slower than that with a smaller bias [18]. In Fig.8, we plot the simulation results of BB-EBMA(,! 4Xw.EH" ) with 100 iterations, for the binary image of RS(255,239), with Ú D w$w, ~ 9`ï "Uww. For log 10 (WER) TSB MLD Upperbound of BMA(3,22) of BB EBMA(2,22,0.05) of BB EBMA(2,22,0.1) of BB EBMA(2,22,0.15) SNR Fig. 7. List error probability of BB-EBMA(É/STÝUòW SËÊ ) for RS(255,239). omparison, we plot the simulation result of the BIAS(2Ø.Œ )- IISR(2Ø )-BMA(-ß4R0š ), with biasing iteration number 2Ø.Œ w, iterative information set redution (IISR)[17] iteration number 2Ø, ontrol band length ˆy, IISR shift width " w, and biasing amplitude š w.eh" [18]. This algorithm ahieves the nearest MLD performane reported so far. We also plot the simulation results of the ADP(2Ø Ý.ÛUÛ? /Ü X2ØÞHII /Ü ) ombined with a hard deision deoder (HDD), with the maximum outer iteration number ŽØ Ý0ÛUÛ? /Ü w, inner iteration number 2ØÞHII /Ü w and the damping oeffiient Þiw,E w [19]. This algorithm is the most effiient reported soft deoding different from diret MRB approahes. We observe that after 10 iterations, BB-EBMA(4s!Xw,EG" ) outperforms BMA(3), BIAS(20)-IISR(3)-BMA(2,22,10,0.1) and ADP(80, 50). After 100 iterations, we observe many MLD errors as reorded in Table I. To effiiently redue the average omputation omplexity, we used the PSC of Setion III with the threshold ë w. The average omputation omplexity of BIAS(20)-IISR(3)-BMA(2,22,10,0.1) and BB-EBMA(,]4w,EG" ) are reorded in Table II, where the same PSC threshold is used for both algorithms. Sine MRB reproessing algorithms are list deoding algorithms, the omplexities are defined as the maximum and average numbers of andidates (or list sizes) per reeived word Y proessed by the algorithm at a given SNR value. We observe from Fig.8 and Table II that the average omputation omplexity of BIAS(20)-IISR(3)-BMA(2,22,10,0.1) is lose to that of BB- EBMA( 4w,EG" ) with 50 iterations. However the performane of BB-EBMA( 4w,EG" ) with 10 iterations is already better than that of BIAS(20)-IISR(3)-BMA(2,22,10,0.1). Note that both algorithms use the same size of memory. Furthermore, BIAS(20)-IISR(3)-BMA(2,22,10,0.1) needs 80 Gaussian eliminations, while BB-EBMA(,3 4w,EG" ) with ŽØ iterations needs 2Ø Gaussian eliminations. We also onduted simulations with enhaned BMA for the

9 ó deoding of binary image of (460,420) Reed Solomon ode defined on the field :T^ˆ0 DÞ< 7. The onept of enhaned BMA with order " is depited in Figure 9 in a similar manner as that in Figure 3. In Figure 10, we plot the simulation results of œ Y ó ý0þ"$$xw.eh" 7 with 10 iterations and œ X ý 0Þ",Xw,EG"/$7 with 15 iterations, with µ D " w$w, nœ ` w$w and the threshold ë# " w$w. We observe that after 10 iterations, the performane of Y Y ó ý0þ"$$xw.eh" 7 an approah that of œ ò œ ý0 7, and the performane of ó ý 0Þ"$,Xw.EH" 7 with 15 iterations has already been better than that of œ ý 0N,$7. Note that the memory used by the an enhaned Y that of œ ý0þ"$ê&,7 algorithm is the same as ý 0Þ"3ð,7, whih is muh smaller than that used by œ ý 0N,],7. REFERENCES [1] M. Fossorier and S. Lin, Soft Deision Deoding of Linear Blok Codes Based on rdered Statistis, IEEE Trans. Inform. Theory., vol. 41, pp , Sept [2] D. Gazelle and J. Snyders, Reliability-Based Code Searh Algorithms for Maximum Likelihood Deoding of Blok Codes, IEEE Trans. Inform. Theory, vol. 43, pp , Jan [3] A. Valembois and M. Fossorier, Box and Math Tehniques Applied to Soft Deision Deoding, IEEE Trans. Inform. Theory., vol. 50, pp , May [4] D. Taipale and M. Pursley, A Improvement to Generalized-minimumdistane Deoding, IEEE Trans. Inform. Theory., vol. 37, pp , Jan [5] A. Valembois and M. Fossorier, A Comparison Between Most Reliable Basis Reproessing Strategies, IEICE Trans. on Fundam., vol. E85-A, pp , July [6] T. Kaneko, T. Nishijima, H. Inazumi, and S. Hirasawa, An Effiient Maximum Likelihood Deoding of Linear Blok Codes with Algebrai Deoder, IEEE Trans. Inform. Theory., vol. 40, pp , Marh [7] M. Fossorier, T. Koumoto, T. Takata, T. Kasami, and S. Lin, The Least Stringent Suffiient Condition on the ptimality of a Suboptimally Deoded Codeword Using the Most Reliable Basis, The Pro IEEE ISIT, pp. 430, Ulm, Germany, June 1997 [8] T Kasami, n Integer Programming Problems Related to Soft Deision Deoding Algorithms, Applied Algebra, Algebri Algorithms and Error-Correting Codes., vol. 1719, pp , Springer-Verlag LNCS, Nov [9] T Kasami, Y. Tang, T. Koumoto, and T. Fujiwara, Suffiient Conditions for Ruling ut Useless Iterative Steps In a Class of Iterative Deoding Algorithms, IEICE Trans. on Fundam, vol. E82-A, pp , t [10] Y. Wu, R. Koetter and C. Hadjiostis, Soft-Deision Deoding of Linear Blok Codes Using Preproessing and Diversifiation, Submitted to IEEE Trans. Inform. Theory, 2005 [11] M. Fossorier, Average and Maximum Computational Complexities for Information Set Deoding of Blok Codes, The Pro Third Asia-Europe Workshop on Coding and Information Theory, Kamogawa, Japan, June [12] M. Fossorier, S. Lin and J. Snyders, Reliability-Based Syndrome Deoding of Linear Blok Codes, IEEE Trans. on Info. Theory, vol. IT-44, pp , Jan [13] A. Ahmed, R. Koetter, and N. R. Shanbhag, Performane Analysis of the Adaptive Parity Chek Matrix Based Soft-Deision Deoding Algorithm, The Pro Asilomar Conf., Paifi Grove, USA, Nov [14] D. Agrawal and A. Vardy. Generalized Minimum Distane Deoding in Eulidean-Spae:Performane Analysis, IEEE Trans. on Inform. Theory, vol. 46, pp , Jan.2000 [15] M. Fossorier and S. Lin, Error Performane Analysis for Reliability- Based Deoding Algorithms, IEEE Trans. on Inform. Theory, vol. 48, pp , Jan.2002 [16] A. Papoulis Probability, Random Variables, and Stohasti Proesses, 3rd ed New York:MGraw-Hill,1991 [17] M. Fossorier, Reliability-Based Soft-Deision Deoding with Iterative Information Set Redution, IEEE Trans. on Info. Theory, vol. 48, pp , De [18] W. Jin and M. Fossorier, Reliability-Based Soft-Deision Deoding with Multiple Biases, IEEE Trans. Inform. Theory., to appear. [19] J. Jiang and K. R. Narayanan, Iterative Soft Input Soft utput Deoding of Reed-Solomon Codes by Adapting the Parity Chek Matrix, IEEE Trans. Inform. Theory., pp , Aug log 10 (WER) TSB MLD Upperbound ADP(80,50) & HDD, α=0.08 of BMA(2,22) BIAS(20) IISR(3) BMA(2,22,10,0,1) of BMA(3,22) BB EBMA(2,22,0.1), k J 1 =100, 10it. BB EBMA(2,22,0.1), k J 1 =100, 50it. BB EBMA(2,22,0.1), k J 1 =100, 100it. MLD lower bound SNR Fig. 8. BB-EBMA(É/STêUòW SŽÌÍÏÎ ) for the deoding of RS(255,239). TABLE I PERCENTAGE F MLD ERRRS TABLE II AVERAGE CMPUTATIN CMPLEXITY

10 & $ # æ å " î í 3. ) 2 - ( 1, ' ä ã % = ì é ð ï òñ èç < 6 9 ô ó õ á þ ý ü â ÿ ö C? A > ß ø ú û! êë 4 / * : ; 8 5 Fig. 9. Conept of DFEHGJIËÎS&TÝUXWLK with enhaned mathing log 10 (WER) of SD(1) of BMA(1) of BMA(2) Simulated SD(1) SNR Simulated BMA(1) Simulated EBMA(1,22) Simulated BM EBMA(1,22,0.15), k J 1 =1700, k J 2 =200, 10 it. Simulated BB EBMA(1,22,0.15), k J 1 =1700, k J 2 =200, 15 it Fig. 10. Simulation results for the deoding of INMPÌ/SMPÉÌPK Reed Solomon ode over RTSUI ÉWVYXPK with enhaned DJEHGFI?ÎZK.