On the Encoding of the Multi-Non-Binary Convolutional Codes

Transcription

1 Buleinul Şiinţific al Universiăţii "Poliehnica" din imişoara Seria ELECRONICĂ şi ELECOMUNICAŢII RANSACIONS on ELECRONICS and COMMUNICAIONS om 53(67), Fascicola -, 008 On he Encoding of he Muli-Non-Binary Convoluional Codes Bala Horia, Alexandre De Baynas, Kovaci Maria Absrac - Recenly, Douillard e al. proposed a new family of muli-binary urbo-codes based on he parallel concaenaion of wo consiuen convoluional codes wih muliple inpus ha has beer global performance han classical urbo-codes. he encoder is based on an r- inpus linear feedback shif regiser (LFSR). In his paper, we show ha he encoder can also be represened by he observer canonical configuraion. his configuraion is essenial o reduce he compuaional complexiy of he code design procedure especially for moderae codeword sizes. Indeed, in his conex, an exhausive search usually provides beer resuls han he EXI char or similar opimizaion ools. We show ha he second configuraion reduces he compuaional complexiy of he search up o 300%. Based on his sraegy, we were able o design a rae-/ urbo-code wih wo inpus and memory m=3 which ouperforms he urbo-code wih same characerisics proposed by Douillard e al. by 0.5 decibels for a frame error rae of 0-4. Keywords: recursive and sysemaic muli-binary convoluional code, generaor marix, urbo-code. I. INRODUCION he muli-binary urbo-codes (MBC) also referred as non-binary urbo-codes ([]-[]) have several advanages compare o he iniial binary urbocodes in [3] like faser convergence and lower error floor. We refer o [4] for a more deailed analysis. he design of he MBC generally requires an exhausive search hrough large ses of feasible codes [5]-[6] and inerleavers [7]. his is paricularly rue for moderae codeword lenghs. In ha case, he asympoic hreshold deermined by EXI char or by similar opimizaion ools may no be accurae due o he lack of randomness of he srucure of he inerleaver and an exhausive search provides ofen beer resuls. In he case of urbo-codes wih muliple binary and nonbinary inpus (MBC [8] and MNBC [9], respecively), his search is a major burden for heir opimizaion since he compuaional complexiy of he search is exponenial wih he number of inpus r. herefore, resricing he search o a limied se of `good M(N)BC is essenial. he MBC proposed in [8] consis of he parallel concaenaion of wo rae-r/(r+) recursive sysemaic convoluional codes (RSC) where r represens he number of inpus of he code. he encoders are based on muliple-inpu linear feedback shif regisers (LFSR). An alernaive encoding mehod for M(N)BC consiss in a generalizaion of he Fibonacci encoder ([0]-[4]). Whereas boh encoding mehods are equivalen for r=, i.e., for he binary single inpu urbo-codes wih and wihou puncuring, hey give wo differen ses of encoder in he case of muliple inpus (r>). In his paper, we compare boh ses according several crieria commonly used in he sysem heory: cardinal number of boh ses, elemen-o-elemen equivalence beween boh ses. his comparison is imporan in order o selec he bes encoding mehod for he design of he MBC and MNBC based on an exhausive search. he res of he paper is organized as follows. Secion II describes he encoding mehod proposed in [8]. In he hird secion, an alernaive encoding mehod based on he generalizaion of a Fibonacci encoder is presened. In Secion IV, he condiions of equivalence beween boh mehods are deermined. Finally, in Secion 5, we presen our opimizaion resuls and compare he packe error rae performance wih he MBC from []. II. CANONICAL FORM OF CONVOLUIONAL ENCODERS BASED ON LINEAR FEEDBACK SHIF REGISER Fig. shows he general srucure of a mulipleinpu recursive sysemaic convoluional encoder ha is used in [8] for each of boh consiuen codes. he encoder is based on an r-inpu linear feedback shif regiser (LFSR). his encoder is generally no decomposable ino r single-inpu encoders, i.e., i canno be represened by he conroller canonical Universiy Poliehnica of imişoara, Faculy of Elecronics and elecommunicaions, V. Parvan, 303 imişoara, Romania, horia.bala@ec.up.ro, maria.kovaci@ec.up.ro Deparmen of Wireless Neworks RWH Aachen Universiy, Aachen, Germany, ade@gollum.mobnes.rwh-aachen.de

2 u r u r- h 0,m Fig. Canonical configuraion of a muliple-inpu rae-r/(r+) RSC convoluional encoder based on linear feedback shif regiser. configuraion []. hroughou he paper, in order o simplify he noaions, his configuraion is simply referred as he canonical form of ype H. he encoder has r inpus, u,... u r and r+ oupus corresponding o he r inpus and one redundan bi u 0 also referred as c. he curren encoder sae is given by he oupus of he m shif regisers s, s,... s m. he inpus u i, i= r are physically conneced o he j-h adder if h ij =; S = [ s m s s ] and U = [ u r ur ] describe he encoder sae a he ime and he inpu vecor of size r, respecively where x denoes he ranspose of he vecor x. he inpu/curren sae and oupu/curren sae relaions of he encoder a he ime can be expressed in he compac form: S+ = H 0 U + S () c = H E S + W S + () where he generaor marix H 0 and he marix are defined as follows: and wih h r, h r-, h, h 0, h 0, S h r, h r-, h, h r+, S hr,m... h,m H = 0 (3) hr,... h, hr,... h, 0( m = ) H h r+, I m R h r,m h r-,m h,m (4) H R = [h 0,m... h 0, h 0, (5) In order o have a decodable code [0], we assume ha H 0 is full rank. In (), he vecor W is equal o [0 0 0 ] m, and: H E = [h r+,m... h r+, h r+, ]. (6) In order o compare his canonical form based on LSFR wih he observer canonical form presened in nex secion, we deermine he ransfer funcion S m h r+,m u r u r- c = u 0 marix of he encoder. Le define he Lauren series of he sequence x as X(D) = = x D. Assuming ha he marix I m +D is inverible, i.e. de(i m +D ) = h 0 (D) 0 wih h i (D)= m j j= h ij D +, he ransfer funcion which corresponds o he redundan sequence C(D) of he encoder and he global ransfer funcion marix of he code M h (D) are respecively equal o: C(D) = M h (D) U(D) (7) M h (D) = D H E (I m +D ) - H 0 + W (I m +D ) - H 0 (8) Afer some algebraic manipulaions, i can be shown ha: ( Im + D ) = Pm ( D ) H R Δm( D) + D Δm( D) b( D ) where P m (D)=[D m- D ] and Δ m (D) is a oepliz marix wih firs row [D D D m- D m ] and firs column [D 0 0 0]. herefore, he ransfer funcion marix M h (D) can be simplified as: hr ( D ) M h (D) = + H R H E + Δ m( D ) H 0 h0( D ) + + [h r, h, h, ] (9) In he nex secion, we inroduce a second canonical form for muliple-inpu encoders based on Fibonacci represenaion. III. OBSERVER CANONICAL FORM OF CONVOLUIONAL ENCODERS WIH MULIPLE INPUS A recursive and sysemaic convoluional encoder wih muliple inpus (MIRSC) is generally no decomposable ino r single-inpu encoders, i.e., his encoder can generally no be represened by an equivalen srucure wih one shif regiser for each inpu. However, we show in his secion ha a realizable srucure consiss o have one shif regiser for he single oupu c as shown in Fig.. In sysem heory, his canonical form is referred as he observer canonical form, [0]. hroughou he paper, for sake of simpliciy, we refer his scheme as he canonical form of ype G. u r u r- u r u r- g r,m g r-,m g r,m- gr-,m- g g,m g r,0 g,m- r-, g g, r, gr-,0 g,0 S m- S 0 c = u 0 g 0,m g 0,m- g 0, g 0,0 = Fig. Observer canonical configuraion of a muliple-inpu raer/(r+) RSC convoluional encoder.

3 Le S = [ sm sm s 0 ] and U = [ u r ur ] denoe he m-componen column vecor describing he encoder sae a he ime and he inpu vecor of size r, respecively. he inpu/curren sae and oupu/curren sae relaions of he encoder a he ime can be expressed in he compac form: where G and are defined as: S+ = G U + S (0) c = G L U + W S () G = G 0 + G R G L () 0 m = GR, (3) I m G L and W are equal o [g r,0 g r-,0 g,0 ] and [0 0 0 ] m, respecively. In order o have a decodable code, we assume ha G is full rank [0]. Moreover, G 0 is defined as: gr,m gr,m... g,m = gr,m gr,m... g,m G (4) gr, gr,... g, and G R = [g 0,m g 0,m- g 0, ]. Afer some algebraic g 0 D as manipulaions and by defining ( ) j ( D) = g g = m j 0 0j D 0, i can be shown ha he ransfer funcion C(D) of redundan sequence and he ransfer funcion marix M g (D) of he encoder are respecively equal o: C(D) = G L U(D) + W S(D) (5) ( D) ( ) ( D P ) m( D) G G (6) L D M g = 0 + g0 Afer deriving he ransfer funcion marices M h (D) and M g (D) for boh canonical forms H and G in (9) and (6) respecively, we invesigae in he nex secion he relaions of equivalence beween he canonical forms H and G. IV. EQUIVALENCE BEWEEN ENCODERS DEFINED WIH HE CANONICAL FORMS H AND G Firs, we define he relaion of equivalence beween he canonical forms H and G: Definiion : he canonical forms H and G are equivalen if for any daa inpu U(D), boh canonical forms give he same oupu C(D), i.e.: M g (D) = M h (D) (7) where M h (D) and M g (D) are defined in (9) and (6), respecively. A. Single inpu classical binary case (r= and g 0 =): We firs invesigae he equivalence beween boh mehods in he single inpu case and when g 0 = as in he classical case [3]. heorem ([] p.0): In he case of single inpu (r=), boh configuraions are equivalen. Proof: For r=, H 0 =W. By muliplying boh sides of (0) by h 0 (D), we have: h 0 (D) M h (D)=(h (D) H R +h 0 (D) H E ) D ( D ) +h 0 (D) P m =h (D) (h 0 (D)+)+h 0 (D) (h (D)+)+h 0 (D)=h (D). Since G 0 is a column vecor and G L is equal o in he single inpu case, (6) can be wrien as: g 0 (D) M g (D) = g,m D m + + g, D + g, D +=g (D). So, if h (D) = g (D) and h 0 (D) = g 0 (D) boh configuraions H and G for he classical binary case are one-o-one equivalen. a) Canonical configuraion for he encoder defined by H=[6 5]. u b) Observer canonical form for he encoder defined by G=[ 3]. Fig.3 he wo canonical forms for he 3/5 8 convoluional RSC We illusrae his resul wih he following example. Suppose he RSC wih he oupu and feedback polynomials equal o 5 8 =+D +D 3 and 3 8 =+D+D 3, respecively. By defining he full generaor marices H and G as follows: and 0 0 H 0 = S 0 S S 3 H R =5 0 H E =6 c = u 0 0 G 0 = S S S 0 c=u 0 0 E R H = [ H H 0 H ] = [h r+ h r h h 0 ] 0 (8) G0 GR G = = [g r g g 0 ] 0, (9) GL H and G are equal o [6 5] and [3 ] in his example. he wo canonical forms H and G for he encoder are shown in Figures 3.a and 3.b, respecively. Since we have a sric equivalence beween boh canonical forms H and G in he classical single inpu case, i.e. r= and g 0 =, he opimizaion of he urbocodes can be performed eiher using he canonical G R =3

4 form G or H. However, we show nex ha i is no rue in he case of muliple inpus. B. Muliple inpu case (r>): We sar wih he following remark. Le G and H denoe he ses of he encoders wih canonical forms G and H, respecively. he sizes of he marices H and G defined in (8) and (9) are m (r+) and (m+) (r+), respecively. Assuming g 0,0 = and m r, he marix H has more enries han G. herefore he se H is larger han he subse G, i.e. he canonical forms H and G are generally no one-o-one equivalen in he muliple inpu case. he condiions for which boh canonical forms are equivalen are summarized in he nex wo heorems. heorem : For any generaor marix H of canonical form H, i exiss a unique equivalen marix G of canonical form G, which is soluion of he following sysem: (α) g 0,0 =, (β) g i,0 = h i, for any i =,..., r, (γ) g 0,i = h 0,i for any i=,... m, (δ) D P m (D) G 0 =(h r+ (D) H R +h 0 (D) H E ) Δ m (D) H (h 0 (D)+) [h r, h, h, ]. Proof: Equaion (α) imposes he srucure o be recursive. Afer some basic manipulaions, i can easily be shown ha (β), (γ) and (δ) are equivalen o (9), (6) and (7). In order o complee he proof, we have o show ha (δ) has a unique soluion G 0. Lemma : Equaion (δ) is equivalen o he equaion: G = A H 0 where he m m marix A is deermined recursively as follows: Iniializaion: H ER = [H E ] H R + [H R ] H E and A(,:) = H ER (,:); Ieraion i=,3, m: A(i,:)=H ER (i,:)+a(i-,:) I m (); where I m (p) is he m m ideniy marix shifed by p- posiions o he righ. M(i,:) denoes he row -vecor of he enries of he i-h row in he marix M (Malab noaions). Proof: We have by definiion: h r+ (D) H R +h 0 (D) H E = [D m D m- D ] ([H E ] H R +[H R ] H E ), and Δ m (D) = I m () D + I m () D + + I m (m) D m. Using (β), (γ) and (), we can now wrie: m i D G i= m+ m i+ p ( i,: ) = D H ( i,: ) I ( p) i= p= or equivalenly erm by erm: G i ER ( i,: ) = H ER ( i,: ) Im ( i k + ) H 0 k= i + k= = H ER ( i,: ) H ER ( i.: ) I m ( i k) Im ( ) H0 = m H 0 (0). () i We define A as: A(i,:) = k= H ER ( i,: ) I m ( i k + ). Using (), he i-h row of A can be expressed as: A(i,:) = H ER (i,:) + A(i-,:) I m () and G (i,:) is equal o A(i,:) H 0, i =,, m or equivalenly: G = A H 0. According o he Lemma, for any encoder defined by is generaor marix H 0, i exiss a unique generaor marix G and herefore a unique equivalen encoder wih a canonical form G which complees he proof of heorem. heorem shows ha for any marix H here is an equivalen marix G. We propose nex o invesigae if he converse is rue. We sar wih he following example: Assume m=r= and a generaor polynomial G = [ 3]. he corresponding encoder is depiced in Fig.4. his example does no belong o he classical binary case because g 0 =0. We have successively: G R =[]=H R, G L =[0]=[h ]=H 0, H=[ H E H 0 H R ] = [x 0 ]. I is easy o show ha he condiion (δ) is []=[+x] [0] which canno be saisfied for a leas one value of x {0, }. hus, he encoder presened in Fig.4 canno be represened in he canonical form H. heorem 3: For any generaor marix in G, i exiss i) none, ii) one or iii) several equivalen generaor marices in H. Proof: A generaor marix H in H which has an equivalen in G verifies he sysem of equaions: (β) [h r, h r-,... h, ]=G L ; (γ) H R =G R ; (δ) A H 0 =G (Lemma ). In order o find he vecor H E and he marix H 0 ha verified (δ), a exhausive compuaional search is performed hrough he (m- ) r+m possible pairs {H E, H 0 }. Equaion (δ) has i) none, ii) one or iii) several soluions: i) We have shown in he previous example ha an encoder in G may no have an equivalen in H. In secion 5.A, we provide several examples. Fig. 4. Encoder for G=[ 3] ii) here is a unique soluion for he single inpu classical binary case as shown in heorem. Moreover, for any r>, one could find a unique soluion H 0 o (δ) for paricular values of G. iii) Clearly we have: H G, where. denoes he cardinal number of a se. However, from heorem, we know ha any generaor marix in H has a unique equivalen in G. hus, here is a leas wo encoders in H wih he same equivalen encoder in G. According o heorem 3, he sysem of equaions (α), (β), (γ) and (δ) esablishes a funcion of equivalence ξ: H G, which is neiher injecive nor subjecive. heorem 3 has a huge impac for he design of C. Indeed, searching a good encoder in G insead of H is no only faser according o he condiion (iii) bu can furhermore lead o a beer soluion which does no exis in H according o he condiion (i). Furhermore, we show in he following heorem ha here are no disinc encoders in G ha are equivalen o each oher. S 0 0 u 0

5 heorem 4: {G, G G } G, G and G are no equivalen i.e. wo disinc marices in G canno be equivalen. Proof: Assume ha wo marices G and G in G are equivalen. According o (6) and (7), i implies ha: ( D) g ( D) g 0 = 0, G R = G R and D P m (D) G 0 + G L = D P m (D) G 0 + G L. he las equaliy is verified if only if: G and G so G = G. 0 G0 L G L C. Exension o he Muli-Non-Binary Convoluional Codes: I is worh noing ha all previous calculaions are valid for binary and non-binary inpus. herefore, all he resuls can be generalized for (u i, s j, h ij, g ij ) in GF( q ), q>. hey can direcly be used for he design of he MNBC [9]. In ha case, each connecion corresponds o a q bis-wide bus; he coefficiens h ij and g ij correspond o a muliplicaion circui over GF ( q ); he shif regisers buffer packes of q bis, and he adders perform modulo symbol-wise addiions. V. SIMULAION RESULS In his secion, we numerically evaluae he imporance of designing he MBC by searching he candidaes among he encoders wih canonical form G insead of he candidaes among he encoders wih canonical form H: ABLE I CARDINAL NUMBER OF HE SES G O, H O AND ξ(h O ) FOR HE CODES OF RAE-r/(r+) WIH MEMORY m=,3,4 AND WIH r INPUS, r=,,3. m r G o H o ξ(h o ) From Secion 4.3, we have H G. herefore, he search for good M(N)BC in G insead of H is faser. However, for pracical values of m and r, i.e. for consiuen codes wih memory 3 or 4, wha is he effecive gain? In heorem 3 we showed ha we could find codes in G ha do no have any equivalen in H. However, for pracical values of m and r, are hese codes among he bes consiuen codes for MBC? A. Cardinal number of he subses G, H and ξ(h) Wihou loss of generaliy, we limi our search o he recursive decodable codes in G and H, i.e. for G R and H R wih a leas one non zero enry. Moreover, swiching he inpus gives an equivalen code. herefore we assume ha he firs r elemens of he encoder marix G are sored in decreasing order. he wo resuling subses are G o and H o. In addiion, we define he subse ξ(h o ) of G o whose each elemen has a leas one equivalen in H o. In able, we compare he cardinal number of he ses G o, H o and ξ(h o ) for pracical values of m and r. he difference G o ξ(h o ) exponenially increases wih respec o r. he raio H o / ξ(h o ) exponenially increases wih respec o m-r. he larger he memory and/or he lower he coding rae, he more ineresing he search among he encoders wih canonical form G insead of H. If he feedback polynomial g 0 (D)=h 0 (D) is a primiive polynomial wih g 0,m =h 0,m 0, which is pracically he case of all `good codes, any marix in G has m-r equivalen marices in H. herefore, even if he search is limied o he encoding marices wih a primiive feedback polynomial, he gain of searching he consiuen codes of he MBC in G insead of H is of he same order, i.e. he compuaional complexiy of an exhausive search decreases exponenially wih respec o m-r. o illusrae his claim, we deermine in able II he equivalen marices for wo rae-/3 codes from [8] of memory m=3 and m=4 wih generaor marices H=[6 7 5] and H=[ ], respecively. B. Opimizaion Resuls In his secion, we presen he opimizaion resuls of an exhausive search among all encoders in G, i.e., all encoders ha can be represened wih observer canonical form. In order o reduce he compuaional complexiy of he MBC based on an exhausive search, we perform he exhausive search in boh following seps: in he firs sep, we selec boh codes and inerleavers based on a quick esimae of he raio beween he average number of ieraions o decode he MNBC and he number of packes ha were ransmied wihou error for a fixed number of packes. Only few packes, say roughly 000 ransmied packes for each code, are needed o accuraely esimae his raio. Moreover, we observe in our simulaions ha he bes code in all cases belongs o he firs per cen only of he codes wih fases convergence which dramaically reduces he code opimizaion. In he second sep, we are simulaing FER a a SNR such ha he FER resuls approximaively mach wih a argeed FER. In our case, we arge a FER around 0-4 and are using codewords o esimae i: he bes code is he code wih he smalles FER. For implemenaion consideraions, we focus on double-binary urbo-codes (MBC wih wo inpus) wih memory m= and 3. In all cases excep if i is noified, we use an S-inerleaver of lengh 75 he rellis-erminaion echnique using ail-bis o drive he encoder o he all-zero sae; moreover, he coding rae is / for all MBC. For m=, he bes code ha we found has for generaor marix G=[7 6 5] which does no have an equivalen in canonical form H. For m>, he codes ha belong o G\H do no have a primiive feedback connecion polynomial g 0 (D) of

6 maximal degree m. Since he bes MBC are based on a primiive feedback polynomial, i is no necessary o consider he codes ha belong o G\H in he MBC design. he canonical form G is sill ineresing for he MBC design in order o simplify he exhausive search based on heorem 4. For m=3 and an S-inerleaver of parameer, he bes MBC ha we found is he code proposed in [8] wih generaor marix G=[3 5 ]. Whereas he S- inerleaver is asympoically opimal in he codeword size [5], we also consider he family of inerleaver proposed in [8] which is simpler o implemen in hardware. hese inerleavers are defined by 4 parameers: P, P P and P3. For more deails on he consrucion of such inerleavers, we refer o [8]. hrough an exhausive search, we found wo bes codes wih generaor marices G=[5 9 ] and G=[5 9 3] and wih inerleavers defined by he parameers P=9, P=376, P=38, P3=96 and P=9, P=376, P=03 and P3=677, respecively. Boh codes have similar Bi Error Rae and Frame Error Rae performance and ouperforms he MBC proposed by [8]. For a frame error rae of 0e-4, he gain is approximaively equal o 0.3 decibels. Finally, for m=4, he bes code ha we found has for generaor marix G=[ 3 5] which is he same ha Douillard proposed in [8]. VI CONCLUSIONS he design of he urbo-codes wih muliple inpus generally requires an exhausive search hrough large ses of feasible codes and inerleavers. his search is a major burden in heir opimizaion since he compuaional complexiy of he search is exponenial wih he number of inpus r. Fig.6 Bi and Frame Error Rae Performance as a funcion of he signal-o-noise raio for hree rae-/ codes wih memory 3: boh codes ha we designed in his paper defined by generaor marices G=[5 9 3] and G=[5 9 ], respecively and he code proposed in [8] wih generaor marix [3 5 ]. he heoreical limis on FER are derived from [6]. In his paper, we showed ha he encoder of such codes could be represened wih he observer canonical configuraion. his configuraion is essenial for reducing he compuaional complexiy of he search (up o 300% for he codes ha we considered). Based on his sraegy, we were able o design a rae-/ urbo-code wih wo inpus and memory m=3 which ouperforms he equivalen urbocode proposed by Douillard e al. by 0.5 decibels for a frame error rae of 0-4. ACKNOWLEDGMEN he auhors are graeful o Professors C. Douillard, Ph.D and M. Jézéquel., Ph. D, from ENS-Breagne for he urbo Codes seminars susained in imişoara, Romania. REFERENCES [] C. Berrou, M. Jézéquel, Nonbinary convoluional codes for urbo coding,elecron. Le.,vol. 35,no.,pp.39 40,Jan [] A. Ghrayeb,. Abualrub, Asympoic performance comparison of concaenaed (urbo) codes over GF(4), In. J. Commun. Sys. 004, 7, [3] C. Berrou, Some clinical aspecs of urbo codes, in Proc. IEEE In. Symp. urbo Codes Relaed op., Bres, France, Sep. 997, pp [4] C. Berrou, M. Jézéquel, C. Douillard, and S. Kerouédan, he advanages of nonbinary urbo codes, in Proc. IEEE Inf. heory Workshop, Cairns, Ausralia, Sep. 00, pp [5] S. Benedeo, R. Garello, and G. Monorsi, A search for good convoluional codes o be used in he consrucion of urbo codes, IEEE ransacions on Communicaions, vol. 46, no. 9, pp. 0 05, Sep [6] A. C. Reid,. A. Gulliver, D. P. aylor, Rae-/ Componen Codes for Nonbinary urbo Codes, IEEE ransacions on Communicaions, vol. 53, No. 9, sep [7] J. Sun, O. Y. akeshia, Inerleavers for urbo Codes Using Permuaion Polynomials Over Ineger Rings, IEEE ransacions on Informaion heory, vol. 5, No.,Jan [8] C. Douillard, C. Berrou, urbo Codes Wih Rae-m/(m+) Consiuen Convoluional Codes, IEEE ransacions on Communicaions, Vol. 53, No. 0, Oc. 005, pp [9] H. Bala, M. Kovaci, A. de Baynas, C. Vladeanu, R. Lucaciu, A Very General Family of urbo-codes: he Muli-Non- Binary urbo Codes, Scienific Bullein of he Poliehnica Univ. of imişoara, Romania, Sep.006, fasc., pp [0] R. Johannesson and K. S. Zigangirov, Fundamenals of Convoluional Coding, ser. Digial, Mobile Commun.. New York: IEEE Press, 999, Chaper. II. [] M. Goresky and A. Klapper, Fibonacci and Galois represenaions of feedback-wih-carry shif regisers. IEEE ransacions on Informaion heory Vol. 48 No. :Nov 00, pp [] R. Johannesson and Z. Wan. A linear algebra approach o minimal convoluional encoders. IEEE ransacions on Informaion heory, Vol. 39 No 4 April 993, pp [3]. Hasegawa, On a Coder of Fibonacci Code for Undermiaer Digial Daa ransmission, IEEE 97 Eng. In he Ocean Environmen Conf., pp [4] W. Xiang and S.S. Pierobon, A New Class of Parallel Daa Convoluional Codes, in Proc. of 6 h Ausralian Communicaions heory Workshop, pp , Feb 005. [5] C. Douillard, C. Berrou, M. Jézéquel, urbo-codes (convoluifs), ech. repor, Mar. 004, imisoara, Romania: hp://hermes.ec.u.ro/docs/cerceare//cari/ccp.pdf [6] C. S. Dolinar, D. Divsalar, and F. Pollara, Code Performance as a Funcion of Block Size, echnical Repor MO Progress Repor 4-33, May 998.