CONVENTIONAL communication systems employ coding

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL On Decodng Bnary Perfect and Quas-Perfect Codes Over Markov Nose Cannels Hader Al-Lawat and Fady Alajaj, Senor Member, IEEE Abstract We study te decodng problem wen a bnary lnear perfect or quas-perfect code s transmtted over a bnary cannel wt addtve Markov nose. After examnng te propertes of te cannel block transton dstrbuton, we derve suffcent condtons under wc strct maxmum-lkelood decodng s equvalent to strct mnmum Hammng dstance decodng wen te code s perfect. Addtonally, we sow a near equvalence relatonsp between strct maxmum lkelood and strct mnmum dstance decodng for quas-perfect codes for a range of cannel parameters and te code s mnmum dstance. As a result, an mproved (complete) mnmum dstance decoder s proposed and smulatons llustratng ts benefts are provded. Index Terms Bnary cannels wt memory, Markov nose, maxmum lkelood decodng, mnmum Hammng dstance decodng, lnear block codes, perfect and quas-perfect codes. I. INTRODUCTION CONVENTIONAL communcaton systems employ codng scemes tat are desgned for memoryless cannels. However, snce most real world cannels ave memory, nterleavng s used n an attempt to spread te cannel nose n a unform fason over te set of receved words so tat te cannel appears memoryless to te decoder. Ts n fact adds more complexty and delay to te system, wle falng to explot te benefts of te cannel memory. Progress as been aceved on te statstcal and nformaton teoretc modelng of cannels wt memory (e.g., see [2], [7], [11], [12]), as well as on te desgn of effectve teratve decoders for suc cannels (e.g., see [3], [4], [8], [9]). However, lttle s known about te structure of optmal maxmum lkelood (ML) decoders for suc cannels. We eren focus on one of te smplest models for a cannel wt memory, te bnary cannel wt addtve Markov nose and analyze te performance of bnary perfect and quas-perfect codes, wc can be useful for complexty and delay constraned applcatons suc as wreless sensor networks. Snce t s well known tat ML decodng of bnary codes over te memoryless bnary symmetrc cannel (wt bt error rate less tan 1/2) s equvalent to mnmum Hammng dstance decodng, t s natural to nvestgate weter a relaton exsts between tese two decodng metods for te Markov nose cannel. For suc Paper approved by M. Skoglund, te Edtor for Source/Cannel Codng of te IEEE Communcatons Socety. Manuscrpt receved August 24, 2007; revsed Marc 31, Ts work was supported n part by te Natural Scences and Engneerng Researc Councl (NSERC) of Canada and te Premer s Researc Excellence Award (PREA) of Ontaro. Parts of ts paper were presented at te Tent Canadan Worksop on Informaton Teory, Edmonton, Alberta, June Te autors are wt te Department of Matematcs and Statstcs, Queen s Unversty, Kngston, Ontaro K7L 3N6, Canada (E-mal: {ader, fady}@mast.queensu.ca). a cannel (wt memory), one would expect tat ML decodng s not equvalent to mnmum dstance decodng for general codes; owever, for certan codes wt good (coset) propertes (suc as perfect and quas-perfect codes), some equvalency may be establsed. Indeed, we provde a partal answer to ts problem by sowng (after elucdatng some propertes of te Markov cannel dstrbuton) tat te strct ML decodng of a bnary lnear perfect code can be equvalent to ts strct mnmum dstance decodng wle te strct ML decodng of a quas-perfect code can be nearly equvalent to ts strct mnmum dstance decodng. As a result, we propose a (complete) decoder wc s an mproved verson of te mnmum dstance decoder, and we llustrate ts performance va smulaton results. In a related work [5], te optmalty of te bnary perfect Hammng codes and te near-optmalty of subcodes of Hammng codes are demonstrated for te same Markov nose cannel. II. SYSTEM DEFINITION AND PROPERTIES We consder a bnary addtve nose cannel wose output symbol Y k at tme k s descrbed by Y k = k Z k, k = 1,2,, were denotes addton modulo-2, k {0,1} s te kt nput symbol and Z k {0,1} s te t nose symbol. We assume tat te nput and nose processes are ndependent of eac oter, and tat te nose process {Z k } s a statonary (frst-order) Markov source wt transton probablty matrx gven by Q = [Q j] =» ε + (1 ε)(1 p) (1 ε)p (1 ε)(1 p) ε + (1 ε)p, (1) were Q j Pr(Z k = j Z k 1 = ),,j {0,1}. Here p = Pr(Z k = 1) s te cannel bt error rate (CBER), and ε Cov(Z k,z k 1 )/V ar(z k ) = [Pr(Z k = 1,Z k 1 = 1) p 2 ]/[p(1 p)] s te correlaton coeffcent of te nose process, were Cov(Z k,z k 1 ) E[Z k Z k 1 ] E[Z k ]E[Z k 1 ] s te covarance of Z k and Z k 1 and V ar(z k ) E[Z 2 k ] E[Z k] 2 s te varance of Z k. We assume tat 0 < p < 1/2 and tat 0 ε < 1, ensurng tat te nose process s rreducble. Wen ε = 0, te nose process becomes ndependent and dentcally dstrbuted (..d.) and te resultng cannel reduces to te (memoryless) bnary symmetrc cannel wt crossover probablty or CBER p (wc we denote by BSC(p)). Note tat ts (memory-one) Markov nose cannel s a specal case of te Glbert-Ellott cannel [7] (realzed wen te probablty for causng an error equals zero n te good state and one n te bad state ). For x n = (x 1,,x n ) {0,1} n and y n = (y 1,,y n ) {0,1} n, te cannel block transton probablty Pr(Y n =

2 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL y n n = x n ) can be expressed n terms of te cannel nose block dstrbuton as follows Pr(Y n = y n n = x n ) = Pr(Z n = z n ) ny = L [z k 1 ε + (1 ε)p] z k k=2 [(1 z k 1 )ε + (1 ε)(1 p)] 1 z k were z k = x k y k, k = 1,,n and L = Pr(Z 1 = z 1 ) = p z1 (1 p) 1 z1. Gven z n = (z 1,,z n ) {0,1} n, let t j (z n ) denote te number of tmes two consecutve bts n z n are equal to (,j), were,j {0,1};.e., t 00(z n ) = (1 z k )(1 z k+1 ), t 11(z n ) = z k z k+1, t 10(z n ) = z k (1 z k+1 ), t 01(z n ) = (1 z k )z k+1. In terms of te t j (z n ) s Pr(Z n = z n ) can be wrtten as Pr(Z n = z n ) = L [ε + (1 ε)(1 p)] t 00 [(1 ε)p] t 01 [(1 ε)(1 p)] t 10 [ε + (1 ε)p] t 11. (2) But from te defnton of te t j (z n ) s, we ave te followng. t 10(z n ) = n 1 w(z n ) t 00(z n ) + z 1 (3) t 01(z n ) = w(z n ) z 1 t 11(z n ), (4) were w(z n ) = n z k s te Hammng wegt of z n. Substtutng (3) and (4) nto (2) yelds te followng expresson for te nose block dstrbuton, wc wll be nstrumental n our analyss:» Pr(Z n = z n ) = (1 ε) () (1 p) n p 1 p» ε + (1 ε)(1 p) (1 ε)(1 p) w(z n ) t00 (z n )» ε + (1 ε)p (1 ε)p t11 (z n ).(5) Te propertes of t 00 (z n ) and t 11 (z n ) n terms of only n and w(z n ) are as follows. 1) If w(z n ) = 0, ten t 00 (z n ) = n 1 and t 11 (z n ) = 0. 2) If 0 < w(z n ) = l n 1, ten t 00 (z n ) n l 1 wt equalty f and only f all te 0 s n z n occur consecutvely. Also t 11 (z n ) l 1 wt equalty f and only f all te 1 s n z n occur consecutvely. 3) If 0 < w(z n ) = l n 2, ten t 00(z n ) max{n 2l 1,0} and t 11 (z n ) 0. 4) If n 2 < w(zn ) = l n 1, ten t 00 (z n ) 0 and t 11 (z n ) 2l n 1. 5) If w(z n ) = n, ten t 11 (z n ) = n 1 and t 00 (z n ) = 0. Wen tere s no possblty for confuson, we wll wrte t 00 (z n ) and t 11 (z n ) as t 00 and t 11, respectvely. We also assume trougout tat te blocklengt n 2. III. ANALYSIS OF THE NOISE BLOCK DISTRIBUTION Lemma 1: Let 0 n be te all-zero word (of lengt n) and let z n 0 n be any non-zero bnary word. Ten Pr(Z n = z n ) < Pr(Z n = 0 n ). Proof: Usng (2), we ave Pr(Z n = z n ) = L [ε + (1 ε)(1 p)] t 00 [(1 ε)p] t 01 [(1 ε)(1 p)] t 10 [ε + (1 ε)p] t 11 < (1 p) [ε + (1 ε)(1 p)] t 00 [ε + (1 ε)(1 p)] t 01 [ε + (1 ε)(1 p)] t 10 [ε + (1 ε)(1 p)] t 11 = (1 p) [ε + (1 ε)(1 p)] t 00+t 01 +t 10 +t 11 = (1 p) [ε + (1 ε)(1 p)] = Pr(Z n = 0 n ), were te strct nequalty olds snce L = p < 1 p f z 1 = 1, and snce p < 1 p wt t 01 > 0 (snce z n 0 n ) f z 1 = 0. Lemma 2: Let z n 1 0 n be a non-zero nose word wt Hammng wegt w(z n 1) < n, t 00 = n w(z n 1) 1 and t 11 = w(z n 1) 1 (.e., z n 1 s of te form ( ) or ( ) ). Let z n 2 be anoter non-zero nose word wt w(z n 2 ) = w(z n 1 ) but wt dfferent t 00 and/or t 11. Ten, f ε > 0, Pr(Z n = z n 1) > Pr(Z n = z n 2). Proof: From (5), we note tat Pr(Z n = z n ) strctly ncreases wt bot t 00 and t 11 wen te wegt s kept constant and ε > 0. Snce z n 1 as maxmum values for bot t 00 and t 11 amongst all nose words of wegt w(z n 1) (but wt dfferent t 00 and/or t 11 ), te strct nequalty above follows. Note tat wen ε = 0, obvously all nose words wt te same wegt ave dentcal dstrbutons (snce te cannel reduces to te BSC(p)). and Lemma 3: Suppose tat ln u < u ln ε+ ε+ 0 < ε < 1 2p 2(1 p). 1 p p ε+(1 ε)p (1 ε)p 1 Let z n be a nose word of wegt w(z n ) = m suc tat 0 m u + 1 n 2. Ten Pr(Zn = z n ) > Pr(Z n = z n ) were z n s any nose word wt wegt w( z n ) = l > m. Proof: Frst, note tat te result drectly olds f m = 0 by Lemma 1. Now let z n be a nose word of nonzero wegt m u+1, and let z n be anoter nose word wt w( z n ) > m.

3 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL Case 1: Assume tat w( z n ) = m + were {1,2,...,n m 1}. Ten by (5), we ave» Pr(Z n = z n m ) ε + (1 ε)(1 p) Pr(Z n = z n ) (1 ε)(1 p)» m+ 1 «ε + (1 ε)p p (1 ε)p 1 p» m 1 ε + (1 ε)(1 p) (1 ε)(1 p)» m «ε + (1 ε)p p f(m). (1 ε)p 1 p Te frst nequalty follows from (5) and by applyng te bounds on t 00 and t 11 descrbed at te end of te prevous secton, wle te second nequalty follows by notng tat te rgt and sde of te frst nequalty decreases n for a fxed m. Snce f(m) s strctly ncreasng n m (wen ε > 0), and m u + 1 < u + 1, we obtan tat f(m) < f(u + 1) = 1 Pr(Zn = z n ) Pr(Z n = z n ) < 1. Case 2: Assume tat w( z n ) = n. Let ẑ n be anoter nose word wt w(ẑ n ) = n 1, t 11 (ẑ n ) = n 2 and t 00 (ẑ n ) = 0. Ten Pr(Z n = z n ) Pr(Z n = z n ) = Pr(Zn = ẑ n ) Pr(Z n = z n ) Pr(Z n = z n ) Pr(Z n = ẑ n ) < Pr(Zn = z n ) Pr(Z n = ẑ n )» «ε + (1 ε)p p = (1 ε)p 1 p» ε + (1 ε)p = < 1 (1 ε)(1 p) were te frst strct nequalty olds snce Pr(Z n = ẑ n ) < Pr(Z n = z n ) by Case 1, and te last strct nequalty olds snce ε < 1 2p 2(1 p). Remark: Note tat te wegt lmt u n te above lemma s ndependent of te codeword lengt. IV. DECODING PERFECT AND QUASI-PERFECT CODES We next study te relatonsp between strct maxmum lkelood (SML) decodng and strct mnmum (Hammng) dstance decodng for bnary lnear perfect and quas-perfect codes sent over te addtve Markov nose cannel. SML decodng s an (ncomplete) optmal decoder were optmalty s n te sense of mnmzng te probablty of codeword error (PCE) wen te codewords are equally lkely (wc we eren assume). Let F n 2 = {0,1} n denote te set of all bnary words of lengt n. A non-empty subset C of F n 2 s called a bnary lnear code f t s a subgroup of F n 2. Te elements of C are called codewords. We usually descrbe C wt te trplet (n,m,d) to ndcate tat n s te blocklengt of ts codewords, M s ts sze and d s ts mnmum Hammng dstance; n oter words, d mn c1,c 2 C:c 1 c 2 d(c 1,c 2 ) were d(c 1,c 2 ) = w(c 1 c 2 ) s te Hammng dstance between c 1 and c 2 and te modulo-2 operaton s appled component-wse on c 1 and c 2. Defnton 1: [10], [6] An (n,m,d) lnear code C s sad to be a perfect code f, for some non-negatve nteger t, t as all patterns (.e., elements of {0,1} n ) of Hammng wegt t or less and no oters as coset leaders. Defnton 2: [10], [6] An (n,m,d) bnary lnear code C s sad to be quas-perfect f, for some non-negatve nteger t, t as all patterns of wegt t or less, some of wegt t + 1, and none of greater wegt as coset leaders. An equvalent defnton for quas-perfectness s tat, for some non-negatve nteger t, C as a packng radus equal to t and a coverng radus equal to t + 1;.e., te speres wt (Hammng) radus t around te codewords of C are dsjont, and te speres wt radus t+1 around te codewords cover {0,1} n. On te oter and, perfectness means tat bot packng and coverng rad are equal. For tese two classes of codes, t = d 1 2 (wt d = 2t + 1 for perfect codes and d = 2t + 1 or d = 2t + 2 for quas-perfect codes). Te (2 m 1,2 2m 1 m,3) Hammng codes (m 2), te (n,2,n) repetton code wt n odd and te (23,2 12,7) Golay code are te only members of te famly of bnary perfect lnear codes. Examples of quas-perfect bnary lnear codes nclude te (n,2,n) repetton codes wt n even, te (2 m,2 2m 1 m,4) extended Hammng codes as well as te (2 m 2,2 2m 2 m,3) sortened Hammng codes (m 2), te (2 m 1,2 2m 1 2m,5) double-error correctng BCH codes (m 3), and te (24,2 12,8) extended Golay code. Perfect codes as well as quas-perfect codes are not powerful error-correctng codes due to ter small Hammng dstances. However, tey can be useful n complexty and delay constraned applcatons were codes wt sort blocklengts are needed. Suppose tat a codeword of a quas-perfect code C s transmtted over te Markov nose cannel and tat y n s receved at te decoder. Te followng are possble decodng rules one can use to recover te transmtted codeword. ML Decodng: y n s decoded nto codeword c 0 C f Pr(Y n = y n n = c 0 ) Pr(Y n = y n n = c) for all c C. If tere s more tan one codeword for wc te above condton olds, ten te decoder pcks one of suc codewords at random. Strct ML (SML) Decodng: It s dentcal to te ML rule wt te excepton of replacng te nequalty wt a strct nequalty; f no codeword c 0 satsfes te strct nequalty, te decoder declares a decodng falure. Mnmum Dstance (MD) Decodng: y n s decoded nto codeword c 0 C f w(c 0 y) w(c y) for all c C. If tere s more tan one codeword for wc te above condton olds, ten te decoder pcks one of suc codewords at random. Strct Mnmum Dstance (SMD) Decodng: It s dentcal to te MD rule wt te excepton of replacng te nequalty wt a strct nequalty; f no codeword c 0 satsfes te strct nequalty, te decoder declares a decodng falure. 1 1 Recall tat te ML and MD decoders are complete decoders.e., tey always select a codeword to decode te receved word wle te SML and SMD decoders are ncomplete decoders as tey declare a decodng falure wen tere are more tan one codeword wt mnmal decodng metrc.

4 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL Lemma 4: Let C be an (n,m,d) perfect code to be used over te Markov nose cannel. Assume tat d 1 ln ε+ 1 p p < 2 ln and ε+ 0 < ε < 1 2p 2(1 p). ε+(1 ε)p (1 ε)p Ten SMD and SML decodng are equvalent. Proof: Frst note tat for perfect codes, te element wtn eac coset of mnmum wegt (.e., te coset leader) s unque. Also notce tat te coset leader s of wegt less tan or equal to (d 1)/2 n/2. Assume tat y n s receved; ten ĉ C wc s unque suc tat w(ĉ y n ) < w(c y n ) c C\{ĉ}. Usng Lemma 3 wt u = (d 1)/2 1, we conclude tat c C\{ĉ} Pr(Z n = ĉ y n ) > Pr(Z n = c y n ) Pr(Y n = y n n = ĉ) > Pr(Y n = y n n = c). Hence, gven a receved word y n, te codeword wt te smallest Hammng dstance to y n wll be te most lkely codeword tat was sent over te cannel amongst all te codewords n C. Terefore, SMD and SML decodng are equvalent. Observatons: Te above lemma also proves tat for perfect codes MD and ML decodng are equvalent under te same assumptons on d,ε and p. Ts s because for suc codes SMD and MD are te same due to te unqueness of ter coset leaders wc results n no tes n te MD decoder. Smlarly, te unqueness of coset leaders coupled wt te proof of te above lemma also mply tat SML and ML are equvalent for te perfect codes under te range of cannel parameters gven n te lemma. In a related work [5], Hamada sowed tat for te Markov cannel wt a non-negatve nose correlaton coeffcent (.e., ε 0) and bt error rate p < 1/2, te bnary perfect Hammng codes (of mnmum dstance 3) are optmal (under ML decodng) n te sense of mnmzng te probablty of decodng error amongst all codes avng te same blocklengt and rate provded tat ε < (1 2p)/2(1 p). Tus, n lgt of te above lemma, for a communcaton system employng codes wt sort blocklengt due to delay constrants, Hammng codes used wt MD decodng wll be optmal over te Markov nose cannel amongst all codes of te same blocklengt and rate. Lemma 5: Let C be an (n,m,d) bnary lnear quas-perfect code to be used over te Markov nose cannel. Assume tat d 1 ln ε+ 1 p p < 1 2 ln and ε+ 0 < ε < 1 2p 2(1 p). ε+(1 ε)p (1 ε)p Ten, for a gven word y n receved at te cannel output, te followng old. (a) If ĉ C suc tat w(ĉ y n ) < w(c y n ) c C\{ĉ}, ten Pr(Y n = y n n = ĉ) > Pr(Y n = y n n = c) c C\{ĉ}. (b) If ĉ C suc tat Pr(Y n = y n n = ĉ) > Pr(Y n = y n n = c) c C\{ĉ}, ten w(ĉ y n ) w(c y n ) c C. Proof: (a) Let ĉ C suc tat w(ĉ y n ) < w(c y n ) c C\{ĉ}. Obvously, ĉ y n s a coset leader, tus w(ĉ y n ) d n 2 snce C s quas-perfect. By Lemma 3, Pr(Z n = ĉ y n ) > Pr(Z n = c y n ) c C Pr(Y n = y n n = ĉ) > Pr(Y n = y n n = c) c C\{ĉ}. (b) Let ĉ C suc tat Pr(Y n = y n n = ĉ) > Pr(Y n = y n n = c) c C\{ĉ}. Assume tat c C\{ĉ} suc tat w( c y n ) < w(ĉ y n ); te exstence of c s always guaranteed by coosng t suc tat c y n s te coset leader of C y n. Tus, we can assume tat w( c y n ) n 2 snce te coset leader as wegt less tan or equal to n 2 (as C s quasperfect). Ten by Lemma 3, Pr(Z n = ĉ y n ) < Pr(Z n = c y n ) Pr(Y n = y n n = ĉ) < Pr(Y n = y n n = c) wc contradcts our assumpton tat ĉ maxmzes Pr(y n c) over all codewords. Hence, w(ĉ y n ) w(c y n ) c C. Note te above lemma mples tat f a quas-perfect code as no decodng falures n ts SMD decoder, ten ts SMD and SML decoders are equvalent under te stated condtons on te Markov cannel parameters (p, ǫ) and te code s mnmum dstance. 2 Inspred by te above result, Lemma 2 and (5), we next propose te followng complete decoder tat mproves over MD decodng. It ncludes SMD decodng and explots te knowledge of t 00 and t 11 to resolve tes (wc occur wen tere are more tan one codeword tat are closest to te receved word). MD+ Decodng: Assume tat y n s receved at te cannel output. Suppose te decoder outputs te codeword c satsfyng te MD decodng condton. If tere s more tan one suc codeword, ten te decoder cooses c tat maxmzes t 00 ( c y n )+t 11 ( c y n ). If tere s stll a te, ten te decoder cooses c tat maxmzes t 11 ( c y n ). Fnally, f tere s stll a te, ten te codeword c s pcked at random. 3 V. SIMULATION RESULTS AND DISCUSSION Gven an (n,m,d) perfect (respectvely, quas-perfect) code and a fxed CBER p, we let ε t 1 (respectvely, ε t ) be te largest ε for wc Lemma 4 (respectvely, Lemma 5) olds, were t (d 1)/2. In Table I, we provde te values of ε t for t = 1,2,3 and dfferent values of p. 2 In contrast, recall tat for te BSC(p) wt p < 1/2, SML and SMD decodng are equvalent for all bnary codes (te same equvalence also olds between ML and MD decodng). j Note k also tat wen ε 0, te condtons n te above lemma reduce to d 1 <, and p < 1 (wc s consstent 2 2 wt wat was just mentoned). 3 Clearly, MD+ and MD decodng are equvalent for te BSC, snce for ts cannel, t does not matter wat codeword te decoder selects wen tere s a te (as long as t s one of te codewords closest to te receved word).

5 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL p ε 0 ε 1 ε 2 ε / / / / / TABLE I VALUES OF ε t FOR DIFFERENT p AND t. LEMMA 4 HOLDS FOR ALL ε ε t 1 AND LEMMA 5 HOLDS FOR ALL ε ε t. PCE MD decodng MD+ decodng ML decodng Cannel Bt Error Rate (p) Fg. 3. PCE vs CBER p under dfferent decodng scemes for te BCH (15, 2 7, 5) code over te Markov cannel wt nose correlaton ǫ = PCE Cannel Bt Error Rate (p) MD (epslon=) ML (epslon=) MD (epslon=0.5) ML (epslon=0.5) Fg. 1. PCE vs CBER p under dfferent decodng scemes for te Hammng (15, 2 11, 3) code over te Markov cannel wt nose correlaton ε =, 0.5. PCE Cannel Bt Error Rate (p) MD decodng MD+ decodng ML decodng Fg. 2. PCE vs CBER p under dfferent decodng scemes for te Hammng (8, 2 4, 4) code over te Markov cannel wt nose correlaton ǫ = We frst examne te perfect (15,2 11,3) Hammng code under dfferent cannel condtons, and sow tat ndeed MD decodng and ML decodng are equvalent for te cannel condtons specfed by Lemma 4, as llustrated n Table I. A large sequence of a unformly dstrbuted bnary..d. source was generated, encoded va one of tese codes and sent over te Markov cannel. Typcal values are sown for ε {, 0.5} n Fg. 1. Note tat ε = satsfes te condtons of Lemma 4 wle ε = 0.5 does not. Te smulaton results sow tat MD and ML are dentcal for te case ε = and almost dentcal at ε = 0.5. We next present smulaton results for decodng two quasperfect codes, te bnary (8,2 4,4) extended Hammng code and te (15,2 7,5) BCH code. For te Hammng code, t = 1; tus te values for ε 1 n Table I provde te largest values of ε for wc Lemma 5 olds for dfferent CBERs p. As a result, we smulated te Hammng system for te 5 values of p lsted n Table I and ε {0.05,,0.2,0.25}. Smlarly, snce t = 2 for te BCH code, te values for ε 2 apply, and te BCH system was smulated for ε = 0.05 and all values of p n Table I except p = A typcal Hammng code smulaton result s presented n Fg. 2 for ε = 0.25, and te BCH code smulaton s sown n Fg. 3 for ε = Te results ndcate tat MD+ performs nearly dentcally to ML decodng and provdes sgnfcant gan over MD decodng. By comparng te two fgures, we also note tat te performance gap between MD and ML decodng decreases wt ε (wc s consstent wt te fact tat MD and ML decodng are equvalent wen ε = 0). Addtonal results are avalable n [1]. As ts work s a basc frst step towards understandng te structure of ML decoders for cannels wt memory, tere are several drectons for future work. For example, note tat one lmtaton of Lemmas 4 and 5 s tat ter condtons are too strngent to accommodate codes wt larger mnmum dstance, unless f te cannel correlaton ε s substantally decreased towards 0, tus renderng te Markov cannel nearly

6 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 4, APRIL memoryless 4 (e.g, see ow ε t decreases as t ncreases n Table I). Te determnaton of less strngent condtons s an nterestng topc for future work. Anoter possble future drecton s to desgn a decoder tat explots te memory between blocks by usng estmates of te prevous nose samples. Ts can result n an mproved performance over te block-by-block ML and MD+ metods (studed ere) at a cost of ncreased complexty. Fnally, extendng ts work to cannels wt Mt order Markovan nose [12] or dden Markovan nose [7], wc are good models for correlated fadng cannels, may be a wortwle endeavor. REFERENCES [1] H. Al-Lawat, Performance Analyss of Lnear Block Codes Over te Queue-Based Cannel, M.Sc. Tess, Matematcs and Engneerng, Department of Matematcs and Statstcs, Queen s Unversty, Kngston, Ontaro K7L 3N6, Canada, Aug Avalable [Onlne] at ttp:// web/publcatons.tml/ [2] F. Alajaj and T. Fuja, A communcaton cannel modeled on contagon, IEEE Trans. Inform. Teory, pp , Nov [3] A. Ecford, F. Kscscang and S. Pasupaty, Analyss of low-densty party-ceck codes for te Glbert-Ellott cannel, IEEE Trans. Inform. Teory, vol. 51, pp , Nov [4] J. Garca-Fras, Decodng of low-densty party-ceck codes over fnte-state bnary Markov cannels, IEEE Trans. Commun., vol. 52, pp , Nov [5] M. Hamada, Near-optmalty of subcodes of Hammng codes on te two-state Markovan addtve cannel, IEICE Trans. Fundamentals Electroncs, Commun. and Comp. Scences, vol. E84-A, pp , Oct [6] F. J. MacWllams and N. J.A. Sloane, Te Teory of Error-Correctng Codes, Nort-Holland, Amsterdam, [7] M. Muskn and I. Bar-Davd, Capacty and codng for te Glbert- Ellott cannel, IEEE Trans. Inform. Teory, vol. 35, pp , Nov [8] V. Nagarajan and O. Mlenkovc, Performance analyss of structured LDPC over te Polya-urn cannel wt fnte memory, n Proc. CCECE, May [9] C. Ncola, F. Alajaj, and T. Lnder, Decodng LDPC codes over bnary cannels wt addtve Markov nose, n Proc. CWIT 05, pp , Montreal, June [10] W. W. Peterson and E. J. Weldon Jr., Error-Correctng Codes, 2nd Ed., MIT, [11] C. Pmentel and I. F. Blake, Modelng burst cannels usng parttoned Frtcman s Markov models, IEEE Trans. Ve. Tecnol., pp , Aug [12] L. Zong, F. Alajaj and G. Takaara, A model for correlated Rcan fadng cannels based on a fnte queue, IEEE Trans. Ve. Tecnol., vol. 57, no. 1, pp , Jan Note tat as ε decreases towards 0, te cannel nose becomes less and less bursty, beavng more and more lke a memoryless process.