Error-Trellis State Complexity of LDPC Convolutional Codes Based on Circulant Matrices

Error -T rellis State Complexit y o f LDPC Con v olutional Codes Based on Circulant Matrices Masato T ajima Dept of Inte llect. Inform. Systems Eng. University of T oyama T oyama 930- 8555, Japa n tajima@eng.u- toyama.ac.jp K oji Okino Inform ation T echno logy Cente r University of T oyama T oyama 930- 8555, Japa n okino@itc.u -toyama.ac.jp T akash i Miyagoshi Dept of Intellect. Info rm. Systems Eng. University of T oyama T oyama 930- 8555, Japa n miyagosi@eng .u-toyama.ac .jp Abstract — Let H ( D ) be the parity-check matrix of an LDPC con volutional code corr esponding to the parity-check matrix H of a QC code obtained using the method of T an ner et al. W e see that the entries in H ( D ) are all monomials and several rows (columns) ha ve monomial factors. Let us cyclically shift the r ows of H . Th en the parity-check matrix H ′ ( D ) corresponding to th e modified ma trix H ′ defines another conv olutional code. Howe ver , its free distance is lower -bounded by the minimum dist ance of the original QC code. Also, each row (column) of H ′ ( D ) has a factor different f rom th e one in H ( D ) . W e sho w th at the state- space complexity of the err or-tr ellis associated with H ′ ( D ) can be significantly reduced by controlling the row shifts applied t o H with th e error -correction capability being preser ved. I . I N T RO D U C T I O N In this pa per, we assume that the under lying field is F = GF (2) . Let G ( D ) be a polyno mial g enerator matrix for an ( n 0 , k 0 , ν ) conv olutio nal code C with mem ory ν . Denote by H ( D ) a corr espondin g p arity-che ck m atrix. Both G ( D ) and H ( D ) are assumed to be canonica l [4], [5]. In this case, the code-trellis mod ule associated with G ( D ) and the erro r-trellis modules [6] associated with the syndrom e f ormer H T ( D ) ( T means tr anspose) have 2 ν states, wh ere the obvious re alization of G ( D ) and the adjoin t-obvious realization [3] of H T ( D ) are assumed, respecti vely . Ariel and Snyders [ 1] presented a constru ction of an erro r-trellis b ased on the scalar check matrix deri ved f rom H ( D ) . They showed that when some ( j th) “co lumn” of H ( D ) has a factor D l (i.e., the j th co lumn is n ot “delay free”), there is a possibility th at state-spac e reduction can be realized . Being motivated by their work, we also examined the same case. W e took notice of a syndr ome generation process. The time- k er ror e k and the time- k syn - drome ζ k are co nnected with th e relatio n ζ k = e k H T ( D ) . From this relation , we noticed [8] that the transforma tion e ( j ) k → D l e ( j ) k = e ( j ) k − l is equiv alent to dividing the j th colum n of H ( D ) b y D l . That is, r eduction can be acco mplished by shifting the “subsequ ence” { e ( j ) k } of the orig inal e rror-path. On the other h and, consider the parity- check m atrix H 1 ( D ) △ =  D 2 D 2 1 1 1 + D + D 2 0  . (1) Since H 1 ( D ) is canonica l and all the column s are delay free, any further reduction seems t o be impossible. In fact, it follo ws from T heorem 1 of [1] that the d imension d 1 of the state space of the er ror-trellis based on H T 1 ( D ) is 4 . Howe ver , a correspo nding generator matrix is given by G 1 ( D ) △ = (1 + D + D 2 , 1 , D 3 + D 4 ) . Note that the third column of G 1 ( D ) h as a factor D 2 . ( Remark: It suffices to d ivide the third colum n by D 2 in ord er to obtain a reduced cod e-trellis.) This fact implies that a reduced e rror-trellis can be con structed [ 1], [8] (i.e., state-sp ace r eduction can be realized ). Th en con sider the reciproca l du al encoder [4] ˜ H 1 ( D ) △ =  1 1 D 2 D 2 1 + D + D 2 0  . (2) Note that the third column of ˜ H 1 ( D ) has a factor D 2 . According ly , d ividing the thir d column of ˜ H 1 ( D ) by D 2 , we can constru ct an err or-trellis with 4 states ( i.e., ˜ d 1 = 2 ) [1], [8]. He re, notice tha t each er ror-path in th e error-trellis based on H T 1 ( D ) can be represented in time-reversed order u sing the error-trellis b ased on ˜ H T 1 ( D ) . Hence, a factor D 2 in the column of ˜ H 1 ( D ) corre sponds to backward-shiftin g by two time units (i.e., D − 2 ) in terms of the original H 1 ( D ) . Actually , by “multiply ing” the third colu mn of H 1 ( D ) by D 2 , we hav e H ′ 1 ( D ) △ =  D 2 D 2 D 2 1 1 + D + D 2 0  . (3) Note that this matrix can be red uced to an equiv ale nt canonical parity-ch eck matr ix H ′′ 1 ( D ) △ =  1 1 1 1 1 + D + D 2 0  (4) by dividing the first “row” by D 2 . Hence, the dimension d 1 can be red uced to 2 . ( R emark: This fact ca nnot be derived from the r esults of [1].) It follows f rom the ab ove argum ent that th ere is a po ssibility tha t a red uced err or-trellis c an be constructed not only using forward-shifte d error su bsequence s but also u sing backward-sh ifted err or subsequences. Now , we remark th at a p arity-ch eck matrix H ( D ) with the form d escribed above appears in [10] . T anner et al. [10] presented a class of algeb raically co nstructed quasi-cyclic (QC) LDPC c odes and th eir c on volutional counterp arts. It is stated that the conv o lutional codes o btained in the paper typically have large constraint length s and therefo re the use of trellis-based decod ing is not feasible. Howev e r , the parity- check m atrices of LDPC conv olutional co des p roposed b y T anner et al. ha ve m onom ial entr ies. Accordingly , the above- mentioned state-space reductio n metho d can be directly ap- plied to those par ity-check ma trices. Then we intended to ev aluate the state-space co mplexity o f the err or-trellis o f an LDPC co n volutional code which ap pears in [10 ]. W e show that the overall co nstraint len gth (abbreviated as “OCL ” in this paper) of the parity-ch eck matrix which specifies an LDPC c onv olution al cod e can be significantly reduce d with the error-correction capability of the conv o lutional code b eing preserved. I I . P R E L I M I N A R I E S A. Err or-T r ellis Construction Using Shifted Err or/Synd r ome Subsequ ences Let H ( D ) be a p arity-chec k matrix for an ( n 0 , k 0 ) co n volu- tional co de C . In th is pape r , we con sider the er ror-trellis based on the synd rome fo rmer H T ( D ) . In this case, the adjoint- obvious realizatio n of H T ( D ) is assumed unless other wise specified. Deno te by e k = ( e (1) k , · · · , e ( j ) k , · · · , e ( n 0 ) k ) and ζ k = ( ζ (1) k , · · · , ζ ( i ) k , · · · , ζ ( r ) k ) the time- k error and the time- k syndrom e, respectively , where r = n 0 − k 0 . T hen we have the relation: ζ k = e k H T ( D ) . (5) Assume that the i th r ow of H ( D ) has the f orm  D l i h ′ i 1 ( D ) D l i h ′ i 2 ( D ) . . . D l i h ′ in 0 ( D )  , (6) where l i ≥ 1 . L et H ′ ( D ) be the mo dified version of H ( D ) with the i th row being replaced by  h ′ i 1 ( D ) h ′ i 2 ( D ) . . . h ′ in 0 ( D )  . (7) Defining ζ ′ ( i ) k as ζ ( i ) k △ = D l i ζ ′ ( i ) k = ζ ′ ( i ) k − l i , we set ζ ′ k △ = ( ζ (1) k , · · · , ζ ′ ( i ) k , · · · , ζ ( r ) k ) . Then we h av e ζ ′ k = e k H ′ T ( D ) . (8) Similarly , assume that the j th column of H ( D ) has the form  D l j h ′ 1 j ( D ) D l j h ′ 2 j ( D ) . . . D l j h ′ r j ( D )  T , (9) where l j ≥ 1 . ( Remark: H ( D ) is n ot basic [2] and then not canonical. ) Let H ′ ( D ) be the mo dified version of H ( D ) with the j th column being replace d b y  h ′ 1 j ( D ) h ′ 2 j ( D ) . . . h ′ r j ( D )  T . (10) Also, le t e ′ k △ = ( e (1) k , · · · , e ′ ( j ) k , · · · , e ( n 0 ) k ) , wh ere e ′ ( j ) k △ = D l j e ( j ) k = e ( j ) k − l j . Then we have ζ k = e ′ k H ′ T ( D ) . (11) Noting these relations [8], [9], in the case where the i th row of H ( D ) h as a factor D l i , by shiftin g th e i th synd rome subsequen ce by l i time units, whereas in the case where the j th colu mn of H ( D ) has a factor D l j , by shifting the j th error subsequen ce b y l j time units, w e can construct an er ror trellis with reduc ed num ber of states. In the following, we call factoring out D l from a row o f H ( D ) a nd from a colum n of H ( D ) “row operation” and “colu mn operation”, respectiv ely . (00) (01) (10) (11) ζ (1) 1 ζ (2) 1 =01 ζ (1) 2 ζ (2) 2 =10 ζ (1) 3 ζ (2) 3 =01 ζ (1) 4 ζ (2) 4 =10 ζ (1) 5 ζ (2) 5 =00 110 011 001 111 101 000 100 101 101 000 010 011 000 010 010 001 t=1 t=2 t=3 t=4 t=5 100 010 000 101 001 011 110 100 111 001 100 001 011 110 000 010 111 101 110 011 100 111 111 110 t=0 Fig. 1. Example error-tr ellis based on H T 2 ( D ) . B. Err or-T r ellis Co nstruction Based o n a Recipr oca l Dual Encoder Consider the (3 , 1 , 2) conv olutio nal code C 2 with c anonical parity-ch eck matr ix giv e n b y H 2 ( D ) △ =  D 0 1 1 1 + D 0  . (12) In this su bsection, we d iscuss using th is specific example. Howe ver, the argument is entirely general. Since the columns of H 2 ( D ) are d elay free, the dimension d 2 of the state space of the error-trellis based on the syndrom e fo rmer H T 2 ( D ) is giv en by 2 (see Th eor em 1 o f [1] ). Fig.1 shows an error-trellis constructed based on H T 2 ( D ) u sing the conventional method [6]. It is assumed th at a transmitted cod e-path is ter minated in the all-zero state at t = 4 an d the correspo nding received d ata is giv en by z = z 1 z 2 z 3 z 4 z 5 = 010 0 11 000 0 01 000 , where z 5 = 0 00 is the im aginary r eceived d ata. Let z be the input of th e syndrome f ormer H T 2 ( D ) , th en we ha ve the syndrom e sequen ce ζ = ζ 1 ζ 2 ζ 3 ζ 4 ζ 5 = 0 1 10 0 1 10 0 0 . The overall erro r-trellis is con structed by co ncatenating five error-trellis mo dules co rrespond ing to ζ k . Note th at the er ror- trellis in Fig.1 is terminated in state (0 0) at t = 5 , which correspo nds to the final syn drome -former state σ 5 = (00) . From Fig.1 (no te that e 5 = 00 0 ), we have fou r ad missible error-paths: e p 1 = 010 011 0 00 001 000 (13) e p 2 = 010 101 1 01 000 000 (14) e p 3 = 100 000 1 00 000 000 (15) e p 4 = 100 110 0 01 001 000 . (16) Next, consider the recipro cal d ual encoder ˜ H 2 ( D ) △ =  1 0 D D 1 + D 0  . (17) Let ˜ z = z 4 z 3 z 2 z 1 z 0 = 00 1 0 00 011 010 00 0 b e the time-reversed r eceiv ed d ata o f { z k } 4 k =1 augmen ted with the imaginary data z 0 = 000 . If ˜ z is inputted to the syndr ome former ˜ H T 2 ( D ) , then the time-reversed syn drome sequence ˜ ζ = ζ 5 ζ 4 ζ 3 ζ 2 ζ 1 = 00 10 0 1 1 0 01 is o btained. The t=0 (00) (01) (10) (11) ζ (1) 5 ζ (2) 5 =00 ζ (1) 4 ζ (2) 4 =10 ζ (1) 3 ζ (2) 3 =01 ζ (1) 2 ζ (2) 2 =10 ζ (1) 1 ζ (2) 1 =01 000 001 010 011 100 101 110 111 100 101 110 111 000 001 011 111 100 111 000 001 010 010 011 111 100 000 110 000 010 010 100 110 110 t=1 t=2 t=3 t=4 t=5 101 001 001 011 101 011 101 Fig. 2. Error-tre llis based on ˜ H T 2 ( D ) . correspo nding err or-trellis is shown in Fig.2, wher e the trellis is terminated in state (00) , which co rrespon ds to the fin al syndrom e-form er state ˜ σ 5 = (00) . From Fig.2 , we ha ve four admissible error-paths: ˜ e q 1 = 001 000 0 11 010 000 ( 18) ˜ e q 2 = 000 101 1 01 010 000 ( 19) ˜ e q 3 = 000 100 0 00 100 000 ( 20) ˜ e q 4 = 001 001 1 10 100 000 . ( 21) Compare the se erro r-paths with th ose in Fig.1. W e o bserve that each err or-path in Fig.2 (re stricted to the section [0 , 4] ) is represented in Fig.1 in time-reversed order . That is, the original error-paths can be r epresented using the erro r-trellis associated with the corre sponding r eciprocal dual encod er . On th e o ther hand, d ividing the third column of ˜ H 2 ( D ) by D , we have the r educed canon ical p arity-check matrix ˜ H ′ 2 ( D ) △ =  1 0 1 D 1 + D 0  . (22) In this case [1 ], [8], erro r-paths associated with ˜ H T 2 ( D ) can be r epresented usin g the error-trellis constructed based on ˜ H ′ T 2 ( D ) . Note that a factor D l in th e colum n of ˜ H ( D ) correspo nds to backward-shif ting by l time units (i.e., D − l ) in terms of the original H ( D ) . This observation implies that error-trellis state-space re duction can be equ ally accomplishe d using backward- shifted err or subsequences. I I I . Q C C O D E S A N D C O R R E S P O N D I N G L D P C C O N V O L U T I O N A L C O D E S A. LDPC Con volutiona l Codes Ba sed on Cir culan t Matrices Each circulant in t he parity-check matrix of a QC block co de can be specified by a unique polynom ial; the polyno mial r epre- sents the entries in the first co lumn of the cir culant matr ix. For example, a circulant matrix whose first column is [1 1 1 0 1 0] T is rep resented by the p olynom ial 1 + D + D 2 + D 4 . Using this correspo ndence, an LD PC conv o lutional cod e is constru cted based on a parity- check matrix H of a giv en QC co de [10]. For example [1 0], let H =   I 1 I 2 I 4 I 8 I 16 I 5 I 10 I 20 I 9 I 18 I 25 I 19 I 7 I 14 I 28   (23) be the parity -check matr ix o f a (15 5 , 6 4) QC c ode ( m = 3 1 ), where I x is a 31 × 31 identity matrix with rows shifted cyclically to the left by x positions. From H we obtain the following p arity-check matrix with polyn omial entr ies: H ( D ) =   D D 2 D 4 D 8 D 16 D 5 D 10 D 20 D 9 D 18 D 25 D 19 D 7 D 14 D 28   . (24) ( Remark: It is stated [10] that the LDPC co n volutiona l code is obtained b y unwrapping the con straint graph (i.e., T anner graph) of the QC cod e.) Note that the polynomials in H ( D ) are all mono mials. In this pape r , we discuss exclusiv ely using th is specific example. Howe ver, the argum ent is entirely general. B. Reor dering Rows of H an d the Corr espond ing H ( D ) Again, consid er th e p arity-chec k matrix H of th e (15 5 , 6 4) QC co de. Let us cyclically shift the first block o f m = 31 rows above by one p osition, th e mid dle bloc k of 31 rows by fi ve positions, an d the last block of 31 rows by 2 5 po sitions. The resulting matrix is g iv en by H ′ =   I 0 I 1 I 3 I 7 I 15 I 0 I 5 I 15 I 4 I 13 I 0 I 25 I 13 I 20 I 3   . ( 25) Clearly , the QC blo ck code and its associated co nstraint graph are unaffected by th ese row shif ts. Howe ver, the co n volutional code obtained based on the above p rocedur e has the parity- check matrix H ′ ( D ) =   1 D D 3 D 7 D 15 1 D 5 D 15 D 4 D 13 1 D 25 D 13 D 20 D 3   . (26) W e see that H ′ ( D ) is not eq uiv alen t to H ( D ) . T wo co n- volutional codes specified by H ( D ) a nd H ′ ( D ) are in fact different. W e also remark that H ( D ) and H ′ ( D ) h av e dif fe rent monom ial entries and according ly , wh en row/column factors are factored out, the resulting matrices h av e different OCLs. On th e other hand , we have the f ollowing importan t fact [10]: Pr operty: The LDPC co n volutional codes ob tained by un- wrapping the con straint graph of th e QC cod es have th eir free distance d f r ee lower -b ounde d by the minimum distance d min of the cor respondin g QC code. It is shown tha t the QC cod e associated with H has a minimum distance d min = 20 . Then d f r ee of the c on volutional code C specified by H ( D ) is lo wer-boun ded by d min = 2 0 . (It is con jectured that C h as a free d istance d f r ee of 24 [10 ].) From the above property , we also have d ′ f r ee ≥ d min = 20 , where d ′ f r ee is the free distan ce of th e con volutional code C ′ specified b y H ′ ( D ) . In gen eral, let H ′ ( D ) be the parity -check matrix associated with H ′ , where H ′ is the parity-ch eck matrix obtained by apply ing cyclic shifts to the rows of eac h block of the or iginal H . Above observations imply th at the OCL of H ′ ( D ) can be contro lled to some extent with its free d istance d ′ f r ee being lower -b ounde d b y th e m inimum distance d min of the QC cod e specified by H . I V . R E D U C T I O N O F O V E R A L L C O N S T R A I N T L E N G T H A. Row/Column Operations a nd Their Equiva lent Rep r esen- tation Again, take the p arity-check matrix H gi ven by Eq.(23 ). Here, let u s cyclically shift the first b lock of 31 rows above by o ne po sition. Then the first block [ I 1 I 2 I 4 I 8 I 16 ] ch anges to [ I 0 I 1 I 3 I 7 I 15 ] . That is, the subscr ipt numb er of each entry decre ases by 1 . A ccording to th is chan ge in H , the first row o f H ( D ) chan ges from [ D D 2 D 4 D 8 D 16 ] to [1 D D 3 D 7 D 15 ] . In terms o f the po we r o f D , th is change is expressed as [1 2 4 8 1 6] → [0 1 3 7 15] . In general, we observe that each entry decr eases by one (m odulo 31 ) when we cyclically shift th e rows above by one position. Con tinuing this proced ure ( Remark: it is ass u med that row operations ha ve been don e), we see that the first row of H ( D ) cor respond s to one of the following five patterns in terms of the power of D :            P 1 = [0 1 3 7 1 5 ] P 2 = [30 0 2 6 14] P 3 = [28 29 0 4 1 2 ] P 4 = [24 25 27 0 8] P 5 = [16 17 19 23 0 ] . (27) Similarly , the second and th ird rows of H ( D ) are r epresented by            Q 1 = [0 5 15 4 13 ] Q 2 = [26 0 10 30 8] Q 3 = [16 21 0 20 29] Q 4 = [27 1 11 0 9 ] Q 5 = [18 23 2 22 0] and            R 1 = [0 25 13 2 0 3] R 2 = [6 0 19 26 9 ] R 3 = [18 12 0 7 2 1 ] R 4 = [11 5 24 0 1 4 ] R 5 = [28 22 10 17 0 ] , (28) respectively . Hen ce, wh en we app ly cyclic shifts to the r ows of each blo ck of the original H , the resulting H ( D ) can be specified by a p attern [ P i , Q j , R k ] T . For example, consider the pattern [ P 1 , Q 1 , R 3 ] T . The c orrespon ding H ( D ) (cf. Eq.(24) ) is given by H ( D ) =   1 D D 3 D 7 D 15 1 D 5 D 15 D 4 D 13 D 18 D 12 1 D 7 D 21   . (29) Since row operatio ns have b een don e, let us app ly column operation s. Th en we have H ′ ( D ) =   1 1 D 3 D 3 D 2 1 D 4 D 15 1 1 D 18 D 11 1 D 3 D 8   . (30) W e see tha t this is equivalent to the transfo rmation from S =   0 1 3 7 15 0 5 15 4 13 18 12 0 7 21   (31) 0 1 2 3 4 5 6 7 30 35 40 45 50 55 60 65 70 75 80 85 90 frequency overall constraint length (μ) Fig. 3. Overa ll constraint length of a reduced H ′ ( D ) . 0 1 2 3 4 5 6 7               frequency overall constraint length (η) Fig. 4. Overa ll constraint length of a reduced ˜ H ′ ( D ) . to S ′ =   0 0 3 3 2 0 4 15 0 0 18 11 0 3 8   , (32) where subtraction is perf ormed in each column of S in or der that the minimum is equal to zero. From S ′ , the OCL of the reduced H ′ ( D ) is obtained as 3 + 15 + 18 = 3 6 . B. Reduction of Overall Constraint Length: Sear ch Results As we h av e seen in the pr evious sectio n, there are 5 × 5 × 5 = 1 25 patterns in total. By apply ing column operations to each p attern, we examined the OCL µ of the cor respond ing reduced pa rity-check ma trix H ′ ( D ) . The r esult is shown in Fig.3, wher e the horizo ntal axis repre sents the OCL µ and the vertical axis represents its freq uency . Obser ve that the minimum OCL µ min is 3 5 , wh ereas the maximum OCL µ max is 83 . T hat is, the values of µ c over a wid e rang e. Next, based o n the argument in Section II-B, we examined th e OCL η using the reciproca l d ual enc oder ˜ H ( D ) associated with H ( D ) . The r esult is shown in Fig.4. W e have η min = 31 an d η max = 85 . Note that in this examp le, the minim um O CL is fur ther reduced using a r eciprocal dual encod er . Moreover , after having applied row/column opera tions to each H ( D ) (denote b y H ′ ( D ) the resu lting m atrix), we took its recip rocal version f H ′ ( D ) . Then again ap plying r ow/column operatio ns to f H ′ ( D ) , we examined the OCL µ ′ of the re sulting matrix. Using this meth od, we have further reduc tion with respect to 0 2 4 6 8 10 12 14 16 18 1 11 21 31 41 51 61 71 81 91 101 111 121 additional memory reduction (Δμ) pattern No. Fig. 5. Addition al overal l-constra int-length reduc tion. the value of µ . The result is shown in Fig.5. W e observe that the maximum reduc tion ∆ µ (= µ − µ ′ ≥ 0) of 16 is realize d. W e remark that µ min = 35 is r educed to µ ′ = 3 4 using this method. C. Efficient Sear ch Method Our aim is the reduction of the OCL o f a par ity-check matrix H ( D ) . Since row oper ations have been d one in a p attern [ P i , Q j , R k ] T , it is desirab le fo r figures in each column to be close together (i.e., th e difference δ j between the m aximum and the minimum in th e j th column is sm all). In th is case, each figure in th e column beco mes small a fter the colum n operation , which finally leads to the redu ction of the OCL. Hence, we search for a pa ttern in which ev ery column has δ j ≤ ∆ , where ∆ is a p redeterm ined search p arameter . For example, set ∆ = 20 . Con sider the p attern [ P 2 , Q 4 , R 3 ] T : S 1 =   30 0 2 6 14 27 1 11 0 9 18 12 0 7 21   . (33) δ j are given b y 12 , 1 2 , 11 , 7 , and 12 , respectively and remain within ∆ = 20 . Applyin g c olumn operatio ns, we have S ′ 1 =   12 0 2 6 5 9 1 11 0 0 0 12 0 7 12   . (34) The OCL is g i ven b y 12 + 11 + 12 = 3 5 . Similarly , consider the pattern [ P 5 , Q 5 , R 5 ] T : S 2 =   16 17 19 23 0 18 23 2 22 0 28 22 10 17 0   . (35) In this case, δ j are given by 12 , 6 , 17 , 6 , and 0 , resp ectiv e ly . Applying colum n o peration s, we have S ′ 2 =   0 0 17 6 0 2 6 0 5 0 12 5 8 0 0   . (36) Again, th e OCL is gi ven by 17 + 6 + 12 = 3 5 . Observe that these pattern s ha ve µ min = 35 . V . C O N C L U S I O N In this paper, we examined the state-space complexity of the error-trellis of an LDPC con volutiona l cod e derived fro m the QC block code specified by a parity-check matrix H . Since the entries in the cor respond ing par ity-check ma trix H ( D ) are all monom ials, we can construct a redu ced err or-trellis using the method of [ 1] or that of [8 ]. W e n oticed that when cyclic shifts are applied to the rows of H , th e QC code remains u nchang ed, whereas the co rrespon ding parity-ch eck matrix H ′ ( D ) , which defines an other conv olu tional co de, has r ow/column factors different from those in the origina l H ( D ) . T hat is, the OCL of H ′′ ( D ) , whe re H ′′ ( D ) is the matrix o btained by factorin g out row/column f ac tors in H ′ ( D ) , varies depending on the row shifts a pplied to H . On the other h and, the free distanc e o f the resulting con volutional c ode is still lo wer-bounded by the minimum d istance of the or iginal QC code. T hese facts imply that the state-space com plexity of the error-trellis associated with H ′ ( D ) can b e contro lled to so me extent with the err or- correction capability being preserved and this is our basic idea. By apply ing o ur method to the examp le in [ 10], we h av e shown tha t the OCL of the parity-ch eck matr ix of an LDPC conv olutio nal cod e can b e significantly reduced compared to the av e rage one. The LDPC co n volution al codes proposed by T ann er e t a l. have large constrain t leng ths. Th erefore , it is stated [ 10] that the u se of trellis-based d ecoding is n ot feasible. W e basically agree o n this point. 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