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2 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION Braided Convoluional Codes: A New Class of Turbo-Like Codes Wei Zhang, Member, IEEE, Michael Lenmaier, Member, IEEE, Kamil Sh. Zigangirov, Fellow, IEEE, and Daniel J. Cosello, Jr., Life Fellow, IEEE Absrac We presen a new class of ieraively decodable urbo-like codes, called braided convoluional codes. Consrucions and encoding procedures for ighly and sparsely braided convoluional codes are inroduced. Sparsely braided codes exhibi good convergence behavior wih ieraive decoding, and a saisical analysis using Markov permuors shows ha he free disance of hese codes grows linearly wih consrain lengh, i.e., hey are asympoically good. Index Terms braided convoluional codes, urbo-like codes, codes on graphs, ieraive decoding, convoluional permuor, free disance. I. INTRODUCTION Braided block codes BBC s) [] were firs inroduced in [2] [3]. These codes can be viewed as a sliding version of produc codes [4] or expander codes [5] [6]. In braided codes, informaion symbols are checked by wo componen encoders, and he pariy symbols of one componen encoder are used as inpus o he oher componen encoder. The connecions beween he wo componen encoders are defined by he posiions where informaion symbols and pariy symbols are sored in a wo-dimensional array. Braided codes form a class of coninuously decodable codes defined on graphs [2], and hus ieraive decoding can be employed. Owing o he coninuously decodable propery of hese codes, he decoder can be implemened using a highly efficien pipeline srucure. Therefore braided codes are well suied for high speed coninuous daa ransmission. In [3], shor block codes such as Hamming codes were employed as componen codes. Two families of BBC s were proposed based on he densiy of he sorage array, i.e., ighly braided block codes TBBC s) and sparsely braided block Manuscrip received May 3, 2006; revised January 30, This work was suppored in par by NSF Gran CCR , Army gran DAAD6-02-C-0057, and NASA gran NNG05GH73G. The maerial of his paper was presened in par a he 2005 IEEE Inernaional Symposium on Informaion Theory, Adelaide, Ausralia, Sepember W. Zhang was wih he Deparmen of Elecrical Engineering, Universiy of Nore Dame, Nore Dame, IN USA. He is now wih QUALCOMM Incorporaed, San Diego, CA wzhang@qualcomm.com). M. Lenmaier was wih he Deparmen of Elecrical Engineering, Universiy of Nore Dame, Nore Dame, IN USA. He is now wih Vodafone Chair Mobile Communicaions Sysems, Dresden Universiy of Technology, 0062 Dresden, Germany Michael.Lenmaier@ifn.e.u-dresden.de). K. Sh. Zigangirov is wih Universiy of Nore Dame, Nore Dame, IN USA, Lund Universiy, Lund, Sweden, and he Insiue for Problems of Informaion Transmission, Moscow, Russia kzigangi@nd.edu). D. J. Cosello, Jr. is wih he Deparmen of Elecrical Engineering, Universiy of Nore Dame, Nore Dame, IN USA Daniel.J.Cosello.2@nd.edu). Communicaed by T. Richardson, Associae Edior for Coding Theory. codes SBBC s), and i was shown ha ieraive decoding performance is grealy improved wih SBBC s. In his paper, we sudy a new class of braided codes, braided convoluional codes BCC s), firs inroduced in [7]. In conras o BBC s, which are described in deail in [], we use convoluional codes as componen codes. Convoluional permuors, an imporan ingredien of BCC s, are inroduced in Secion II, and code consrucions are described in Secion III. Analogous o BBC s, a ighly braided convoluional code TBCC) resuls when a dense array is used o sore he informaion and pariy symbols. Sparsely braided convoluional codes SBCC s) are hen proposed o overcome he shor cycles in he Tanner graph represenaion [8] of TBCC s. The sorage array of SBCC s has a lower densiy, resuling in improved ieraive decoding performance. In Secion IV a syndrome former marix is defined, and SBCC s are shown o be a ype of low densiy pariy check LDPC) convoluional code. Then in Secion V a pipeline decoder archiecure for high speed coninuous daa ransmission is presened. In Secion VI, a blockwise version of BCC s is proposed for applicaions involving packeized daa. The performance of rae R = /3 SBCC s is hen evaluaed by compuer simulaion in Secion VII. By means of a Markov permuor analysis [9], a numerical mehod is developed in Secion VIII o compue a lower bound on free disance for he ensemble of BCC s. The free disance bound shows linear growh in free disance as a funcion of consrain lengh. This implies ha BCC s, in conras o urbo codes or serially concaenaed codes, are asympoically good in erms of disance growh. Finally, we presen some conclusions in Secion IX. II. CONVOLUTIONAL PERMUTORS An essenial par of he encoder for BCC s is a convoluional permuor also called a convoluional scrambler [0]). In his secion, we briefly review he basic heory of muliple convoluional permuors given in []. A symmeric muliple convoluional permuor MCP) of mulipliciy k can be described by a semi-infinie marix P = p, ),, Z +, which has k ones in each row and in each column saring from he h column. The oher enries are zeros. The marix P also saisfies he causaliy condiion, i.e., p, = 0, <. )

3 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 2 We use he following represenaion for P : These permuaion funcions are sored in a ROM for implemenaion. p 0,δ p 0,δ+ p 0, To reduce he sorage space required by he permuaion P = p,+δ p,+δ+ p,+, funcions, periodic permuors are assumed. In his case,.... p.., = p +T, +T,,. 8) 2) and we assume ha p,+δ = for a leas one value of and p,+ = for a leas one value of. The parameer δ 0 is called he minimal permuor delay and 0 is called he maximal permuor delay. As in convoluional coding, we call he maximal delay he memory m of he permuor, i.e., m =. The value w = δ + is called he permuor widh. A single convoluional permuor has mulipliciy k =. If w = and p, =, a single permuor is he ideniy permuor. If p,+δ =, a single permuor is he delay permuor wih delay δ. If a muliple permuor is described by he marix P, he inverse permuor is described by he ranspose marix [P] T. Wih his marix represenaion, we can describe a single convoluional permuor as follows. Le x = x 0, x,...) be he inpu sequence o he permuor. Then he oupu sequence y = y 0, y,...) is given by y = xp. 3) The minimal T for which 8) is saisfied is called he period of a periodic convoluional permuor. In [], [], a mehod was proposed o consruc periodic muliple convoluional permuors from muliple block permuors. A T T block permuor of mulipliciy k is described by a T T square marix having k ones in each row and each column. A periodic muliple convoluional permuor wih period T is hen consruced from he basic muliple block permuor of size T T and mulipliciy k using he so-called unwrapping procedure []. Example : The consrucion of a single convoluional permuor wih period T = 6, minimal delay δ = 0, and maximal delay = 5, from a 6 6 basic block permuor of mulipliciy k = is illusraed in Figure. Firs divide he 6 6 permuaion marix describing he basic block permuor below he diagonal as shown in Figure a), hen unwrap he lower par of he marix as shown in Figure b), and finally replicae he unwrapped marix diagonally as shown in Figure c). In his way, he mapping beween he inpu and oupu is defined as y = x, where is deermined by he permuaion funcion f P ) associaed wih P, i.e., = f P ). 4) Equaion 3) describes he operaion of a single convoluional permuor, bu he operaion of a muliple k > ) convoluional permuor can be described as he muliplicaion of a vecor by a marix. In he case of a muliple permuor, he enries in he marix P represen memory unis ha can sore an inpu symbol. The inpu sequence X enering he MCP is divided ino k-uples, i.e., X = x 0,x,...,x,...), where x = x,, x,2,..., x,k ) T. The blocks x, = 0,,..., are wrien o he memory unis row by row. The oupu sequence Y = y 0,y,...,y,...), where y = y,, y,2,..., y,k ) T, is read ou column wise. Since here are he same number of ones in each row and column, every inpu symbol occurs once and only once in he oupu sequence. To describe he operaion of a muliple convoluional permuor, a marix permuaion operaor or permuaion ensor P can be inroduced. Refer o [] for deails.) Similar o a single convoluional permuor, we define he mapping beween inpus and oupus as y,i = x,i, i k, 5) where and i are deermined by he permuaion funcions f P, ) and g P, ) associaed wih he permuaion operaor P as follows = f P, i), 6) i = g P, i). 7) Fig.. a) Basic block permuor. b) Unwrapped block permuor. c) Convoluional permuor. Consrucion of a single periodic convoluional permuor. The convoluional permuor inroduced in Example is a single periodic convoluional permuor. Single convoluional permuors are used in his paper o describe rae R = /3 BCC s. An example of an MCP wih mulipliciy k = 2 and period T = 5 consruced using he unwrapping procedure is shown in Figure 2. From he unwrapping procedure, we see ha a single periodic convoluional permuor consruced as described above may no always have minimal delay δ = 0 and maximal delay memory) = m = T. In oher words, is widh is no necessarily T. However, as shown in [], if a block permuor of mulipliciy k is chosen randomly, hen wih probabiliy

4 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 3 Fig. 2. a) Muliple block permuor. b) Unwrapped block permuor. c) Muliple convoluional permuor. Consrucion of a muliple periodic convoluional permuor. /e) k he maximal delay memory) of he unwrapped muliple convoluional permuor of mulipliciy k equals T. The memory m is an imporan parameer characerizing he behavior of a convoluional permuor. Anoher imporan parameer is is overall consrain lengh M. For a given, we inroduce he se P = {p i,j : i, j > }, Z +. 9) The overall consrain lengh of he convoluional permuor is hen defined by M = w H P ), 0) where w H P ) is he Hamming weigh of he se P. I follows ha M is equal o he maximum number of symbols ha is sored in a realizaion of he permuor a any ime, analogous o he definiion of overall consrain lengh for convoluional codes [0], [2]. For single convoluional permuors, since each row and column of P have only a single, he weigh of P does no depend on he ime index, and we can omi in defining M. Thus he overall consrain lengh is independen of for single convoluional permuors. Example 2: Figure 3 illusraes a single convoluional permuor wih he same parameers, T = 6, δ = 0, and = 5, as he convoluional permuor shown in Figure c). Is overall consrain lengh is M = 4. By conras, he convoluional permuor in Figure c) has overall consrain lengh M =. Fig. 3. A single periodic convoluional permuor wih T = 6, δ = 0, = 5, and M = 4. For w >, he overall consrain lengh of a single convoluional permuor mus saisfy 0 M T 2. ) The single convoluional permuors for he BCC s considered in his paper were consruced from a basic block permuor permuaion marix) chosen randomly, assuming ha all T! possible permuaion marices of size T T are equiprobable. The delays of he corresponding convoluional permuors hen saisfy 0 δ T, and we noe ha he ideniy permuor has parameers T = and δ = = M = 0. Muliple convoluional permuors of mulipliciy k for BCC s can be consruced from ses of k 2 permuaion marices by using he operaions of row- and column-inerleaving and unwrapping see [] for deails). Convoluional permuors consruced from T T block permuors canno have period larger han T. Their periods can be T, T/2, T/3,..., and so on. If he period is T/2 T even), hen he T/2 + i)-h row of he basic block permuor is a cyclic shif of he i-h row, for i T/2. Similar argumens are valid for periods of T/3, T/4,..., and so on. The probabiliy ha he cyclic shif condiion is saisfied goes o zero as T for randomly chosen permuors. An MCP of mulipliciy k consruced from a T T block permuor is called ypical [] if i has period T, maximal delay memory) = T, and overall consrain lengh M = kt )/2. 2) Shifing a ypical MCP of mulipliciy k by a > 0 symbols, i.e., p, p +a, +a, we obain an MCP wih addiional delay a. For his permuor, he minimal delay is δ + a, he maximal delay is + a, and he overall consrain lengh is M = ka + kt )/2. 3) In general, a single convoluional permuor wih maximal delay can be implemened wih a shif regiser of lengh. The permuaion funcion f P ) associaed wih he permuor is sored in a conroller o indicae he oupu indices of he regiser sages. A each ime uni, he permuor selecs an oupu from one of he sored symbols according o he permuaion funcion. Then i delees he righ mos symbol and shifs all oher symbols one sage o he righ. The new inpu symbol is placed ino he lef mos posiion. III. CONSTRUCTION OF BRAIDED CONVOLUTIONAL CODES In his secion, we describe he consrucion of BCC s. In general, braided codes, including BBC s [2] [3] and BCC s, represen a sliding version of classic produc codes [4]. As illusraed in Figure 4, produc codes are consruced based on a recangular array ha sores he coded symbols. The k k 2 informaion sysemaic) symbols are locaed in he upper-lef corner of he array. The symbols in each row form a codeword of a horizonal componen code C n, k ). Meanwhile, he symbols of each column form a codeword of a verical componen code C 2 n 2, k 2 ). In conras, braided codes are consruced on an infinie wo-dimensional array. Furhermore,

5 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 4 he horizonal and verical encoders are linked hrough pariy feedback. In his manner, ÁÒ ÓÖÑ Ø ÓÒ he sysemaic and pariy symbols are braided ogeher. Ø ¾ Ò½ ½µ ÓÐÙÑÒ ½Ò½ Ô Ö ØÝ È Ö ØÝ Ô Ö ØÝ Ô Ö ØÝ ÓÒ Fig. 4. An n n 2, k k 2 ) produc code. ؼ A. Rae R = /3 Braided Convoluional Codes Ø Ú ¾µ Ø ½ÙØ Ú ½µ Ú ¾µ Ø ½ ÙØ ½ Ú ½µ Ú ¾µ Ø È ¾µ Ì È ¼µ È ½µ Fig. 5. Array represenaion of a rae R = /3 TBCC. Ò¾ ¾ÊÓÛ ½ Ò½ ½µ Depending on he densiy of he array, we can disinguish wo ypes of BCC s TBCC s and SBCC s. An example of a rae R = /3 TBCC is illusraed in Figure 5. Similar o urbo codes, recursive sysemaic convoluional RSC) encoders wih rae R = 2/3 are used as horizonal and verical componen encoders. The array consiss of hree diagonal ribbons, each of widh one symbol. Each enry in he array is characerized by a pair of posiion indices, ): he verical posiion and he horizonal posiion, +. The informaion symbols u are placed in he cenral ribbon wih posiion indices, ), where =, corresponding o an ideniy permuor P 0). The pariy symbols ˆv ) of he horizonal encoder encoder ) are sored in he upper ribbon wih posiion indices, + ). We may consider ha he upper ribbon is described by a delay- permuor and is denoed P ). The pariy symbols ˆv 2) of he verical encoder encoder 2) are sored in he lower ribbon wih posiion indices +, ). The lower ribbon corresponds o he ranspose of a delay- permuor and is denoed [P 2) ] T. The dark enries in he array indicae he previous inpus and oupus of he encoders ha are known a ime. Noe ha a ime 0, when he firs informaion symbol arrives, he previous pariy symbols are assumed o be 0, i.e., ˆv ) and ˆv 2) are zeros for < 0. A ime, he horizonal encoder encodes he curren informaion symbol u and is lef neighbor ˆv 2). The oupu symbol ˆv) depends on ˆv 2), u, and he convoluional encoder sae. The verical encoder performs is encoding analogously. So he -h row of he array conains ˆv 2), u, and ˆv ), and he -h column of he array conains ˆv ), u, and ˆv 2). The encoding procedure coninues in his fashion as he horizonal and verical encoders slide down and o he righ along he diagonal. The code sequence of he horizonal encoder is v ) = v ) 0,v),...,v),...), where v ) = v ),, v),2, v),3 ), v ), = u, v ),2 = ˆv2), and v),3 = ˆv). The code sequence of he verical encoder is v 2) = v 2) 0,v2),...,v2),...), where v 2) = v 2),, v2),2, v2),3 ), v2), = u, v 2),2 = ˆv), and v 2),3 = ˆv2). The code sequence ransmied over he channel is v = v 0,v,...,v,...), where v = v,, v,2, v,3 ), and u, i = v,i = ˆv ), i = 2. 4) ˆv 2), i = 3 The rae of he TBCC is R = /3. During he encoding process, wo previously encoded pariy bis are sored in he array, and hus he overall consrain lengh is M = 2. Shor cycles are generaed in he Tanner graph of TBCC s due o heir dense array srucure. Thus ieraive decoding performance can be improved if he cycle lengh is increased. This moivaes he consrucion of SBCC s, in which informaion symbols and pariy symbols are spread ou in a sparse array. An example of he array represenaion of a rae R = /3 SBCC is illusraed in Figure 6. Each row and column of he array conains one informaion symbol, one pariy symbol from he verical encoder, and one pariy symbol from he horizonal encoder. Analogous o TBCC s, he sparse array reains he hree-ribbon srucure and hree corresponding convoluional permuors. We assume ha he permuors P j) = p j) i,k ) are periodic wih periods T j, j = {0,, 2}, and ha hey are consruced using he unwrapping procedure described in Secion II, wih he widh of each ribbon equal o he period of he corresponding permuor. Thus he widhs of he cenral, upper, and lower ribbons are T 0, T, and T 2, respecively. All he enries in he array are again indexed by coordinaes, ), where and represen he imes of he horizonal and verical encodings, respecively, as shown in Figure 6. The informaion symbols u are placed in he cenral ribbon. The srucure of he cenral ribbon is defined by he permuor P 0). If p 0), =, hen he -h inpu symbol u of he encoder is placed in he array enry wih index, ). This means ha u eners he horizonal encoder a ime, and he permued symbol ũ eners he verical encoder a ime. Based on he analysis in Secion II, a ypical permuor P 0) has an overall consrain lengh of M 0 = T 0 )/2. The pariy symbols of he horizonal encoder are placed in he -h row of he upper ribbon. The srucure of he upper ribbon is defined by permuor P ). To mach he ribbon srucure of he array, his permuor has an addiional delay of T 0 symbols, and is overall consrain lengh is M = T 0 + T )/2. If p ) ˆv ), =, hen he pariy symbol ˆv ) is placed in he posiion wih index, ). Since p ), = 0 for >, he permued pariy

6 Î ÖØ Ð ÒÓ Ò Ø¼ Ú ½µ Ø IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 5 Ø Ú ¾µ ÙØ The permued pariy sequence ṽ ) a he oupu of convoluional permuor P ) is fed back o he second inpu of encoder 2, and he permued pariy sequence ṽ 2) a he oupu Ø ÙØ Ú ½µ Ø ÀÓÖ ÞÓÒØ Ð Ú ¾µ ÒÓ Ò of convoluional permuor P 2) is fed back o he second inpu of encoder. The informaion sequence u and he pariy sequences ˆv ) and ˆv 2) are muliplexed ino he oupu 00 Ø sequence of he encoder v = v È ¼µ È ½µ 0,v,...,v,...), where v = v,, v,2, v,3 ), and u, i = ÙØ v,i = ˆv ), i = 2. 7) ˆv 2), i = È ¾µ Ì ÒÓ Ö½ Ê Ø ¾» ÅÍ ÚØ È ¼µ Ú ½µ È ½µ Ú ½µ ÙØ Ú ¾µ Ø Fig. 6. Array represenaion for SBCC s. ÒÓ Ö¾ Ê Ø ¾» È ¾µ Ú ¾µ Ø symbol ṽ ) will ener he verical encoder a ime when i leaves permuor P ). The pariy symbols ˆv 2) of he verical Fig. 7. Encoder for a rae R = /3 braided convoluional code. encoder are placed in he -h column of he lower ribbon, whose srucure depends on permuor P 2). To mach he array srucure, P 2) has minimal delay, maximal delay T 2, and overall consrain lengh M 2 = T 2 )/2 +. If p 2), =, is placed in he posiion wih index, = 0 for >, he permued pariy symbol ṽ 2) will ener he horizonal encoder a ime when i leaves permuor P 2). The memory of he encoder is defined as he maximal number of ime unis ha a symbol says in he encoder. The overall consrain lengh M of an SBCC encoder is defined as he oal number of symbols sored in he encoder. Thus, if all permuors P 0), P ), and P 2) are ypical, hen hen he pariy symbol ˆv 2), ). Since p 2) M = T T 2 + T T ) If he permuors are all ypical and T 0 = T = T 2 = T, he oal widh of he hree ribbons in a BCC is 3T, and he oal number of symbols sored in he memory of he permuors is given by M = 5T )/ ) The implemenaion of a rae R = /3 BCC encoder is shown in Figure 7. The encoder consiss of wo rae R cc = 2/3 RSC componen encoders, he horizonal encoder encoder ) and he verical encoder encoder 2), and hree convoluional permuors P 0), P ), and P 2) are employed. The informaion sequence u = u 0, u,..., u,...) eners he firs inpu of encoder direcly, and he permued informaion sequence ũ a he oupu of convoluional permuor P 0) eners he firs inpu of encoder 2. Encoder generaes he pariy sequence ˆv ) = ˆv ) 0, ˆv),..., ˆv),...) and encoder 2 generaes he pariy sequence ˆv 2) = ˆv 2),..., ˆv2),...). 0, ˆv2) B. Generalized Braided Convoluional Codes Generalizing he rae R = /3 BCC s in Secion III-A o oher raes is sraighforward. In principle, we can use differen componen encoders for he horizonal and verical encodings. If we employ a rae R ) cc = horizonal encoder and a rae k 0) + k 2) k 0) + k ) + k 2) 8) R 2) cc = k 0) + k ) k 0) + k ) + k 2) 9) verical encoder, where k 0), k ), and k 2) are posiive inegers, he rae of he resuling BCC is k 0) R =. 20) k 0) + k ) + k 2) The array represenaion is shown in Figure 8. The cenral ribbon is described by an MCP P 0) of mulipliciy k 0), and he upper and lower ribbons are described by MCP s P ) and [P 2) ] T of mulipliciy k ) and k 2), respecively. Horizonal and verical encoding proceeds by row and column in he same fashion as for rae R = /3 BCC s. If he convoluional permuors are consruced from block permuors as described in Secion II and hey are ypical, hen he overall consrain lengh of he encoder is given by M = k0) T 0 ) 2 + k) T ) 2 + k2) T 2 ) 2 +k ) T 0 +k 2), 2)

7 v e),, ve),2,...,ve),k 0) +k ) +k 2) ). Here, he mapping rules be- ÑÙÐØ ÔÐ Øݹ ½µ È ½µ IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 6 ÑÙÐØ ÔÐ Øݹ ¾µ È ¾µ Ì ÑÙÐØ ÔÐ Øݹ ¼µ ween he inpus and oupus of each generalized BCC componen encoder can be described by È ¼µ u,i, i k 0) v ),i = ṽ 2), i k 0) k 2),i k 23) 0) ˆv ), i k 0) k 2) k ),i k 0) k 2) and ũ,i, i k 0) v 2),i = ṽ ), i k 0) k ),i k 0) ˆv 2) Fig. 8. Array represenaion of generalized BCC s. where T 0, T, and T 2 are he periods of P 0), P ), and P 2), respecively. As illusraed in Figure 9, he srucure of he encoder for generalized BCC s is similar o he rae R = /3 case, excep ha he permuors may now be MCP s. The horizonal encoder encoder ) has k 0) + k 2) inpus. A ime insan, he k 0) -uple informaion block u = u,, u,2,...,u,k 0)) of he informaion sequence u = u 0,u,...,u,...) eners he firs k 0) inpus of he horizonal encoder. Meanwhile, he verical encoder produces a block of k 2) pariy symbols ˆv 2) = ˆv 2),, ˆv2),2,..., ˆv2),k 2) ) ha eners he MCP P 2). The oupu ṽ 2) = ṽ 2),, ṽ2),2,..., ṽ2) ) of P 2) appears in he,k 2) -h row of he lower ribbon and provides he remaining k 2) inpus o he horizonal encoder. In parallel, he informaion sequence u = u 0,u 2,...,u,...) eners he MCP P 0). The oupu sequence of P 0) is ũ = ũ 0,ũ 2,...,ũ,...), where ũ = ũ,, ũ,2,..., ũ,k 0)). The verical encoder encoder 2) has k 0) + k ) inpus. The block ũ eners he firs k 0) inpus of verical encoder a he ime insan. This block appears in he -h column of he cenral ribbon. Meanwhile, he horizonal encoder produces a block of k ) pariy symbols ˆv ) = ˆv ),, ˆv),2,..., ˆv),k ) ) ha eners he MCP P ). The oupu ṽ ) = ṽ ),, ṽ),2,..., ṽ) ) of P ) appears in he -,k ) h column of he upper ribbon and provides he remaining k ) inpus o he verical encoder. The combinaion of he blocks u, ˆv ), and ˆv 2), consising of k 0) + k ) + k 2) bis, forms he oupu code block v = v,, v,2,..., v,k 0) +k ) +k 2)) of he generalized BCC encoder. The muliplexing rule is defined as u,i, i k 0) v,i = ˆv ), i k 0) k ),i k. 22) 0) ˆv 2), i k 0) k ) k 2),i k 0) k ) We can also denoe he oupu code sequences of he horizonal e = ) and verical e = 2) encoders as v e) = v e) 0,ve),...,ve),...), where v e) =,i k 0) k ), i k 0) k ) k 2). 24) A he receiver, hese mapping rules deermine he demuliplexing requiremens of he componen decoders. Fig. 9. u ũ P 0) u ) ṽ ) ṽ 2) Encoder R = k0) +k 2) k 0) +k ) +k 2) P ) P 2) ˆv ) ˆv 2) R = k0) +k ) k 0) +k ) +k 2) Encoder 2 Encoder for generalized BCC s. v MUX IV. SYNDROME FORMER REPRESENTATION OF BRAIDED CONVOLUTIONAL CODES In his secion, we derive a canonical represenaion of BCC s using he syndrome former marix. The syndrome former is useful for inerpreing he srucural properies of BCC s. In paricular, we show ha he sparsiy of he permuors in he BCC encoder insures ha he overall BCC syndrome former is sparse, hus making BCC s suiable for ieraive decoding. We consider firs some examples of he consrucion of syndrome formers for convoluional codes. Example 3: Consider a rae R cc = /2 RSC encoder wih generaor marix GD) = + D + D 2 ). 25) The inpu sequence of he encoder is u = u 0, u,...,u,...) and he oupu sequence is v = v 0, v,..., v,...). We denoe he wo individual oupus of he encoder by v 0) = v 0) 0, v0),...,v0),...) and v ) = v ) 0, v),..., v),...). Since he encoder is sysemaic, v 0) = u. A pariy check marix for his encoder is given by HD) = H ) D) ) ) = + D + D 2. Corresponding o H ) D), we inroduce

8 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 7 he semi-infinie marix [ H )] T =......, 26) which we call he parial syndrome former marix. Then he encoder s pariy consrain is described by he following equaion v 0) I + v )[ H )] T = 0, 27) where I is a semi-infinie ideniy marix. In order o obain he usual descripion of a convoluional syndrome former, we will use he operaions of row- and column- inerleaving. These operaions were inroduced in [0] for wo marices and generalized in [] for a larger number of marices. The row-inerleaving ) of he se of marices P ),P 2),...,P k) see Definiion 2.2 in []) we designae as P = P ),P 2),...,P k)). 28) Analogously, he column-inerleaving ) of he se of marices P ),P 2),...,P k) see Definiion 2.3 in []) we designae as P = P ),P 2),...,P k)). 29) In Example 3, he oupu code sequence v = v 0,v,...,v,...), where v = v 0), v ) ), can be represened as an inerleaved version of sequences v 0) and v ). If we row-inerleave he marices I and [ H )] T, hen we obain he syndrome former H T = I, [ H )] ) T of he encoder in Example 3, i.e., vh T = 0. Example 4: Consider a rae R cc = /2 RSC encoder wih generaor marix GD) = + D 2 + D + D 2 ). 30) Wih inpu sequence u = u 0, u,..., u,...), he oupu sequence v = v 0,v,...,v,...), where v = v 0), v ) ), can be represened as an inerleaved version of sequences v 0) and v ), where v 0) = u and v ) = v ) 0, v) 0,...,v),...). A pariy check marix is given by HD) = H 0) D) H ) D) ) = + D 2 + D + D 2). Then we have where [ H 0) ] T = v 0)[ H 0)] T + v ) [ H )] T = 0, 3) ) corresponds o H 0) D), and [ H )] T is defined in 26). The syndrome former in he convenional form is hen given by [H H T 0) = ] T [ ] ), H ) T, and vh T = 0. Example 5: Consider a rae R cc = 2/3 RSC encoder wih generaor marix GD) = D + D 2 + D 2 + D + D 2. 33) The inpu sequences are denoed as u 0) = u 0) 0, u0),..., u0),...) and u ) = u ) 0, u),..., u),...). The oupu sequence is v = v 0,v,v 2,...,v,...), where v = v 0), v ), v 2) ). Since he encoder is sysemaic, v 0) = v 0) 0, v0),..., v0),...) = u 0), v ) = v ) 0, v),..., v),...) = u ), and v 2) = v 2) 0, v2),...,v2),...) is he pariy sequence. A pariy check marix is given by HD) = H 0) D) H ) D) ) = + D 2 + D + D 2). Then we have v 0) I + v )[ H 0)] T + v 2) [ H )] T = 0, 34) where I is an semi-infinie ideniy marix and [ H 0)] T and [ H )] T are defined in 32) and 26), respecively. The syndrome former is hen given by H T = I, [ H 0)] T [, H ) ] ) T. 35) We now describe he consrucion of he syndrome former for he BCC of Figure 7. For simpliciy, we assume ha componen encoders and 2 are given by he generaor marix in 33). Le u = v 0) be he informaion sequence and ˆv e) = ˆv e) 0, ˆve),..., ˆve),...), e {, 2}, where ˆv e) = ˆv e),, ˆve),2, ˆve),3 ), be he oupu pariy sequences of encoder horizonal) and encoder 2 verical), respecively. Then hey mus saisfy he following pariy consrains: v 0) I + ˆv )[ H )] T + ˆv 2) P 2)[ H 0)] T = 0, 36) v 0) P 0) + ˆv ) P )[ H 0)] T + ˆv 2) [ H )] T = 0. 37) Equaion 36) describes he horizonal encoder. The syndrome former H T hor of he horizonal encoder is H T hor = I, [ H )] T,P 2) [ H 0)] ) T, 38) and i follows ha vh T hor = 0, where v is he oupu sequence of he BCC encoder shown in Figure 7. Similarly, 37) describes he verical encoder. Is syndrome former is H T ver = P 0),P )[ H 0)] T [, H ) ] ) T, 39) and vh T ver = 0. I follows ha he syndrome former H T of he rae R = /3 BCC in Figure 7 wih rae R cc = 2/3 componen encoders given by 33) is ) H T = H T hor,ht ver, 40)

9 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 8 and hence vh T = 0. Now we have a convenional represenaion of he syndrome former marix. If he periods T 0, T, and T 2 of permuors P 0), P ), and P 2) are large enough, P 0), P 2)[ H 0)] T [, and P ) H 0)] T are also sparse. Thus he syndrome former marix H T is sparse, and he corresponding BCC can be considered as a special case of an LDPC convoluional code []. The syndrome former for generalized BCC s can be expressed in a similar way by making use he row and column inerleaving operaions. The model we have considered so far assumes he ransmission of an infinie lengh informaion sequence. Since real communicaion sysems ransmi finie lengh informaion sequences, he encoding of BCC s should be erminaed so ha he informaion bis a he end of he inpu sequence are adequaely proeced. In convoluional coding, he normal mehod of erminaion is o add a ail o he informaion sequence ha forces he encoder o he zero sae. The ail depends boh on he encoder srucure and he encoder sae. The ail bis can be compued by a simple erminaion circui if he encoder is based on a parial syndrome realizaion, as developed for LDPC convoluional codes in [3] and applied o BBCs in []. Given a syndrome former represenaion of a specific code, he parameers for his erminaion circui can be precompued by solving a sysem of linear equaions. For he urbo-like encoder srucure shown in Figure 7, he sae of he BCC encoder depends no only on he saes of he componen encoders, bu also on he saes of he convoluional permuors. The deerminaion of ail bis ha drive he overall encoder o he zero sae is in his case no sraighforward. A subopimal bu simple way of erminaing such an encoder is o append a ail of zero bis o he informaion sequence. In his case, only he pariy bis in he ail mus be ransmied. For BCC s wih period T convoluional permuors, a lengh 2T zero ail has been deermined o be sufficien in pracice. In his case, if he lengh of he informaion sequence is L for a rae R = /3 BCC, he resuling code rae of he erminaed code is given by R = L 3 L + 4T/3. 4) V. PIPELINE DECODER ARCHITECTURE A pipeline decoder archiecure for LDPC convoluional codes was firs proposed in [], where he coninuously decodable propery of hese codes was exploied o accelerae he decoding speed. By employing a number of processors equal o he number of ieraions o execue he decoding algorihm in parallel, he pipeline decoder yields esimaed oupus a each execuion cycle afer some iniial decoding delay. Since BCC s are a special class of LDPC convoluional codes, hey can be decoded using he pipeline archiecure. In his secion, we describe he pipeline srucure for coninuous decoding of BCC s. Assume ha he generalized BCC encoder described in Secion III-B is used. The code sequence is v = v 0,v,...,v,...), where v = v,, v,2,...,v,k 0) +k ) +k 2)). Afer ransmiing over a memoryless channel, such as an addiive whie Gaussian AWGN) channel, he received sequence is r = r 0,r,...,r,...), where r = r,, r,2,..., r,k 0) +k ) +k 2)). Using he condiional probabiliy pr v) of receiving he signal r given he ransmied signal v, we can calculae he channel loglikelihood raio s LLR s) l = l 0,l,...,l,...), where l = l,, l,2,..., l,k 0) +k ) +k 2)), for he coded bis: l,i = log pr,i v,i = 0) pr,i v,i = ), 0, i k0) + k ) + k 2). 42) According o he mapping rules 23) and 24), hese LLR s are demuliplexed ino wo sreams. For componen encoder e, e {, 2}, he channel LLR s corresponding o he oupus v e) = v e) 0,ve),...,ve),...), where v e) = v e),, ve),2,..., ve) given by l e) = l e) 0,le),...,le) l e),k 0) +k ) +k 2) ), are,...), where = l e),, le),2,...,le) ).,k 0) +k ) +k 2) Le L0) = L 0 0),L 0),...,L 0),...), where L 0) = L, 0), L,2 0),..., L,k 0) +k ) +k 2)0)), be he se of apriori LLR s for he code sequence v. In his way, we denoe he apriori LLR for he coded bi v,k as L,k 0). The apriori LLR s for he code sequence v are given by L,i 0) = {, < 0 0, 0, i k0) + k ) + k 2). 43) Analogously, le L ) 0) and L 2) 0) be he se of apriori LLR s for he code sequences v ) and v 2) from he horizonal and verical encoders, respecively. Since here is a one-one mapping beween he symbols of he sequences v and v ) and v 2) according o 22), 23), and 24), we can also find he values for L ) 0) and L 2) 0). When he ransmied signals arrive a he receiver, he channel LLR s are calculaed and placed ino parallel buffers along wih he apriori LLR s. The componen codes are hen decoded using a parallel bank of 2I a poseriori probabiliy APP) processors using he windowed BCJR algorihm [4] [5], where I is he number of ieraions o be performed. Based on he channel LLR s l ) and he apriori LLR s L ) 0), he firs APP processor B ) obains he exrinsic LLR s L ) ) for a window of W coded symbols of he sequence v ) from he horizonal encoder. Then he exrinsic LLR s L ) ) are reordered o L 2) ) according o he order of he code sequence v 2) of he verical encoder, based on he mapping rules in 23) and 24). During he reordering, he exrinsic LLR s in L ) ) for u, ṽ 2), and ˆv ) are permued by P 0), [ P 2)] T, and P ), respecively. L 2) ) is used as apriori LLR s for he code sequence v 2) by he APP processor B 2) 2. In he same manner as for he firs processor B ), processor B2) 2 calculaes he exrinsic LLR s L 2) 2) for a window of W symbols of he sequence v 2). The exrinsic LLR s L 2) 2) are hen reordered o L ) 2) according o he order of he code sequence v ) of he horizonal encoder, based on he mapping rules in 24) and 23). During he reordering, he exrinsic LLR s in L 2) 2) for ũ, ṽ ), and are permued by [ P 0)] T [ ], P ) T, and P 2), respecively. ˆv 2) The hird APP processor B ) 3 hen uses L ) 2) as apriori LLR s. The following APP processors work in a similar

10 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 9 fashion as described above. A pipeline decoder comprised of 2I APP processors o perform I ieraions of decoding is shown in Figure 0. Processors B ) 2j and B2) 2j, j I, perform horizonal and verical componen decoding, respecively. Each processor performs he windowed BCJR algorihm on a window of size W, where W should be large compared o he consrain lengh of he componen encoder [6]. In order o avoid differen processors working on overlapping ses of coded bis a he same ime, a separaion delay of τ coded symbols is imposed beween adjacen processors so ha he apriori values are updaed wihou memory conflics. If T is he period of all he permuors, i is sufficien o se τ = 3T. 44) Evenually he received sequence flows hrough he series of processors B ), B2) 2, B) 3,..., B2) 2I, which updae he apriori values for he coded bis 2I imes. The las processor B 2) 2I makes hard decisions for he informaion bis based on is oupu APP values. Using his pipeline srucure, we can process 2I informaion symbols in parallel, hus achieving high speed decoding. l L0) Fig. 0. W τ Buffer Demux Demux Demux Demux l ) l 2) l ) l 2) L ) 0) B ) B 2) 2 B ) 3 L ) ) L 2) 2) L ) 3) B 2) 2I hard decisions L 2) ) L ) 2) L 2) 2I ) Pipeline decoder for BCC s. This procedure is similar o he decoding of urbo codes. The major difference is ha he pipeline decoder uses a windowed BCJR decoder and calculaes APP values for all he code symbols insead of only he informaion symbols. A drawback of pipeline decoding is ha i has a large iniial decoding delay. Only afer he las processor in he pipeline has filled up does he decoder sar making hard decisions on he informaion bis. Thus here is an iniial delay laency) of 2IW + τ) coded symbols, or abou 2.5I imes he overall consrain lengh of he encoder. Neverheless, we obain coninuous decoding oupus afer his iniial delay. In he nex secion, we consider blockwise BCC s. In his case, we assume ha he informaion sequence eners he encoder in a block by block manner wih a relaively large block size. This corresponds o many pracical applicaions in which he daa sream is ransmied in finie lengh packes. In his sense, he BCC s inroduced in he previous secions are referred o as biwise BCC s. VI. BLOCKWISE BRAIDED CONVOLUTIONAL CODES To encode a blockwise BCC he informaion sequence is divided ino blocks of lengh N symbols, i.e., u = u 0,u,...,u,...), where u = u,, u,2,...,u,n ). To simplify he descripion, we suppose ha he whole block u is sen o he encoder a ime insan. If we allow for some change of noaion, a rae R = /3 blockwise BCC encoder can sill be described by Figure 7. In paricular, P 0), P ), and P 2) now denoe block permuors of size N raher han convoluional permuors. The informaion symbol u a he encoder inpu is replaced by he block u, he pariy symbol ˆv ) of he horizonal encoder is replaced by he pariy block ˆv ) = ˆv ),, ˆv),2,..., ˆv),N ), and he pariy symbol ˆv 2) of he verical encoder is replaced by he pariy block ˆv 2) = ˆv 2),, ˆv2),2,..., ˆv2),N ). As componen encoders we consider now rae R = 2/3 ail-biing convoluional encoders ha sar from and end in he same sae. This way he rellises are decoupled beween differen blocks and he componen decoding can be performed independenly for differen ime insans. A erminaion of he encoders o he zero sae wihin each ime insan migh slighly improve he performance bu a he cos of a loss in rae. A he 0-h ime insan, informaion block u 0 and is permued version ũ 0 = u 0 P 0) ener he firs inpus of encoder and encoder 2, respecively. Meanwhile, blocks ṽ 2) and, consising of N zeros each, ener he second inpus of encoder and encoder 2, respecively. Encoders and 2 hen generae he lengh N pariy blocks ˆv ) 0 and ˆv 2) 0. Blocks v 0) 0 = u 0, v ) 0 = ˆv ) 0, and v2) 0 = ˆv 2) 0 are sen over he channel. A he -h ime insan, pariy block v ) is calculaed ṽ ) by encoder as a funcion of u and ṽ ) = v 2) Similarly, pariy block v 2) funcion of ũ = u P 0) and ṽ 2) = v ) v 0) = u, v ) = ˆv ), and v 2) = ˆv 2) he code sequence where P2). is calculaed by encoder 2 as a P). The blocks are muliplexed ino v = v 0,v,...,v,...), 45) v = v 0), v), v2), v0) 2, v) 2, v2) 2,..., v0) N, v) N, v2) N ). 46) In he following example, we use parial syndrome former marices o describe he encoding process for blockwise BCC s. Example 6: Consider he rae R = 2/3 encoder wih generaor marix given by 33). In Examples 3 5, 27), 3), and 34) describe he consrains implied by he encoders given in 25), 30), and 33). Suppose ha he encoder in 33) is used as a ail-biing rae R = 2/3 encoder o encode he lengh N informaion sequences u ) and u 2). The parial syndrome formers are N N marices [ ] H0) T = )

11 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 0 and [ ] H) T =..., 48) where he overbars on H 0) and H ) indicae he ail-biing versions of he syndrome formers. Then he code sequence v 0),v ),v 2) ), where v 0) = u 0) and v ) = u ), saisfies he consrain v 0) + v )[ H0) ] T + v 2) [ H) ] T = 0. 49) We assume ha wo such ail-biing convoluional encoders are used in he rae R = /3 blockwise BCC encoder. Then v 0) + v 2) P2)[ ] H0) T )[ ] + v H) T = 0, 50) v 0) P 0) + v ) P)[ ] H0) T 2)[ ] + v H) T = 0. 5), and v2), 50) and 5) define he code blocks v ) and v 2). These equaions for v ) and v 2) can Given v 0), v ) be uniquely solved if and only if he marix [ H) ] T has a righ inverse G ). Then v ) = v 0) G ) + v 2) P2)[ H0) ] T G ), 52) v 2) = v 0) P 0) G ) + v ) P)[ H0) ] T G ). 53) We can use he same echniques as in Secion IV o consruc he syndrome former for blockwise BCC s. The following marices are derived from he row-inerleaving operaion, H T hor = Ī, [ H) ] T,P 2) [ H0) ] T ), 54) H T ver = P 0),P )[ ] H0) T [ ] ), H) T, 55) where Ī is he N N ideniy marix. By means of he Kronecker produc [7], he syndrome former marices for he horizonal and verical componen codes is hen given by and H T hor = I H T hor 56) H T ver = I H T ver, 57) respecively, where I is he semi-infinie ideniy marix so ha he block marices H T hor and H T ver are replicaed infiniely along he diagonal. Corresponding o he code sequence v given by 45) for a rae R = /3 blockwise BCC, he syndrome former is obained by column-inerleaving he marices H T hor and HT ver, i.e., ) H T =. 58) H T hor,h T ver If N is large, he syndrome former marix H T of he blockwise BCC is sparse, and blockwise BCC s can be considered as special cases of LDPC convoluional codes. Similar o biwise BCC s, erminaion is used o give proecion o he informaion blocks a he end of he inpu sequence for blockwise BCC s. To reduce he encoding complexiy, we again use erminaion wih a ail of all-zero blocks for blockwise BCC s. In his case, afer he informaion blocks u [0,L ] = u 0,u,...,u L ) 59) ener he blockwise BCC encoder, Λ addiional all-zero blocks u L,...,u L+Λ ener he encoder. Since hese Λ blocks are no sen over he channel, he componen encoders have, in fac, rae R = /2 insead of R = 2/3. The resuling rae of he BCC including he ail is R = L 3 L + 2Λ/3, 60) where a ail lengh Λ = 2 blocks 2N bis) has been deermined o be sufficien in pracice. VII. SIMULATION RESULTS In his secion, he bi-error-rae BER) performance of rae R = /3 BCC s is evaluaed on an addiive whie Gaussian noise AWGN) channel using compuer simulaion. We consider firs biwise SBCC s wih wo idenical rae R cc = 2/3, memory m cc = 2, low complexiy 4-sae) RSC componen encoders. The generaor marix of he componen encoders is given by GD) = D + D 2 + D 2 + D + D 2. 6) The hree convoluional permuors P 0), P ), and P 2) used in he encoder were consruced randomly wih he same period T. We assumed ha ransmission consiss of an informaion sequence of lengh 50T and a ail of 2T zero ail bis. Thus we have a rae loss of 2.67%, i.e., he effecive rae is abou In he pipeline BCJR decoder, a window lengh of T and I = 00 decoding ieraions were used. The resuls are presened in Figure, where we view he effec of he period T of he convoluional permuors on he error performance as a funcion of he signal-o-noise raio SNR) E b /N 0. We see ha he performance of ieraive decoding improves dramaically as he permuor period increases, an effec equivalen o he inerleaver gain of urbo codes [8]. The SBCC achieves a BER of 0 5 a an E b /N 0 of 0.4dB wih permuor period T = 8000, which is abou db from he capaciy of he binary-inpu AWGN channel wih code rae We also sudied he performance of rae R = /3 blockwise BCC s. The ail-biing version of he encoder whose generaor marix is given in 6) was employed. The hree block permuors used in he encoder were chosen randomly wih he same size N. As above, he ransmission of 50 informaion blocks is erminaed wih 2 all-zero blocks. The parameers for decoding are he same as for he biwise SBCC case, A value of W = T was chosen for convenience, bu in a pracical implemenaion, a much smaller value of W can be chosen o minimize laency [6].

12 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION wih a separaion delay τ = N. The BER performance is shown in Figure 2, where we changed he size of he block permuors from N = 00 o Similar o he biwise case, he performance of blockwise BCC s improves as we increase he size of he block permuors. Furhermore, we see ha he performance of blockwise BCC s is close o he biwise case when he block permuor size equals he convoluional permuor period. Finally, he blockwise BCC was compared o a rae R = /3 urbo code wih 4-sae [, 5/7] ocal forma) componen encoders and permuor size 892. The urbo code exhibis an error floor a a BER of 0 6 and E b /N 0 = 0.5dB. By conras, he blockwise BCC s achieve a BER of 0 6 a E b /N 0 = 0.3dB wih permuor size N = 8000 and error floor did no show in he simulaion. These resuls sugges ha BCC s have good minimum disance properies. In he nex secion, we presen a disance analysis for he ensemble of BCC s ha confirms his observaion. Figure 3 shows he performance of he same blockwise BCC s for a coninuous pipeline decoder wihou any erminaion. The corresponding densiy evoluion hreshold a 0.98dB has been esimaed by racking he probabiliy densiy funcions of he decoder oupu LLR s wih Mone Carlo mehods, as described in [9]. Alhough a differen, proograph-based BCC ensemble [20] is considered in [9], he srucure of he compuaion ree and, consequenly, he asympoic hreshold are he same as for our biwise and blockwise ensembles 2. Already for permuor size N = 500 he blockwise BCC s achieve BER levels below 0 5 a an E b /N 0 ha is less han 0.02dB away from he esimaed hreshold. For larger permuors, like for BBCs [], i can be observed ha erminaed blockwise BCC s have beer performance and even ouperform he hresholds of coninuous BCC s. This again indicaes ha erminaed convoluional codes have beer hresholds han heir non-erminaed counerpars, as was shown in [2]. VIII. STATISTICAL ANALYSIS OF BRAIDED CONVOLUTIONAL CODES One of he mos imporan performance measures of a convoluional code is is minimum free disance d free, since is large SNR performance wih maximum likelihood decoding depends on d free. Also, wih ieraive decoding, a large d free proecs agains he appearance of an error floor a low BER s. In his secion, we describe a mehod o compue a lower bound on he free disance of BCC s wih sufficienly large overall consrain lengh. Using a numerical analysis for a randomized ensemble of BCC s, we obain a lower bound on d free ha grows linearly wih overall consrain lengh M as M goes o infiniy. A. Markov Permuors In [9], a sochasic device called a Markov permuor was inroduced o analyze he disance properies of LDPC convoluional codes. A Markov permuor is a ime-varying nonperiodic permuor wih minimal delay δ = and maximal 2 The hreshold has been esimaed o be a.0db in [20]. This value was improved o 0.98dB by improving he resoluion in he represenaion of he esimaed probabiliy densiy funcions. BER Permuor period 00 Permuor period 500 Permuor period 000 Permuor period E b /N 0 db) Fig.. Error performance of rae R = /3 erminaed sparsely braided convoluional codes on an AWGN channel. 0 0 Permuor size 00 Permuor size Permuor size 000 Permuor size 8000 Turbo code [ 5/7] permuor size Fig. 2. Error performance of rae R = /3 erminaed blockwise braided convoluional codes and urbo codes on an AWGN channel. delay =. I sores a fixed number of symbols M, i.e., he overall consrain lengh of he Markov permuor is M. To find a lower bound on free disance for he ensemble of BCC s based on Markov permuors, we define he sae of he Markov permuor as he number of s sored in he permuor. A each ime uni, he Markov permuor chooses one symbol from he sored symbols as is oupu symbol. The probabiliy ha a given sored symbol in he Markov permuor becomes he oupu symbol is /M. Based on hese assumpions, he probabiliy disribuions of he oupus and sae ransiions can be derived. In his fashion, he Markov permuor characerizes an ensemble of randomly chosen convoluional permuors wih overall consrain lengh M.

13 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 2 BER Permuor size 00 Permuor size 500 Permuor size 000 Permuor size 8000 Threshold u Muliple Markov Permuor u 0) ṽ 0) ṽ ) ṽ 2) Rae 2/3 Encoder Rae 2/3 Encoder 2 v 0) v ) v 2) MUX v 0 5 Fig. 4. Rae R = /3 BCC encoder wih a muliple Markov permuor E b /N 0 db) Fig. 3. Error performance of rae R = /3 coninuous blockwise braided convoluional codes on an AWGN channel. I follows ha he average delay of a symbol is given by i ) i = M. 62) M M i= This means ha a Markov permuor sores each inpu symbol an average of M ime insans in is memory. Noe ha, in conras o fixed convoluional permuors, where a symbol canno be held longer han he maximal delay, a Markov permuor can sore symbols, in principle, for an infinie ime.) Consider as an example he rae R = /3 BCC encoder in Figure 7, bu replace each convoluional permuor wih a Markov permuor having overall consrain lengh M/3. The bound o be derived below can be exended o generalized BCC s in a sraighforward manner.) A ime insan, = 0,,..., each permuor chooses randomly one symbol from among he M/3 symbols ha are sored in is memory and passes his symbol o he permuor oupu. The permuor P 0) replaces his symbol wih a new informaion symbol. The permuors P ) and P 2) replace heir oupus wih new pariy symbols v ) and v 2), respecively. The ensemble of BCC encoders wih Markov permuors can be sudied analyically o deermine an average disance specrum and, consequenly, a lower bound on free disance for BCC s. The problem involves solving a sysem of recursive equaions whose variables represen he pah weighs and he saes of he permuors and he componen encoders. However, his approach is quie difficul for numerical calculaion. To simplify he analysis, we replace he hree Markov permuors wih one muliple Markov permuor MMP) of overall consrain lengh M and mulipliciy 3 see Figure 4). By definiion, an MMP of mulipliciy k has k inpus and k oupus per ime insan. Iniially, he MMP sores M zero symbols. A each ime insan 0, he permuor chooses uniformly hree symbols ṽ 0), ṽ ), and ṽ 2) from among he M symbols in is memory. As shown in Figure 4, he permuor sends his hree-uple ṽ = ṽ 0), ṽ ), ṽ 2) ) ogeher wih he informaion symbol u o encoders and 2. Based on he inpus, he componen encoders calculae he pariy symbols. The code symbols v 0) = u, v ), and v 2) are hen fed o he MMP inpu, and code block v = v 0), v ), v 2) ) is sen over he channel. Consider he ensemble of BCC s using an MMP of mulipliciy 3, as shown in Figure 4. By definiion, he sae µ of he MMP a he -h ime insan is he number of s sored in is memory, and µ {0,,..., M}. 63) We assume componen encoder e has memory m e) cc, e {, 2}. Le σ e) denoe he sae of componen encoder e a he -h ime insan, where σ e) {0,,..., 2 me) cc }. 64) The composie sae of he wo componen encoders a ime is denoed σ = σ ), σ 2) ). Combining he saes of he MMP and he componen encoders, he sae of he BCC encoder is defined as µ, σ ). As shown in Figure 5, a super rellis for he encoder ensemble can be consruced for analyzing he sae ransiions during encoding. The branches of he super rellis are labeled wih u /ṽ v. The oupu block v of he encoder a ime and he composie sae of he wo componen encoders a ime + are funcions of he composie sae σ, he inpu symbol u, and he oupu of he MMP ṽ : v = Gσ, u,ṽ ), 65) σ + = Fσ, u,ṽ ). 66) The funcions of G ) and F ) depend on he componen encoders. The code symbols v are hen fed back o he MMP, and he nex sae of he MMP is given by µ + = µ + w H v ) w H ṽ ). 67) Condiioned on he permuor sae µ, we can find he probabiliy disribuion of he permuor oupu ṽ. From a populaion of n symbols, he number of ordered samples of size i ha can be formed wihou replacemen is given by [22] n) i nn ) n i + ) = n!/n i)!. 68) Thus, he oal number of ordered samples of he oupus from he mulipliciy-3 MMP wih overall consrain lengh M is

14 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 3 M 3 ). Among hem, here are µ ) M µ w H ṽ ) 3 w H ṽ )) ordered samples wih he same weigh number of s) as ṽ. Under he assumpion ha he oupu symbols are randomly seleced from he MMP, we have 0, if 3 w H ṽ ) > M µ or w H ṽ ) > µ ) ) ) Pṽ µ ) = µ M µ M, oherwise w H ṽ ) 3 w H ṽ ) 3 69) which follows from he fac ha he number of 0 s or s in x canno exceed he number of 0 s or s in sorage. This probabiliy disribuion is used in he nex secion o recursively calculae he average disance specrum of an ensemble of BCC s. B. Calculaion of he Average Disance Specrum In his secion, we analyze he average disance specrum of he codes in he ensemble of BCC s based on he Markov permuors described above. Since BCC s are linear, his specrum coincides wih he average weigh specrum of he codes in he ensemble. We assume ha iniially he BCC encoder is in he zero sae, i.e., µ 0 = 0, σ 0 = 0. Assume an informaion symbol u 0 = eners he encoder. Correspondingly, he MMP ransiions o he sae µ = and he componen encoders o a sae σ 0. The encoding process hen coninues from sae µ, σ ). Ulimaely, wih probabiliy, he BCC encoder will reurn o he zero sae µ l, σ l ) = 0,0) a some l-h ime insan. For he purpose of bounding he free disance, we are ineresed in he weigh disribuion of he encoder oupu sequence beween he wo ime insans when he encoder is in he zero sae. Le ād, i, l) = E[ad, i, l)] denoe he expecaion of he number of pahs wih codeword weigh d and informaion weigh i ha depar from he all-zero pah a ime insan 0 and remerge wih he all-zero pah a ime l, l. The se {ād, i, l)}, 0 d, i, l, is called he average exended weigh specrum AEWS) of he encoder. The AEWS is derived using a backward recursion on he super rellis. In he backward recursion, we mus consider runcaed pahs ha sar from non-zero saes, i.e., µ, σ ) 0,0), where he AEWS from sae µ, σ ) is denoed as āµ, σ ), d, i, l). Now we describe he backward recursion. As shown in Figure 5, we assume ha he encoder is in sae µ, σ ). Wih inpu u = {0, } and random oupus x from he MMP, several successive saes µ +, σ + ) are possible in a one sep ransiion. Wih u known, i follows direcly from 66) ha he ransiion probabiliy is Pσ σ + µ ) = Pṽ µ ), 70) where Pṽ µ ) is given by 69). All pahs saring from hese successor saes are exensions of he pahs passing hrough sae µ, σ ). In summary, he AEWS s from he successor saes µ +, σ + ) conribue o he AEWS from sae µ, σ ) in a probabilisic summaion. I follows ha āµ, σ ),d, i, l) = Pṽ µ ) u =0 ṽ 7) āµ +, σ + ), d wv ), i u, l ), where v, σ +, and µ + are given by 65), 66), and 67), respecively. Noe ha he codeword weighs, informaion weighs, and pah lenghs of he AEWS s from he successor saes mus be decreased o ake ino accoun he weighs on he ransiion branch. Fig µ, σ ) u = 0/ṽ v u = /ṽ v + Sae ransiions on a super rellis. µ +, σ + ) In he super rellis, he pah ha diverges from he all-zero pah is unique since i can be caused only by an informaion symbol u 0 = enering he encoder. Thus he probabiliy associaed wih his ransiion is uniy. Le µ, σ ) denoe he corresponding successor sae of he encoder, and d denoe he weigh of he ransiion from 0,0) o µ, σ ). Subsiuing hese values in 7), we obain ād, i, l) = āµ, σ ), d d, i, l ) 72) On he basis of he AEWS, he average weigh specrum AWS) is defined as l ād) = ād, i, l). 73) l= i= As in 73), if we sum over all i and l in 7), we obain he following sysem of recursive equaions for he AWS from sae µ, σ ): āµ, σ ), d) = Pṽ µ ) u =0 ṽ āµ +, σ + ), d wv )). 74) Finally, he AWS can be compued using following seps: ) Se he overall consrain lengh M and generae he super rellis {µ, σ ) µ +, σ + )} according o 66) and 67) for he componen encoders. 2) Find µ, σ ) and d. 3) Se he boundary condiions { d = 0 ā0,0), d) = 75) 0 d and āµ, σ), d) = 0, d < 0. 76) For µ = 0 o M 4) For d = 0 o d max For µ = 0 o M For all σ, calculae āµ, σ), d) based on 74) and boundary condiions;

15 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 4 End End ād) = āµ, σ ), d d ); C. A Lower Bound on Free Disance Afer deriving he AWS for given componen encoders wih consrain lengh M, a free disance lower bound can be obained using he usual Gilber-Varshamov see, e.g., [0]) argumen, as saed in he following heorem. Theorem : If ˆd is he larges ineger value of δ ha saisfies δ ād) <, 77) d= hen a leas one code in he ensemble mus have free disance no less han ˆd. We calculae ˆd, and i follows from Theorem ha here exiss a leas one code in he ensemble for which d free is lower bounded by ˆd. The free disance bound implied by 77) is a funcion of he componen encoders and he overall consrain lengh M of he MMP. Recall ha in Secion III we showed ha a BCC encoder wih hree convoluional permuors of widh T has an overall consrain lengh of M = 5T )/2+. Solving for ˆd for differen values of M hen gives us a numerical lower bound on d free. We plo ˆd as a funcion of M, 0 < M 000, in Figure 6. Three rae R = /3 SBCC s wih idenical RSC componen encoders of memory m cc = 2, 3, and 4 were considered in he calculaion. We see ha he free disance bounds exhibi essenially linear growh as a funcion of he overall consrain lengh M of he MMP. Free disance lower bound m cc =2 m cc =3 m cc = Permuor overall consrain lengh Fig. 6. Lower bounds on he free disance of BCC s wih differen componen encoders. Alhough he numerical resuls ploed in Figure 6 exend only o M = 000, hey provide srong evidence o conjecure ha asympoically, as M goes o infiniy, he raio of he free disance of hese rae R = /3 BCC s o heir overall consrain lengh is lower bounded by γ bc, where γ bc is he average slope of he corresponding curves in Figure 6. Values of γ bc derived from Figure 6 are given in Table I for BCC s wih rae R cc = 2/3 componen encoders of memory m cc = 2, 3, and 4. The generaor polynomials are denoed in ocal form. TABLE I FREE DISTANCE BOUND FOR RATE R = /3 BCC S WITH DIFFERENT COMPONENT ENCODERS. Componen encoder memory Generaor marix Asympoic raio γ bc m cc = 2 0 4/7 0 5/ m cc = 3 0 7/5 0 3/ m cc = / / I is ineresing o compare his bound wih he Cosello bound [23] on he free disance of he ensemble of convoluional codes. The Cosello bound saes ha here exiss rae R = b/c convoluional encoders of memory m wih free disance lower bounded by he following inequaliy ) d free cm R log 2 [2 R ] + O log2 m, 78) m which can also be wrien as d free bm log 2 [2 R ] + O log2 m m ). 79) Since he overall consrain lengh M of a convoluional encoder is upper bounded by he inequaliy M bm, we can wrie log2 M d free M log 2 [2 R ] + O M ). 80) Asympoically, as M goes o infiniy, we have d free γ cos M, where γ cos = /log 2 [2 R ]). In paricular, for R = /3, γ cos = Noe ha he coefficiens γ bc for BCC s are roughly a facor of 2 less han he raio γ cos in he Cosello bound. This is consisen wih he ypical reducion in disance growh rae observed when comparing Gallager s minimum disance bound [24] for LDPC block codes o he Gilber-Varshamov [0] minimum disance bound for he ensemble of block codes. IX. CONCLUSIONS In his paper, we proposed a new class of urbo-like codes, namely, braided convoluional codes, ha are suiable for high speed coninuous daa ransmission. We presened a consrucion mehod for ighly and sparsely braided convoluional codes. For applicaions involving packeized daa, we also inroduced a blockwise encoding srucure. Compuer simulaion resuls show ha sparsely braided convoluional codes achieve good convergence performance wih ieraive decoding. Furhermore, he simulaion resuls suggess ha braided convoluional codes have good disance properies, in conras o convenional urbo codes. This observaion was heoreically confirmed by an analysis of braided convoluional

16 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 5 codes using a saisical Markov permuor model. For his model, we showed ha braided convoluional codes have a free disance ha grows linearly wih overall consrain lengh, i.e., braided convoluional codes are asympoically good. ACKNOWLEDGMENT Some of he resuls in Secion VIII were obained wihin he 2004 IMA Summer Program in Coding and Crypology a he Universiy of Nore Dame. We would like o hank Bodo Blume, Ali Pusane, and Zeying Wang for heir conribuions. Furhermore, we would like o hank Marcos Tavares for he compuaion of he densiy evoluion hreshold shown in Figure 3. REFERENCES [] A. J. Felsröm, M. Lenmaier, D. V. Truhachev, and K. S. Zigangirov, Braided block codes, IEEE Trans. Inform. Theory, vol. 55, no. 6, pp , Jun [2] M. Lenmaier, D. V. Truhachev, and K. S. Zigangirov, Ieraive decodable sliding codes on graphs, in Proceedings of ACCT-VIII, S- Peersburg, Russia, Sep. 2002, pp [3] D. V. Truhachev, M. Lenmaier, and K. S. Zigangirov, On braided block codes, in Proceedings of he IEEE Inernaional Symposium on Informaion Theory, Yokohama, Japan, Jun. 2003, p. 32. [4] P. Elias, Error free coding, IRE Transacions on Informaion Theory, vol. 4, no. 4, pp , Sep [5] M. Sipser and D. A. Spielman, Expander codes, IEEE Trans. Inform. Theory, vol. 42, no. 6, pp , Nov [6] G. Zemor, On expander codes, IEEE Trans. Inform. Theory, vol. 47, no. 2, pp , Feb [7] W. Zhang, M. Lenmaier, K. Sh. Zigangirov, and D. J. Cosello, Jr., Braided convoluional codes, in Proceedings of he IEEE Inernaional Symposium on Informaion Theory, Adelaide, Ausralia, Sep. 2005, pp [8] R. M. Tanner, A recursive approach o low complexiy codes, IEEE Trans. Inform. Theory, vol. 27, no. 5, pp , Sep. 98. [9] K. Engdahl, M. Lenmaier, and K. S. Zigangirov, On he heory of low-densiy convoluional codes, in Proceedings of he Symposium on Applied Algebra, Algebraic Algorihms, and Error-Correcing Codes, Honolulu, Hawaii, June 999, pp [0] R. Johannesson and K. S. Zigangirov, Fundamenals of convoluional coding. New York, NY: IEEE Press, 999. [] A. J. Felsröm and K. S. Zigangirov, Periodic ime-varying convoluional codes wih low-densiy pariy-check marices, IEEE Trans. Inform. Theory, vol. 45, no. 5, pp , Sep [2] S. Lin and D. J. Cosello, Jr., Error conrol coding, 2nd ed. Upper Saddle River, NJ: Pearson Prenice Hall, [3] A. Pusane, A. J. F. Felsrom, A. Sridharan, M. Lenmaier, K. S. Zigangirov, and D. Cosello, Jr., Implemenaion aspecs of ldpc convoluional codes, IEEE Trans. Commun., vol. 56, no. 7, pp , July [4] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Opimal decoding of linear codes for minimizing symbol error rae, IEEE Trans. Inform. Theory, vol. 20, no. 2, pp , Mar [5] S. Benedeo, G. Monorsi, D. Divsalar, and F. Pollara, A sof-inpu sof-oupu maximum a poseriori map) module o decode parallel and serial concaenaed codes, JPL TDA Progress Repor, vol. 42, no. 27, pp. 20, Nov [6] A. J. Vierbi, An inuiive jusificaion of he MAP decoder for convoluional codes, IEEE J. Selec. Areas Commun., vol. 6, no. 2, pp , Feb [7] R. Horn and C. R. Johnson, Topics in marix analysis. Cambridge, UK: Cambridge Universiy Press, 99. [8] S. Benedeo and G. Monorsi, Unveiling urbo codes: some resuls on parallel concaenaed coding schemes, IEEE Trans. Inform. Theory, vol. 42, no. 2, pp , Mar [9] M.B.S. Tavares, M. Lenmaier, G.P. Feweis, and K.Sh. Zigangirov, Asympoic disance and convergence analysis of braided proograph convoluional codes, in Proceedings of he 46h Annual Alleron Conference on Communicaion, Conrol, and Compuing, Monicello, IL, Sep [20] M.B.S. Tavares, M. Lenmaier, K.Sh. Zigangirov, and G.P. Feweis, LDPC convoluional codes based on braided convoluional codes, in Proceedings of he IEEE Inernaional Symposium on Informaion Theory, Torono, Canada, July [2] M. Lenmaier, A. Sridharan, K. S. Zigangirov, and D. J. Cosello, Jr., Terminaed LDPC convoluional codes wih hresholds close o capaciy, in Proceedings of he IEEE Inernaional Symposium on Informaion Theory, Adelaide, Ausralia, Sep. 2005, pp [22] H. Sark and J. W. Woods, Probabiliy, Random Processes, and Esimaion Theory for Engineers, 2nd ed. Upper Saddle River, NJ: Prenice Hall, 994. [23] D. J. Cosello, Jr., Free disance bounds for convoluional codes, IEEE Trans. Inform. Theory, vol. 20, no. 3, pp , May 974. [24] R. Gallager, Low-Densiy Pariy-Check Codes. Cambridge, MA: MIT Press, 963. Wei Zhang was born on Augus 3, 975 in Shijiazhuang, P.R.China. He received h B.E. degree in 998 from Xidian Universiy, Xi an, P. R. China, M.S. degree in 998 from Tsinghua Universiy, Beijing, P. R. China, and Ph.D. degree in 2006 from Universiy of Nore Dame, Nore Dame, IN. Since 2006 he is wih he Corporae Research and Developmen Deparmen, QUAL- COMM Incorporaed, working on 3GPP sandards and advanced wireless receiver design. His research ineress include digial ransmission, channel coding, and wireless communicaions. Michael Lenmaier received he Dipl.-Ing. degree in elecrical engineering from Universiy of Ulm, Ulm, Germany in 998, and he Ph.D. degree in elecommunicaion heory from Lund Universiy, Lund, Sweden in He hen worked as a Pos-Docoral Research Associae a Universiy of Nore Dame, Indiana, and a Universiy of Ulm, Germany. From 2005 o 2007 he was wih he Insiue of Communicaions and Navigaion of he German Aerospace Cener DLR) in Oberpfaffenhofen, working on high resoluion channel esimaion echniques for mulipah miigaion in saellie navigaion receivers. Since January 2008 he is a senior researcher and lecurer a he Vodafone Chair Mobile Communicaions Sysems a Dresden Universiy of Technology TU Dresden). His research ineress include design and analysis of coding sysems, graph based ieraive algorihms and Bayesian mehods applied o decoding, deecion and esimaion. Daniel J. Cosello, Jr. was born in Seale, WA, on Augus 9, 942. He received he B.S.E.E. degree from Seale Universiy, Seale, WA, in 964, and he M.S. and Ph.D. degrees in Elecrical Engineering from he Universiy of Nore Dame, Nore Dame, IN, in 966 and 969, respecively. Dr. Cosello joined he faculy of he Illinois Insiue of Technology, Chicago, IL, in 969 as an Assisan Professor of Elecrical Engineering. He was promoed o Associae Professor in 973, and o Full Professor in 980. In 985 he became Professor of Elecrical Engineering a he Universiy of Nore Dame, Nore Dame, IN, and from 989 o 998 served as Chair of he Deparmen of Elecrical Engineering. In 99, he was seleced as one of 00 Seale Universiy alumni o receive he Cenennial Alumni Award in recogniion of alumni who have displayed ousanding service o ohers, excepional leadership, or uncommon achievemen. In 999, he received a Humbold Research Prize from he Alexander von Humbold Foundaion in Germany. In 2000, he was named he Leonard Beex Professor of Elecrical Engineering a Nore Dame. Dr. Cosello has been a member of IEEE since 969 and was eleced Fellow in 985. Since 983, he has been a member of he Informaion Theory Sociey Board of Governors, and in 986 he served as Presiden of he BOG. He has also served as Associae Edior for Communicaion Theory for he IEEE Transacions on Communicaions, Associae Edior for Coding Techniques for he IEEE Transacions on Informaion Theory, and Co-Chair of he IEEE Inernaional Symposia on Informaion Theory in Kobe, Japan 988), Ulm, Germany 997), and Chicago, IL 2004). In 2000, he was seleced by he IEEE Informaion Theory Sociey as a recipien of a Third-Millennium Medal. He was co-recipien of he 2009 IEEE Donald G. Fink Prize Paper Award, which recognizes an ousanding survey, review, or uorial paper in any IEEE publicaion issued during he previous calendar year.

17 IEEE TRANSACTIONS ON INFORMATION THEORY, ACCEPTED FOR PUBLICATION 6 Dr. Cosello s research ineress are in he area of digial communicaions, wih special emphasis on error conrol coding and coded modulaion. He has numerous echnical publicaions in his field, and in 983 he co-auhored a exbook eniled Error Conrol Coding: Fundamenals and Applicaions, he 2nd ediion of which was published in Kamil Sh. Zigangirov was born in he U.S.S.R. in 938. He received he M.S. degree in 962 from he Moscow Insiue for Physics and Technology,Moscow, U.S.S.R., and he Ph.D. degree in 966 from he Insiue of Radio Engineering and Elecronics of he U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. From 965 o 99, he held various research posiions a he Insiue for Problems of Informaion Transmission of he U.S.S.R. Academy of Sciences, Moscow, firs as a Junior Scienis, and laer as a Main Scienis. During his period, he visied several universiies in he Unied Saes, Sweden, Ialy, and Swizerland as a Gues Researcher. He organized several symposia on informaion heory in he U.S.S.R. In 994, he received he Chair of Telecommunicaion Theory a Lund Universiy, Lund, Sweden. From 2003 o 2009, he has been a Visiing Professor a he Universiy of Nore Dame, Nore Dame, IN, Dresden Technical Universiy, Dresden, Germany, and a he Universiy of Albera, Edmonon, AB, Canada. His scienific ineress include informaion heory, coding heory, deecion heory, and mahemaical saisics. In addiion o papers in hese areas, he published a book on sequenial decoding of convoluional codes in Russian) in 974. Wih R. Johannesson, he coauhored he exbook Fundamenals of Convoluional Coding Piscaaway, NJ: IEEE Press, 999). His book Theory of CDMA Communicaion was published by IEEE Press in 2004.