Soft decision decoding of Reed-Muller codes: recursive lists

Transcription

1 1 Soft decision decodin of Reed-Mulle codes: ecusive lists Ilya Due, Senio Mebe IEEE and Kiill Sabunov, Mebe IEEE Abstact Recusive list decodin is consideed fo Reed-Mulle (RM) codes. Te aloit epeatedly eleates itself to te sote RM codes by ecalculatin te posteio pobabilities of tei sybols. Inteediate decodins ae only pefoed wen tese ecalculations eac te tivial RM codes. In tun, te updated lists of ost plausible codewods ae used in subsequent decodins. Te aloit is fute ipoved by usin peutation tecniques on code positions and by eliinatin te ost eopone infoation bits. Siulation esults sow tat fo all RM codes of lent 56 and any subcodes of lent 51, tese aloits appoac axiu-likeliood (ML) pefoance witin a ain of 0.1 db. As a esult, we pesent tit expeiental bounds on ML pefoance fo tese codes. Index tes Maxiu-likeliood pefoance, Plotkin constuction, posteio pobabilities, ecusive lists, Reed-Mulle codes. I. INTRODUCTION Te ain oal of tis pape is to desin feasible eocoectin aloits tat appoac ML decodin on te odeate lents anin fo 100 to 1000 bits. Te poble is pactically ipotant due to te void left on tese lents by te best aloits known to date. In paticula, exact ML decodin as ue decodin coplexity even on te blocks of 100 bits. On te ote and, cuently known iteative (essaepassin) aloits ave been efficient only on te blocks of tousands of bits. To acieve nea-ml pefoance wit odeate coplexity, we wis to use ecusive tecniques tat epeatedly split an oiinal code into te sote ones. Fo tis eason, we conside Reed-Mulle (RM) codes, wic epesent te ost notable exaple of ecusive constuctions known to date. Tese codes - denoted below ª - ave lent n = and Hain distance d =. Tey also adit a siple ecusive stuctue basedonteplotkin constuction (u, u + v), wic splits te oiinal RM code into te two sote codes of lent 1. Tis stuctue was efficiently used in ecusive decodin aloits of []-[4], wic deive te coupted sybols of te sote codes u and v fo te eceived sybols. Tese ecalculations ae ten epeated until te pocess eaces te epetition codes o full spaces, weeupon new infoation sybols can be etieved by any poweful aloit - say, ML decodin. As a esult, ecusive aloits acieve bounded distance decodin wit a low coplexity ode of n in{, },wic ipoves upon te coplexity of ajoity decodin [1]. Ilya Due is wit te Collee of Enineein, Univesity of Califonia, Riveside, CA 951; e-ail: due@ee.uc.edu. Kiill Sabunov is wit te XVD Copoation, San Jose, CA, 95134; e-ail: ksabunov@xvdcop.co. Tis eseac was suppoted by NSF ant CCR We also ention two list decodin aloits of [5] and [6], wic substantially educe te eo ates at te expense of a ie coplexity. In bot aloits, RM codes ae epesented as te enealized concatenated codes, wic ae epeatedly decoposed into te sote blocks siilaly to te Plotkin constuction. In all inteediate steps, te aloit of [5] ties to estiate te Euclidean distance to te eceived vecto and ten etieves te codewods wit te sallest estiates. To do so, te aloit cooses soe nube L of codewods fo bot constituent codes u and v. Ten te poduct list is constucted fo te oiinal code. Tese lists ae ecusively eevaluated and updated in ultiple uns. Te second tecnique [6] is based on a novel sequential scee tat uses bot te ain stack and te copleentay one. Te idea ee is to lowe-bound te iniu distance between te eceived vecto and te best code candidates tat will be obtained in te futue steps. Tis look-aead appoac ives low eo ates and educes te decodin coplexity of [5]. Recently, new ecusive aloits wee consideed in [8] and [9]. In paticula, fo lon RM codes of fixed code ate R, ecusive decodin of [8] coects ost eo pattens of weit (d ln d)/ instead of te foe tesold of d/. Tis is done witout any incease in decodin coplexity. Howeve, te new decodin tesold is still infeio to tat of a uc oe poweful ML decodin. In te sequel, we advance te aloit of [8], also applyin list decodin tecniques. Tis appoac ostly follows [9] and diffes fo te pio esults in a few ipotant aspects. Fist, we use exact posteio pobabilities in ou ecusive ecalculations instead of te distance appoxiations eployed befoe. Tis allows us to desin a tee-like ecusive aloit tat can bette sot out all plausible candidates in inteediate steps and avoid ultiple decodin uns. Second, we sall see tat te output eo ate sinificantly vaies fo te diffeent infoation sybols deived in te ecusive pocess. Teefoe, we also conside subcodes of RM codes obtained by eovin te least potected infoation bits. Finally, decodin will be ipoved by applyin a few peutations on code positions. As a esult, we closely appoac te pefoance of ML decodin on te lents 56 and 51, wic was beyond te eac of te foe tecniques. Te ateial is oanized as follows. In Section, we biefly suaize soe ecusive popeties of RM codes and tei decodin pocedues. In Section 3, we descibe ou list decodin aloit Ψ (L). Finally, in Section 4 we discuss te ipoveents obtained by eliinatin te least potected infoation bits and usin peutation tecniques.

2 II. RECURSIVE ENCODING AND DECODING FOR RM CODES A. Encodin Te followin desciption is detailed in [10]. Let any codewod c of RM code ª be epesented in te fo u, u + v wee u ª n o 1 and v 1 1. We say tat c is split onto two pats u and v. By splittin bot pats, we obtain fou pats tat lead to RM codes of lent, and so on. In eac step i of ou splittin, we assin te pat value ξ i =0to a new v-coponent and ξ i =1to a new u-coponent. All pats end at te epetition codes ª n o 0 o full spaces, wee =1,...,, =1,...,. Tus, we can conside a specific binay pat ξ def =(ξ 1,..., ξ ) tat leads fo te oiin ª to soe left-end code ª n o 0.Fo any it-end node, tesaepocessivesasubpatξ of lent : ξ def =(ξ 1,..., ξ ). A siila decoposition can be pefoed on te block a of k infoation bits tat encode te oiinal vecto c. In tis way, any left-end pat ξ ives only one infoation bit associated wit its end node ª 0. Any it-end pat o ives infoation bits associated wit te end code. We can also n add an abitay binay suffix of lent to te it-end pats, and obtain a one-to-one appin between te extended pats ξ and k infoation bits a(ξ). B. Basic decodin wit posteio pobabilities Let any binay sybol a be apped onto ( 1) a.tenany codewod of RM code belons to {1, 1} n and as te fo c =(u, uv). Tis codewod is tansitted ove a eoyless cannel Z. Te eceived block x consists of te two alves x 0 and x 00, wic ae te coupted iaes of vectos u and uv. Te decode fist takes te sybols x 0 i and x00 i fo any position i =1,..., n/, and finds te posteio pobabilities of tansitted sybols u i and u i v i : q 0 i def =P{u i =1 x 0 i}, qi 00 def =P{u i v i =1 x 00 i }. To siplify ou notation, below we use te associated quantities yi 0 def =qi 0 1, yi 00 =qi (1) Note tat yi 0 is te diffeence between te two posteio pobabilities qi 0 and 1 q0 i of 1 and 1 in position i of te left alf. Siilaly, yi 00 is obtained on te it alf. Te followin basic ecusive aloit is descibed in [8] and Section IV of [10] in oe detail. Step 1. Let qi v =P{v i =1 x 0 i,x00 i } be te posteio pobability of any sybol v i of te codewod v. We find te coespondin quantity yi v =qv i 1, wic is (see foula (18) in [10]) yi v = yiy 0 i 00. () Sybols yi v fo te vecto y v of lent n/. Ten we use soe soft-decision o decode Ψ v (y v ) tat ives a vecto ˆv and its infoation block â v. n 1 1 Step. Nowweassuetat Step1ivescoect vecto ˆv = v. Let qi u =P{u i =1 x 0 i,x00 i } be te posteio pobability of a sybol u i. Ten te coespondin quantity y u i =qu i 1 is (see foula (19) in [10]) y u i =(y 0 i +ŷ i )/(1 + y 0 iŷ i ), (3) wee ŷ i = yi 00ˆv i. Te sybols yi u fo te vecto yu of lent n/. We use soe (soft decision) ª decodin aloit Ψ u (y u ) and its infoation block â u. to obtain a vecto û 1 In a oe eneal scee Ψ,vectosy v and y u ae not decoded but used as ou new inputs y. Tese inputs ae ecalculated ultiple ties accodin to () and (3). Finally, we eac o. Hee we pefo axiu- te end nodes ª n 0 and likeliood (ML) decodin as follows. At any node { }, ou input is a newly ecalculated vecto y of lent wit te iven diffeences y i between posteio pobabilities of two sybols c i = ±1. Rewitin definition (1), we assin te posteio pobability P(c i y i )=(1+c i y i )/ to a sybol c i = ±1. In tis way, we can find te posteio pobability P (c y) = Y i=1 (1 + c i y i )/ (4) of any codewod c { }, and coose te ost pobable codewod ĉ,wee c { } : P (ĉ y) P (c y). (5) Te decoded codewod ĉ { } and te coespondin infoation block â ae now obtained as follows (ee opeations () and (3) ae pefoed on vectos coponentwise). Aloit Ψ fo an input vecto y. 1. If 0 <<, execute te followin Calculate vecto y v = y 0 y 00. Decode y v into vecto ˆv =Ψ 1 1 (yv ). Pass ˆv and â v to Step Calculate vecto y u =(y 0 + ŷ)/(1 + y 0 ŷ). Decode y u into vecto û =Ψ 1 (y u ). Output decoded coponents: â := (â v â u ); ĉ := (û ûˆv).. If =0, use ML-decodin (5) fo 0ª. 3. If =, use ML-decodin (5) fo ª. Note tat tis aloit Ψ diffes fo te siplified aloit Φ of [10] in tee aspects. Fistly, we use exact ecalculations (3) instead of te foe siplification y u =(y 0 + ŷ)/. (6)

3 3 Secondly, we use ML decodin instead of te iniu distance decodin tat cooses ĉ wit te axiu inne poduct: c :(ĉ, y) (c,y). Tidly, we eploy a diffeent ule and stop at te epetition codes 0ª instead of te biotoonal codes used in [10]. Tis last cane will ake it easie to use te list decodin descibed in te followin section. Finally, note tat ecalculations () equie n/ opeations, wile ecalculations (3) can be done in 5n/ opeations. Teefoe ou decodin coplexity satisfies ecusion 1 Ψ + +3n. Ψ 1 Ψ 1 Siilaly to [10], tis ecusion ives decodin coplexity Ψ 6n in(, )+n. Tus, coplexity Ψ as axiu ode of 3n lo n, wic is twice te coplexity Φ of te aloit Φ of [10]. III. LIST DECODING To enance aloit Ψ, we sall use soe lists of L = p o fewe codewods obtained on any pat ξ. Tis aloit called Ψ (L) inceases te nube of opeations at ost L ties and as te oveall coplexity ode of Ln lo n. Given any intee paaete A, we say tat te list ave size A, if decodin outputs eite all available ecods o A ecods, wiceve is less. Tis aloit pefos as follows. At any step s =1,..., k of te aloit Ψ (L), ou input consists of L ecods A =(ā, ρ(ā), y(ā)). Eac ecod is foed by soe infoation block ā, its cost function ρ(ā), and te coespondin input y(ā), wic is updated in te decodin pocess Tese tee enties ae defined below. Decodin stats at te oot node ª. Hee we set s =0 and take one ecod ā =, ρ(ā) =1, y(ā) =y, (7) wee y is te input vecto. Decodin takes te fist pat (denoted ξ =1)to te leftost code ª 0 and ecalculates vecto y(ā) siilaly to te aloit Ψ. Howeve, now we take bot values a 1 =0, 1 of te fist infoation sybol and conside bot codewods 1 d and 1 d of lent d = in te epetition code c(a 1 ). Te posteio pobabilities (4) of tese two vectos will also define te cost function of te new infoation block ā = a 1 : Y ρ(ā)= i=1 1+c i (a 1 )y i (ā) In ou list decodin, we epesent te two outcoes ā as te initial edes apped wit tei cost functions P (ā). Ten we poceed to te next code ª 1 0, wic coesponds to te subsequent pat denoted ξ =. Given two diffeent decodin esults v = c(a 1 ), ou ecusion (), (3) ives two diffeent vectos y(ā) aivin at tis node. Teefoe, decodin is pefoed two ties and ives te full tee of dept. Moe eneally, at any step s, decodin is executed as follows. Suppose tat te fist s 1 pats ae aleady pocessed. Tis ives L infoation blocks ā =(a 1,...,a s 1 ) of lent s 1 and te coespondin ecods A. Eac vecto y(ā) is ten ecalculated on te new pat ξ = s usin foulas () and (3), in te sae way it was done in Ψ. Let tis pat end on soe left-end code ª 0. Decodin of te new infoation sybol a s =0, 1 yields L extended blocks ā := ā,a s of dept s, aked by tei cost functions ρ(ā) :=ρ(ā) Y i=1 1+c i (a s )y i (ā). (8) Step s is copleted by coosin L blocks wit te iest cost functions ρ(ā). Te decodin on te it-end nodes is siila. Te only diffeence is tat te full spaces ª include codewods defined by infoation blocks a s of lent a s =. In tis case, we can coose te two ost pobable vectos c(a s ) (in essence, akin bit-by-bit decisions) and set = in ou cost calculations (8). Anote - oe efined vesion of te aloit - cooses fou diffeent vectos of te code ª weneve. Te best ecod is cosen at te last node ª. Moe eneally, te aloit is executed as follows. Aloit Ψ (L). Input: L ecods A =(ā, ρ(ā), y(ā)), counte s =0. 1. If 0 <<, fo all vectos y(ā) : 1.1. Set y(ā) :=y 0 (ā)y 00 (ā). Pefo decodin Ψ 1 1 (y(ā)). Pass L new ecods A to Step Set y(ā) := y0 (ā)+ŷ(ā) 1+y 0 (ā)ŷ(ā). Pefo decodin Ψ 1 (y(ā)). Output L new ecods A.. If =0, take bot values a s =0, 1. Calculate costs (8) fo eac (ā,a s ). Coose L best blocks ā := (ā,a s ). Set s := s +1and etun L ecods A. 3. If =, coose 4 best blocks a s. Calculate costs (8) fo eac (ā,a s ). Coose L best blocks ā := (ā,a s ). Set s := s + a s and etun L ecods A.

4 WER 4 Discussion. Conside te above aloit on te AWGN cannel N (0, σ ). Usin te esults of [10], it can be poven tat on tis cannel, te v-coponent is decoded on te cannel wit te new noise powe k = 9, L( Δ) = 16 k = 64, L( Δ) = 16 k = 99, L( Δ) = 8 σ v ax{σ, σ 4 }. Te fist appoxiation is tit fo vey sall σ (tou te cannel is no lone Gaussian), wile te second one pefos well on te bad cannels wit σ À 1. Tus, te noise powe always inceases in te v-diection; te oe so te wose te oiinal cannel is. By contast, te u-cannel can be appoxiated by te salle powe σ /. Tese obsevations also sow tat te fist infoation sybol - wic is obtained on te binay pat 0 - is potected te least, and ten te decodin adually ipoves on te subsequent pats. Now we see tat te aloit Ψ (L) wit te list of size L = p delays ou decision on any infoation sybol by p steps, akin tis decision bette potected. In te paticula case of a bad cannel, it can be veified tat te fist sybol a 1 is now decoded wen te noise powe is educed p ties. Fo tis eason, tis list decodin substantially educes te output wod eo ates (WER) even fo sall size L. Fo L = +1, te aloit Ψ (L) pocesses all te codewods of te fist biotoonal code ª +1 1 and is siila to te aloit Φ of [10]. On te ote and, aloit Ψ (L) updates all L cost functions, wile Φ cooses one codewod on eac end node. Teefoe Ψ (L) can be consideed as a enealization of Φ tat continuously updates decodin lists in all inteediate steps. Te esult is a oe poweful decodin tat coes alon wit a ie coplexity. Siulation esults. Below we pesent ou siulation esults fo te AWGN cannels. Hee we also counted all te instances, wen fo a iven output te decoded vecto was oe pobable tan te tansitted one. Obviously, tese specific eventsalso epesent te eos of ML decodin. Tus, te faction of tese events ives a lowe bound on te ML decodin eo pobability. Tis lowe bound is also depicted in te subsequent fiues fo all te codes tested. Ou siulation esults sow tat fo all RM codes of lents 18 and 56, te aloit Ψ (L) apidly appoaces ML pefoance as te list size L ows. Fo RM codes of lent 18 and distance d > 4, we suaize tese esults in Fi. 1. Fo eac RM code, we pesent tit lowe bounds fo te eo pobability of ML decodin. To easue te efficiency of te aloit Ψ (L), we also exibit te actual list size L( ) at wic Ψ (L) appoaces te optial ML decodin witin a sall ain of =0.5 db. Tis pefoance loss is easued at te output wod eo ate (WER) P =10 4 ; oweve, we found little to no diffeence fo all ote WER tested in ou siulation. In Table 1, we copleent tese lists sizes L( ) wit te two ote elevant paaetes: te sinal-to-noise atios (SNR pe infoation bit) at wic aloit Ψ (L) ives te WER P =10 4 ; te coplexity estiates Ψ (L) counted as te nube of floatin point opeations. Fi. 1. Tit lowe bounds on WER of ML decodin fo tee RM codes of lent 18. Te leend ives te list size L( ) fo wic te aloit Ψ (L) pefos witin =0.5 db fo tese bounds. RM Code 7 ª 7 ª 7 3 4ª List size L( ) Coplexity Ψ(L) at Table 1. RM codes of lent 18: te list size L( ), decodin coplexity, and te coespondin SNR at wic aloit Ψ (L) pefos witin =0.5 db fo ML decodin at WER Fo RM codes of lent 56, we skip ost decodin esults as tese will be ipoved in te next section by te peutation tecniques. In ou sinle exaple in Fi., we pesent te esults fo te (n = 56,k = 93) code ª 8 3. Tis code ives te lowest ate of conveence to te ML decodin aon all RM codes of lent 56. In ote wods, all ote codes equie te salle lists to acieve te sae pefoance loss. Tis exaple and ote siulation esults sow tat te aloit Ψ (L) pefos witin 0.5 db fo ML decodin on te lents 18 and 56 usin lists of sall o odeate size. IV. FURTHER IMPROVEMENTS A. Subcodes of RM codes Moe detailed esults also sow tat any codes of lent n 56 equie lists of lae size L 104 to appoac ML decodin witin te sall ain of 0.5 db. Teefoe fo n 56, we also eploy a diffeent appoac. Naely, te decodin pefoance can be ipoved by eliinatin tose pats, wee ecusive decodin fails oe often. Hee we use te esults of [10], wic sow tat te leftost pats ae te least potected. Recall tat eac left-end pat ξ coesponds to one infoation sybol. Teefoe, decodin on tese pats can be eliinated by settin te coespondin infoation bits as zeos.

5 WER WER WER 5 L = 1 L = L = 4 L = 16 L = 64 L = 56 L = 104 ML, lowe bound k = 93, L = 1 k = 78, L = 1 k = 78, L = k = 78, L = 4 k = 78, L = 8 k = 78, L = 16 k = 78, ML, lowe bound Fi.. (56, 93)RMcode 8 3ª. WER fo te aloit Ψ (L) wit lists of size L. Fi. 3. (56, 78)-subcode of te (56, 93) RM code 8 3ª. WER fo te aloit Ψ (L) wit lists of size L. In tis way, we eploy te subcodes of te oiinal code ª. Note tat ou decodin aloit Ψ (L) uns vitually uncaned on subcodes. Indeed, te sinle diffeence aises wen soe infoation block a s takes only one value 0 on te coespondin left node (o less tan values on te it node). Teefoe, on eac step s, we can poceed as befoe, by takin only te actual blocks a s left at tis node afte expuation. In te aloit Ψ (L), tis expuation stats wit te least potected infoation pat 0 tat ends at te node ª 0. It can be sown tat fo lon RM codes of fixed ode, eliinatin even te sinle weakest pat 0 inceases te adissible noise powe 1/ ties. Tus, te lowest odes =, 3 yield te biest ain (10lo 10 )/ db, wic equals 0.75 db and db, espectively. To poceed fute, we eliinate te next weakest pat Howeve, te teoetical analysis becoes oe coplicated on te subsequent bits and it is unclea wic bits sould be eliinated fist. Fo tis eason, we optiized tis punin pocess in ou siulation by akin a few ad oc tials and eliinatin subsequent bits in diffeent ode. Te coespondin siulation esults ae pesented in Fiue 3 fo te (56, 93)-code ª 8 3 and its (56, 78)-subcode. We see tat punin substantially ipoves code pefoance. It is also inteestin to copae Fiues and 3. We see tat te subcode appoaces te optial ML pefoance uc faste tan te oiinal code does. In paticula, te sae ain of =0.5 db can be eaced wit only L =16codewods instead of L = 104 codewods needed on te code. In all ote exaples, te subcodes also deonstated a uc faste conveence, wic leads to a lesse coplexity. In Fi. 4, we pesent siila esults fo te (51, 101)- subcode of te (51, 130)-code 9 3ª. HeeinTable,wealso ive a few list sizes L, te coespondin SNRs needed to eac te output WER P = 10 4, and te coplexity estiates Ψ (L) counted by te nube of floatin point opeations. Siila esults wee also obtained fo te subcodes of ote RM codes of lent 51. k = 130, L = 1 k = 101, L = 1 k = 101, L = 4 k = 101, L = 16 k = 101, L = 64 k = 101, L = 56 k = 101, ML, lowe bound Fi. 4. (51, 101)-subcode of te (51, 130) RM code 9 3ª. WER fo te aloit Ψ (L) wit lists of size L. List size L Coplexity SNR at Table. (51, 101)-subcode of te (51, 130) RM code 9 3ª. List sizes L, te coespondin SNRs, and coplexity estiates Ψ (L) needed at WER Tese siulation esults sow tat cobinin bot tecniques - eliinatin te least potected bits and usin sall lists of codewods - ives a ain of 3 to 4 db on te lents n 51 ove te oiinal non-list decodin aloit Ψ. Fo subcodes, we also appoac ML decodin wit te lists educed 4 to 8 ties elative to te oiinal RM codes.

6 WER 6 B. New peutation tecniques Te second ipoveent to te aloit Ψ (L) utilizes te ic syety oup GA() of RM codes [7] tat includes O() peutations of n positions i =(i 1,...,i ). To eploy fewe peutations, we fist peute te indices (1,,..., ) of all n positions i =(i 1,...,i ). Tus, we fist take a peutation (1,,..., ) 7 π (π(1),...,π()) of indices and conside te coespondin! peutations π(i) of positions i : π(i) :(i 1,...,i ) (i π(1),...,i π() ). (9) Reak. Note tat te indices epesent te diffeent axes in E. Tus, any peutation of indices is te peutation of axes of E. Fo exaple, te peutation (, 1, 3, 4,...,) of indices leaves uncaned te fist and te fout quates of all positions 1,..., n, but canes te ode of te second and te tid quates. Given a peutation π, conside te subset of oiinal indices (axes) π 1 {1,..., } tat wee tansfoed into te fist axes 1,..., by te peutation π. We say tat two peutations π and η ae equivalent if tese iaes fo te identical (unodeed) subsets π 1 {1,..., } = η 1 {1,..., }. Now conside any subset T of peutations (9) tat includes exactly one peutation fo eac equivalent class. Tus, T includes ( ) peutations, eac of wic specifies a subset of te fist indices. Recall tat tese indices coespond to te axes tat ae pocessed fist on te subpat 0 (fo exaple, we can stat wit te axis i instead of i 1, in wic case we fist fold te adjacent quates instead of te alves of te oiinal block). Tus, tis subset T specifies all possible ways of coosin unodeed axes tat will be pocessed fist by te aloit Ψ. Given soe positive intee l (wic is salle tan te foe paaete L), we ten incopoate tese peutations π(i) into te list decodin Ψ (l). Naely, we fo all peutations y π(i) of te eceived vecto y and apply aloit Ψ (l) to eac vecto y π(i). Howeve, at eac step of te aloit, we also cobine diffeent lists and leave only l best candidates in te cobined list, eac counted once. Note tat tis tecnique akes only ainal canes to ou conventional list decodin Ψ (l). Indeed, te sinle vecto y in ou oiinal settin (7) is eplaced by ( ) peutations y π(i). Tus, we use paaete ( ) in ou initial settin but keep paaete l fo all decodin steps. If l<( ), ten te nube of ecods dops to l alost iediately, afte te fist decodin is pefoed on te pat 0. Also, infoation bits ae now decoded in diffeent odes dependin on a specific peutation π(i). Note tat we ay (and often do) et te sae enties epeated any ties. Teefoe, in steps and 3 we ust eliinate identical enties. Tis is done in all steps by applyin invese peutations and copain te coespondin blocks a. Tis peutation-based aloit is called Υ (l) below and as coplexity siila to Ψ (l) fo all te codes tested. Te otivation fo tis aloit is as follows. Te specific ode of ou axes also defines te ode in wic te decodin aloit folds te oiinal block into te subblocks of lents n/, ten n/4, and so on. Now note tat tis foldin pocedue will likely accuulate te eos weneve eoneous positions substantially disaee on te two alves (coespondinly, quates, and so on). Tis can also appen if te eos ae unevenly spead ove te two alves of te oiinal block. By usin any peutations, we ake it oe likely tat te eo positions ae spead oe evenly even if tey et accuulated in te oiinal settin π(i) =i o any ote specific settin. In tis way, peutation tecniques seve te sae functions as inteleavin does on te busty cannels. Siulation esults fo te odeate lents 56 and 51 sow tat te aloit Υ (l) appoaces te optial ML pefoance even wen te cobined list of l ost pobable candidates is educed two to eit ties elative to te oiinal aloit Ψ (L). Fo RM codes of lent 56, we suaize tese esults in Fi. 5. Fo eac RM code, we fist pesent te lowe bounds fo te ML decodin eo pobability. Siilaly to Fi. 1, we ten find te iniu size l( ) tat akes te aloit Υ (l) pefo only witin =0.5 db away fo ML decodin. Tese sizes and coplexity estiates Υ (l) ae also iven in Table 3. Note tat bot aloits ive salle listsoncetispefoanceloss is slitly inceased. In paticula, te esults in Table 4 sow tat te lists ae educed two ties fo =0.5dB. In suay, te peutation aloit Υ (l) pefos witin 0.5 db fo ML decodin on te lent 56, by pocessin l 64 vectos fo all RM codes. To date, bot tecniques - peutation decodin Υ (l) of coplete RM codes and list decodin Ψ (L) of tei subcodes - yield te best tade-offs between nea-ml pefoance and its coplexity known on te lents n 56. k = 37, l( Δ) = 64, L( Δ) = 18 k = 93, l( Δ) = 18, L( Δ) = 104 k = 163, l( Δ) = 18, L( Δ) = 104 k = 19, l( Δ) = 16, L( Δ) = 3 Fi. 5. Tit lowe bounds on WER of ML decodin fo fou RM codes of lent 56. Te leend ives te list sizes l( ) and L( ) fo wic te aloits Υ (l) and Ψ (L) pefo witin =0.5 db fo tese bounds.

7 7 RM Code ª List size l( ) Coplexity Υ (l) SNR at Table 3. RM codes of lent 56: te list sizes, coplexities, and te coespondin SNRs, at wic te peutation aloit Υ (l) pefos witin =0.5 db fo ML decodin at WER RM Code ª List size l( ) Coplexity Υ (l) SNR at Table 4. RM codes of lent 56: te list sizes, coplexities, and te coespondin SNRs, at wic te peutation aloit Υ (l) pefos witin =0.5 db fo ML decodin at WER Note, oweve, tat te aloit Υ (l) ives alost no advantae fo te subcodes consideed in te pevious subsection. Indeed, tese subcodes ae obtained by eliinatin te leftost (least potected) infoation bits. Howeve, any new peutation π(i) assins te new infoation bits to tese leftost nodes. Tus, te new bits also becoe te least potected. Anote unsatisfactoy obsevation is tat inceasin te size of te peutation set T - say, to include all! peutations of all indices - elps little in ipovin decodin pefoance. Moe eneally, tee ae a nube of ipotant open pobles elated to tese peutation tecniques. We nae a few: find te best peutation set T fo te aloit Υ (l); analyze te aloit Υ (l) analytically; odify te aloit Υ (l) fo subcodes. [] S.N. Litsyn, On decodin coplexity of low-ate Reed-Mulle codes, Poc. 9 t All-Union Conf. on Codin Teoy and Info. Tansission, Pat 1, Odessa, USSR, pp. 0-04, 1988 (in Russian). [3] F. Heati, Closest coset decodin of u u + v codes, IEEE Selected Aeas Coun., vol. 7, pp , [4] G.A. Kabatyanskii, On decodin of Reed-Mulle codes in seicontinuous cannels, Poc. nd Int. Woksop Aleb. and Cob. Codin Teoy, Leninad, USSR, 1990, pp [5] R. Lucas, M. Bosset, and A. Daann, Ipoved soft-decision decodin of Reed-Mulle codes as enealized ultiple concatenated codes, Poc. ITG Conf. on Souce and Cannel Codin, Aaen, Geany, 1998, pp [6] N. Stolte and U. Soe, Soft-decision stack decodin of binay Reed- Mulle codes wit Look-Aead tecnique, Poc. 7 t Int. Woksop Aleb. and Cob. Codin Teoy, Bansko, Bulaia, June 18-4, 000, pp [7] F.J. MacWillias, N.J.A. Sloane, Te Teoy of Eo-Coectin Codes, Not-Holland, Asteda, [8] I. Due, Recusive decodin of Reed-Mulle codes, Poc. 37 t Alleton Conf. on Coun., Cont., and Coputin, Monticello, IL, Sept. -4, 1999, pp [9] I. Due and K. Sabunov, Recusive constuctions and tei axiu likeliood decodin, Poc. 38 t Alleton Conf. on Coun., Cont., and Coputin, Monticello, IL, USA, 000, pp [10] I. Due, Soft decision decodin of Reed-Mulle codes: a siplified aloit, IEEE Tans. Info. Teoy, tis issue. PLACE PHOTO HERE Ilya Due (M 94 SM 04) eceived te M.Sc. deee fo te Moscow Institute of Pysics and Tecnoloy, Russia, in 1976 and te P.D. deee fo te Institute fo Infoation Tansission Pobles of te Russian Acadey of Sciences, in Fo 1983 to 1995, e was wit te Institute fo Infoation Tansission Pobles. Since 1995, e as been a Pofesso of Electical Enineein at te Univesity of Califonia, Riveside. Duin , e was a Royal Society Guest Reseac Fellow at Manceste Univesity, Manceste, U.K., and duin , an Alexande von Huboldt Fellow at te Institute fo Expeiental Mateatics in Essen, Geany. His eseac inteests ae in codin teoy and its applications. PLACE PHOTO HERE Kiill Sabunov (S 00 M 04) eceived te M.Sc. deee fo te St. Petesbu State Univesity of Aeospace Instuentation, Russia, in 1996 and te P.D. deee fo te Univesity of Califonia, Riveside, in 004. He is cuently a Reseac Eninee at te XVD Copoation, San Jose, CA. His eseac inteests ae in codin teoy, video copession, and tei applications. V. CONCLUDING REMARKS In tis pape, we consideed ecusive decodin aloits fo RM codes tat can povide nea-axiu likeliood decodin wit feasible coplexity fo RM codes o tei subcodes on te odeate lents n 51. Ou study yet leaves any open pobles. Fistly, we need to titly estiate te eo pobabilities p(ξ) on te diffeent pats ξ. To optiize ou punin pocedues fo specific subcodes, it is ipotant to find te ode in wic infoation bits sould be eoved fo te oiinal RM code. Finally, it is yet an open poble to analytically estiate te pefoance of te aloits Ψ (L) and Υ (l). REFERENCES [1] I.S. Reed, A class of ultiple eo coectin codes and te decodin scee, IEEE Tans. Info. Teoy, vol. IT-4, pp , 1954.