The Design of Efficiently-Encodable Rate-Compatible LDPC Codes

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1 The Desgn of Effcently-Encoable Rate-Compatble LDPC Coes Jaehong Km, Atya Ramamoorthy, Member, IEEE, an Steven W. McLaughln, Fellow, IEEE Abstract We present a new class of rregular low-ensty party-check (LDPC) coes for moerate block lengths (up to a few thousan bts) that are well-sute for rate-compatble puncturng. The propose coes show goo performance uner puncturng over a we range of rates an are sutable for usage n ncremental reunancy hybr-automatc repeat request (ARQ) systems. In aton, these coes are lnear-tme encoable wth smple shft-regster crcuts. For a block length of 00 bts the coes outperform optmze rregular LDPC coes an extene rregular repeat-accumulate (eira) coes for all puncturng rates 0.6~0.9 (base coe performance s almost the same) an are partcularly goo at hgh puncturng rates where goo puncturng performance has been prevously ffcult to acheve. Key Wors: Effcent encong, low-ensty party-check (LDPC) coe, puncturng, rate-compatble coe. I. INTRODUCTION Low-ensty party-check (LDPC) coes are consere goo canates for next-generaton forwar error control n hgh throughput wreless an recorng applcatons. Ther excellent performance an parallelzable ecoer make them approprate for technologes such as DVB-S, IEEE 80.6e, an IEEE 80.n. Whle semconuctor technology has progresse to an extent where the mplementaton of LDPC coes has become possble, many ssues stll reman. Frst an foremost, there s a nee to reuce complexty wthout sacrfcng performance. Secon, for applcatons such as wreless LAN, the system throughput epens upon the channel contons Jaehong Km s wth the epartment of Electrcal & Computer Engneerng, Georga Insttute of Technology, Atlanta, GA USA (e-mal: onl@ece.gatech.eu). Atya Ramamoorthy s the epartment of Electrcal & Computer Engneerng, Iowa State Unversty, Ames, IA, USA (e-mal: atyar@astate.eu) Steven W. McLaughln s wth the epartment of Electrcal & Computer Engneerng, Georga Insttute of Technology, Atlanta, GA USA (e-mal: swm@ece.gatech.eu).

2 an hence the coe nees to have the ablty to operate at fferent rates. Thr, whle the LDPC ecoer can operate n lnear tme, t may be har to perform low-complexty encong of these coes. Whle the encong tme of rregular LDPC coes can be reuce substantally usng the technques presente n [] at long block lengths, ther technques may be har to apply at short block lengths. The other opton s to resort to quas-cyclc (QC) LDPC constructons that can be encoe by shft regsters []. However such constructons are typcally algebrac n nature an usually result n coes wth regular egree strbutons. An mportant problem s the esgn of LDPC coes that can be easly encoe an have goo puncturng performance across a we range of rates. In ths work, we ntrouce a new class of LDPC coes calle Effcently-Encoable Rate-Compatble (E RC) coes that have a lnear-tme encoer an have goo performance uner puncturng for a we varety of rates. Secton II overvews pror work n rregular LDPC coes an rate-compatble puncturng. In secton III, we present the E RC constructon algorthm. The shft-regster base encoer structure for the E RC coes s explane n secton IV. Secton V compares the puncturng performance of the E RC coes wth that of other rregular LDPC coes an secton VI outlnes the conclusons. II. BACKGROUND AND RELATED WORK LDPC coes can be efne by a sparse bnary party-check matrx of sze M N, where M an N are the number of party symbols an coewor symbols respectvely. The party-check matrx can equvalently be consere as a bpartte graph (calle the Tanner graph of the coe [3]), where columns an rows n the party-check matrx correspon to varable noes an check noes on the graph, respectvely. The strbuton of varable (check) noes n the graph can be represente as a polynomal λ( x) = λx ( ρ( x) ρ x varable (check) noes of egree. = ), where λ ( ρ ) s the fracton of eges ncent to In ths paper we work wth systematc LDPC coes. Thus, assumng that the party-check matrx s full-rank, we have K columns corresponng to nformaton bts an M columns corresponng to party bts, where K + M = N. In the sequel we shall refer to the submatrx of the party-check matrx corresponng to the K nformaton bts, the systematc part an the submatrx

3 3 corresponng to the party bts, the nonsystematc part. We shall enote the party-check matrx by H, the systematc part by H an the nonsystematc part by A. Extene Irregular Repeat Accumulate Coes H. Thus, H [ H H ] =. A promsng class of LDPC coes calle Irregular Repeat Accumulate (IRA) coes was ntrouce by Jn et al. n [4]. These coes have several esrable propertes. Frst, IRA coes can be encoe n lnear tme lke Turbo coes. Secon, ther performance s superor to turbo coes of comparable complexty an as goo as best known rregular LDPC coes [4]. The columns corresponng to the egree-two noes n the party check matrx of IRA coes has b-agonal structure shown below (). H =, () The class of extene IRA (eira) coes was ntrouce by Yang et al. n [5]. The eira coes acheve goo performance by assgnng egree- noes to nonsystematc bts an ensurng that the egree- noes o not form a cycle amongst themselves. Furthermore, they avo cycles of length four an make the systematc bts correspon to varable noes of egree hgher than two. They ensure effcent encong by formng the party n the b-agonal structure lke IRA coes as shown n (). For more etals we refer the reaer to [5]. It s nterestng to see whether there exst other ways of placng the egree- noes wthout any cycles nvolvng only egree- noes. We present an example of such a placement below n the case when M = 8.

4 4 H = () Observe that the column egree of each noe s except the last column an that there oes not exst any cycle n ths matrx. We shall see later that ths constructon can be generalze an the resultng matrces can be use to construct LDPC coes that can be effcently encoe an have goo puncturng performance over a we range of rates. B. Rate-Compatble Puncturng In wreless channels where the channel contons vary wth tme, usng systematc coes an puncturng the party bts s an effcent strategy for rate-aaptablty, snce the system requres only one encoer-ecoer par. Rate-compatble puncture coes (RCPC) were ntrouce by Hagenauer [6] as an effcent channel cong scheme for Incremental Reunancy (IR) Hybr-Automatc Repeat request (ARQ) schemes. In an RCPC famly, the party bts of a hgher-rate coe are a subset of the party bts of the lower-rate coe. If the recever cannot ecoe the coe base on the current receve bts, t requests the transmtter for atonal party bts untl t ecoes correctly. Thus, the subset property that the party bts of coes of fferent rate satsfy s useful. Rate-compatble puncturng of LDPC coes was consere by Ha et al. [7]. They erve the ensty evoluton equatons for the esgn of goo puncturng egree strbutons uner the Gaussan Approxmaton. Ha et al. also propose an effcent puncturng algorthm for a gven mother coe n [8-9]. For fnte length (up to several thousan symbols) LDPC coes, Yazan et al. construct rate-compatble LDPC coes usng puncturng an extenng [0]. The algorthm of [8-9] takes as nput a partcular mother coe Tanner graph an a set of target

5 5 rates. It then performs a search to entfy the set of coewor symbols that shoul be puncture to acheve those target rates. Snce our coe constructon technque s nspre by t, we present a bref escrpton of the algorthm below. Suppose that the Tanner graph of the mother coe s enote by G = ( V C, E), where V enotes the set of varable noes, C enotes the set of check noes an E enotes the set of eges. Let S V be a subset of the varable noes. Then the set of check noe neghbors of S shall be enote by N ( S). Smlar notaton shall be use to enote the set of varable noe neghbors of a subset of the check noes. The set of unpuncture noes s enote by V0 an the set of puncture varable noes s enote by V \ V 0 (usng stanar set-theory notaton). Defnton [-step recoverable noe]: A puncture varable noe p V \ V0 s calle a -step recoverable (-SR) noe f there exsts c N ({ p}) such that N ({ c}) \{ p} V0. -step recoverable noes are so name because n the absence of any channel errors these noes can be ecoe n one step of teratve econg. Ths efnton can be generalze to k-step recoverable (k-sr) noes (see Fg. ). Let V be the set of -SR noes among the puncture varable noes. Smlarly, let V k be the set of k-step recoverable noes, whch are efne as follows: Defnton : A puncture varable noe p V \ V0 s calle k-step recoverable (k-sr) noe f k there exsts c N ({ p}) such that {} c \ p V where q k. V N ( ) { } an that there exsts q ({ c} )\{ p} = 0 N, From the above two efntons, note that V = V (V represents the set of noes that cannot = 0 be recovere by erasure econg). Uner these contons, note that the k-sr noe wll be recovere after exactly k teratons of teratve econg assumng that the channel oes not cause any errors. So a large number of low-sr noes are ntutvely lkely to reuce the overall number of teratons, whch results n goo puncturng performance. The general ea of the puncturng algorthm n [8-9] s to fn a goo ensemble of sets V k s maxmzng V k for small k > 0. The

6 6 puncturng algorthm n [8-9] works well for any gven mother coe. However, the maxmum puncturng rate s often lmte when ths algorthm s apple, so that hgh puncturng rates are ffcult to acheve. Ths s because t s ffcult to fn enough number of low-sr noes from a ranomly constructe matrx. In aton, [8-9] o not aress the problem of mother coe esgn for puncturng,.e., they o not present a technque for the esgn of a mother coe n whch the party check matrx has a large number of varable noes that are k-step recoverable wth low values of k. Ths s the focus of ths paper. III. A NEW CLASS OF IRREGULAR LDPC CODES In ths work we are ntereste n esgnng rate-compatble puncture coes that exhbt goo performance across a we range of cong rates. To ensure goo performance over the fferent cong rates we attempt to esgn the mother coe matrx to have a large number of k-sr noes wth low values of k. From a practcal perspectve the requrement of low-complexty encong s also mportant. Lke puncture RA, IRA an eira coes, these coes are esgne to recover all the puncture bts when the channel s error-free even when they acheve the maxmum puncturng rate by runnng suffcent teratons of teratve econg. Thus, encong of these coes s also relatvely smple. A. Coe Constructon Algorthm Before escrbng our esgn algorthm, we efne a k-sr matrx. Let v enote the -th column of the party-check matrx H, where 0 < N. We shall use t nterchangeably to enote the varable noe corresponng to the -th column n the Tanner graph of H. Defnton 3: The matrx ( ) S { 0,,, N }. = s calle a k-sr matrx, f v s V k for all s S, where P vs s S In the propose E RC coes, we construct the party-check matrx by placng several k-sr matrces as shown n Fg.. We assgn all the egree- noes to the nonsystematc part. Noes havng egree hgher than two are elements of the 0-SR matrx whch conssts of message noes an party noes that shall not be puncture. Conser the submatrx of 0-SR matrx forme by the

7 7 hgh egree noes n the nonsystematc part. We enote such submatrx of 0-SR matrx as L, an the number of columns n L as l as epcte n Fg. (a). Defnton 4: The epth s the number of fferent types of k-sr matrces that have egree- columns n a party-check matrx. Defnton 5: The functon γ ( k) s the number of columns n the k-sr matrx n a party-check matrx,.e., γ ( k) = V, where k > 0. k From Defnton 5, note that the sze of the k-sr matrx s M γ(k). Let N () represent the number of varable noes of egree. Fg. (a) shows the case when N ( ) v v < M, an we shall elaborate on the esgn of such coes n subsecton III.B. Other than that, we assume that N () = M throughout the paper. When N () > M we cannot guarantee the cycle-free v property among the egree- noes, whch s an mportant esgn rule that wll be explane later. When N () = M, there wll be no 0-SR noes n the nonsystematc part,.e. l = 0. In ths case, v we nsert a egree- noe n the last column of nonsystematc part, an assgn all the varable noes of the nonsystematc part to egree- noes except the last egree- noe as shown n Fg. (b). Example : For M = 8 an N v() = 7, we can construct the nonsystematc part H as n (). In (), the frst four columns form the -SR matrx, the next two columns form the -SR matrx, an the next one column forms the 3-SR matrx. Thus, epth = 3, γ() = 4, γ() =, an γ(3) =. We can also regar the last egree- column as 4-SR matrx. However, our conventon n ths paper s to only conser egree- columns to calculate the epth. From now on, we refer to the last egree- column n H as (+)-SR matrx snce the connectons wth other k-sr matrces makes v t (+)-SR noe. Let S k k = γ ( j). Thus, S k represents the sum of the number of columns n the submatrx j= forme by the placng the -SR, -SR, an k-sr matrces next to each other. We set S 0 to 0. We shall represent the poston of the ones n a column belongng to a k-sr matrx by the powers

8 8 of a polynomal n D. Accorng to our constructon, the j-th column of k-sr matrx can be represente by the followng polynomal γ k ( ) h = D + D k j k an j + S k ( ) k, j, where,0 γ ( ) M h = D. + In the sequence, D represents the poston of nonzero element n a column,.e., -th element of the column s nonzero, where 0 M. For Example, we note that the epth can be obtane by settng log M log8 3 = = = an ( k) M k γ = for k, ( ) γ + =. In general, M nee not be a power of two. We present the algorthm for constructng H for general M below. E RC Coe Constructon Algorthm STEP [Fnng Optmal Degree Dstrbuton] Fn an optmal egree strbuton for the esre coe rate wth constrant that Nv() < M. STEP [Parameter Settng] For a gven esgn parameter, M (number of party symbols), obtan the epth an γ(k). The computaton of an γ(k) s explane below. The sze of the k-sr matrx s set to be M γ(k). STEP 3 [Generatng k-sr matrx] The j-th column of the k-sr matrx has the followng sequence: γ k ( ) j + ( ) S k D + D, for k hk, j=, where 0 j γ ( k). M D, for k = + STEP 4 [Constructng matrx T] Construct the matrx T as follows: [ ] T = -SR matrx -SR matrx -SR matrx. STEP 5 [Formng matrx H ] A a egree- noe to T an form H [ T ] = ( + )-SRmatrx. STEP 6 [Ege Constructon] Construct the matrx H by matchng the egree strbuton obtane n STEP as closely as possble. STEP 7 [Constructng matrx H] Assgn H as the systematc part an H as the nonsystematc part: [ ] H = H H.

9 9 In STEP, we frst fn an optmal egree strbuton for the esre mother coe rate, say R L, usng the ensty evoluton []. When we etermne the egree strbuton, the number of egree- noes, N ( ) v, s an mportant factor. The E RC coes are esgne so that all the egree- noes n the nonsystematc part can be puncture. Ths wll gve us the hghest achevable puncturng rate, say H R. Then, RH K ( N Nv( ) ) =. Thus, the E RC coes can prove an ensemble of rate-compatble coes of rate R ~ R. When N () = M all the party L H v bts have egree two an can be puncture so that R H =.0. In STEP, we set the esgn parameters. We try to obtan a large number of low-sr noes whle constranng the ncrease n the row egree. In fact, we esgn the functon γ(k) such that t assgns approxmately half of the partes as -SR noes, an approxmately the half of the remanng partes as -SR noes, an so on. The epth s gven as = log M, an γ(k) as k γ( k) = M γ( ) for k, γ ( + ) =, an γ (0) M (3) = 0 where an are the celng functon an the floor functon, respectvely. We observe that the functon γ(k) s such that the followng facts hol true. Fact : The functon γ ( k) s such that γ ( k) for k. S = γ () = M, where = log M. Furthermore = Proof: See Appenx. From the generaton sequence n STEP 3, we can notce that the k-sr matrx s compose of only egree- varable noes except for the last (+)-SR matrx. Note that every column n k-sr matrx has egree two. In partcular, when N () = M, all the columns of the nonsystematc v part have egree two except the last column whch has egree one. After generatng the k-sr matrces, we put them together to form the matrx T n STEP 4. Then n STEP 5, we construct the nonsystematc part H [ T ] = ( + )-SRmatrx by ang a egree- column at the en of H. Example shows an example of the constructon of a H matrx usng the propose algorthm.

10 0 Example : For M = 7 an N v() = 6, the epth = 3, an γ() = 3, γ() =, γ(3) =. H = In STEP 6 the matrx H s constructe by tryng to match the egree strbuton obtane from STEP. Note that the egree strbuton of the nonsystematc part s alreay fxe by the constructon algorthm. Ths may cause some check noes to have egrees hgher than those specfe by the optmal egree strbuton. In ths case we try to match the optmal egree strbuton as closely as possble. Snce we have some hgh egree check noes, we compensate t to match the average rght egree by enlargng the number of lower egree check noes or placng some lower egree check noes. Fnally, H an H are combne to make the whole party-check matrx n STEP 7. We now present some propertes of the coes that are constructe usng the prevous algorthm. In the subsequent statements an scusson, unless otherwse specfe, H shall represent the nonsystematc part of a party-check matrx an shall be assume to have been generate by the E RC constructon algorthm. Lemma : In the matrx H, any column n a k-sr matrx s connecte to at least one row of egree-k. Furthermore, ths row has exactly one connecton to a column from each l-sr matrx, where l < k. Proof: See Appenx. From Lemma, t s possble to fn the exact number of rows wth egree-k except the last row. We efne ζ as the row egree of the last row. Observaton : The row egree ζ of the last row n the matrx H can be obtane as [ γ() S S]. ζ = + + =

11 Proof: Conser the connectons of the last row wth each k-sr matrx. It s easy to see that f M = γ(), there s a connecton between the -SR matrx an the last row, otherwse, there s no connecton. Smlarly, f M = γ()+ γ(), there s a connecton between the -SR matrx an the last row, an so on. Thus, we can get ζ as ( ( M () )) ( ( M () ())) ( ( M () () ( ) )) ζ = γ + γ γ + + γ γ γ + -SR matrx -SR matrx -SR matrx [ γ S M ] = () + ( ) + = [ γ S S] = () + +, = ( +)-SR matrx snce we have S = M from Fact. From Observaton, we can obtan ζ [ γ S ] 3 = () = 3for Example. Snce we know ζ, = we are reay to get the rght egree strbutons for H. Observaton : The number of egree-k rows n the matrx H s γ ( k) δ( k ζ) + for k, f = 0, where δ () =. 0 otherwse Corollary : The rght egree strbuton (noe perspectve) of the matrx H s as follows: ρ( x) + = ˆ ρx, where = γ () + δ( ζ) δ ( ζ ) ˆ ρ = for k an ˆ ρ + =. M M Proof: Conser the k-sr matrx when k. From Lemma, f we pck a column n the k-sr matrx, the frst element of the column s nclue n a row of egree k, an the secon element has row egree greater than k. The number of columns n the k-sr matrx s γ(k) an each column s connecte to one egree-k row. Thus, the number of rows havng egree k s at least γ(k) except the last row. For a (+)-SR matrx, there s only one egree-ζ row. From Fact, summng the number of rows havng egree-k results n γ () + γ() + + γ( ) + = M. Therefore, the number of rows of egree k except the last row s exactly γ ( k ). The result follows. Once the optmal egree strbutons for the whole coe for a esre coe rate have been foun, we can get the egree strbutons for the H matrx whle fxng the egree strbutons obtane

12 from the constructon algorthm for H. In general, matchng the optmal egree strbuton for the whole coe may not be possble because of the constructon algorthm. For the systematc part, namely the H matrx, we choose varable noes of hgher egree greater than two. Beses fnng the optmal egree strbutons, there are three atonal esgn rules for fnte-length LDPC coes propose n []: (a) Assgn egree- varable noes to nonsystematc bts; (b) Avo short cycles nvolvng only egree- varable noes. (c) Avo cycles of length four. The propose E RC coes meet the esgn rule (a) as state above. For esgn rule (b), we show that there are no cycles nvolvng only egree- varable noes. Lemma : Suppose that there exsts a length-s cycle n a matrx whch conssts of only weght two columns. Conser the submatrx forme by the subset of columns that partcpates n the cycle. Then, all the partcpatng rows n the cycle must have egree two n that submatrx. Proof: To have a length-s cycle, the number of columns partcpatng n the cycle nees to be s an the number of rows partcpatng n the cycle nees to be s. Let us enote the submatrx forme by the columns partcpatng n the cycle by U. Then, the number of eges n U s s snce each of the columns has egree two. Each row partcpatng n the cycle must have a egree greater than or equal to two n U snce each row has to lnk at least two fferent columns n U. Suppose there s a row havng egree strctly greater than two n U. Then there shoul be a row havng egree less than two n U, snce the average row weght n U s two (the number of eges / the number of rows = s / s = ). Ths s a contracton because a row that has egree less than two n U cannot partcpate n a cycle wth the columns n U. Thus, every partcpatng row must have egree two n U. Usng Lemma, we prove that the propose matrx H s cycle free. Lemma 3: The matrx H constructe by the E RC constructon algorthm s cycle free. Proof: Suppose that there exst s columns v, v,, vs n H that form a cycle of length s. We form the M s submatrx forme by the columns. Let us enote ths submatrx by H s. Suppose

13 3 that column v belongs to the k -SR matrx n H. Let k mn = mn k. Applyng Lemma, we have {} that v k mn has exactly one connecton to each l-sr matrx, where l < k mn, an no connecton to m-sr matrces where m > k mn,.e., there s a check noe connecte to v k mn that s sngly-connecte n the submatrx H s. Applyng Lemma, we realze that a cycle cannot exst amongst the s columns. The matrx H has a hgh fracton of egree- noes. In fact, f N () = M then ( M )/ M fracton of the noes n H are of egree-, an there s only one egree- noe. The constructon algorthm also nuces a sprea n the check noe strbuton. Ths may cause the constructe coes to have hgher error floors. To reuce these effects, we can use methos such as those presente n [-5] when we construct the H matrx. By ong so, the E RC coes can meet the esgn rule (c). B. Low-Rate E RC Coe Desgn Conserng E RC mother coe esgn for low rate ( R < 0.5 ) s a natural step. In ths case, we shoul conser a esgn that allows some porton of the noes n the nonsystematc part to have egree greater than two snce t s har to obtan a goo egree strbuton that has all the party bts of egree two. Ths s the reason why we conser the case when N () < M. We wll brefly explan the fferences n the constructon algorthm for ths case compare to the case consere earler. Recall that we puncture only the egree two noes. The matrx L that has l columns shown n Fg. (a) conssts of those party bts that have egree hgher than two an shall not be puncture. Snce N () = M l < M, we set the epth as the maxmum such that S v < N ν (), an obtan γ(k) as before for k <. The prevous settngs for γ(k) s are esgne to match S = Nv() = M. In ths case, however, we set γ ( ) = Nv() S so that they can satsfy S = N (). To generate the sequence of -SR matrx, we set v δ = M γ(). = 0 Then, the j-th column of k-sr matrx of STEP 3 has the followng sequence: v v

14 4 γ ( k ) ( ) δ ( ), j+ S k D + D, for k < hk, j=, where 0 j γ ( k). j+ Sk D + D for k = We formulate T n the same way as before an set H [ L T] =, where varable noes n the matrx L have egree hgher than two. Note that we o not put the egree- noe n H. Then, we nee to construct eges for the matrx L an H by tryng to match the target egree strbuton an by avong cycles of length four. Note that the submatrx forme by the columns of N v() s cycle free (the proof s very smlar to the prevous proof). For the propose coes, rate-compatblty can be easly obtane by puncturng the egree-two noes from left to rght n the H matrx. For a esre coe rate R p to be obtane from puncturng the mother coe of rate R L, the number of puncturng symbols s p N( RL Rp ) =, where N s the coe length an RL Rp RH. Any esre coe rate can be acheve by frst puncturng noes from the -SR matrx, then from the -SR matrx an so on. Thus the coes of fferent rates can be apple to IR Hybr-ARQ systems. IV. EFFICIENT ENCODER IMPLEMENTATION In ths secton we show that E RC coes can be encoe n lnear tme. We start by presentng an effcent shft regster base technque that can be apple to other smlar block coes as well. Frst, we wll explan the case when N () M v =. For the party-check matrx H [ H H ] = of an E RC coe obtane from the propose constructon algorthm, let a coewor c [ m p] where m s the systematc symbols, an p s nonsystematc symbols. Then, we have [ ] [ ] T T H c H H m p = T T T T = Hm + Hp = 0. Let s = Hm, then we have T T T Hp = Hm = s. Snce H s sparse s can be foun effcently. =,

15 5 H h h h M p s h h h p s M h h h p s. h h h p s 3 3 3M 3 3 T p = = M M MM M M Let H ( h, j) =, then, j M M snce j s = hp = hp+ p j j j j j= j= h = for = j an h = 0 for j < j (snce H s lower trangular) n the constructon of the E RC coes. Snce all noes n H are egree-, the elements between the two entres of the sequence are 0. Ths means that for j γ (), h j, = j or = j+γ () =. 0, otherwse Then we have p s, for γ () = + j j, γ () + j= s hp for M. The above results tell us that we can get p for γ () rectly from s. By usng the obtane γ () p s, we can get p one by one for γ () + M, whch enables us to mplement the E RC encoer by usng γ () shft regsters. The followng example llustrates the encong metho. Example 3: For M=7, we can construct H matrx as follows: H p s p s p3 s3 T p = p4 = s p5 s p6 s p 7 s 7

16 6 Smplfyng we get: p = s, p = s, p3 = s3, p+ p4 = s4, p + p5 = s5, p3 + p4 + p6 = s6, an p5 + p6 + p7 = s7. Then, we can obtan p s by usng p j s, where j < : p = s, p = s, p3 = s3, p4 = p+ s4, p5 = p + s5, p6 = p3 + p4 + s6, an p7 = p5 + p6 + s7. We only nee γ () = 3 memory elements for the encoer n Fg. 3. The coeffcents for multplcaton n Fg. 3 can be obtane from the slng wnows hghlghte as squares n the matrx equaton. For ths reason, we wll refer to ths encong metho as slng wnow metho. The coeffcent g s are tme varyng. Assumng that the wnow starts from the frst row at ntal tme t=0, g 0 wll be on urng t=3~5, g wll be on urng t=5~6, an g wll be on at t=6. From the Example 3, we can generalze the shft-regster encoer mplementaton of E RC coes. The encoer can be represente as vson crcut as shown n Fg. 3. The vson crcut can be specfe by a generator polynomal g( x) = g + g x+ g x + + g x + x γ() γ() 0 γ (). By observng the matrx H, we can obtan the coeffcents of the polynomal. As n Fg. 4, conser the wnow of sze w. As we sle the wnow from the frst row to the last row, we can get party-check equatons one by one. The coeffcents n the wnow wll change or stay between 0 an for each row. If we trace the tme-varyng coeffcents, then we can mplement the shft-regster encoer of Fg. 3. We set the wnow sze w as γ () snce the largest stance between nonzero elements n a row of H s γ (). The wnow sze can be set fferently for other coes. In the slng wnow, the frst entry correspons to g 0, an the last entry to g γ (). Let us efne the tme, t = 0 when the wnow starts from the frst row. The ntal status of coeffcents s 0. In the coe constructon, note that g can exst only f = γ () γ ( k) for k. In other wors, we only have to conser coeffcents an other than those are all zero. For a such coeffcent g, t s on at tme t = S k, an wll last untl the wnow reach the last row ( t = S ) f there s a connecton for k-sr matrx n the last row. Otherwse, t wll be off at the last row. Fg. 5 shows the tmng agram of coeffcents. From Observaton, note that there s a connecton for k-sr matrx n the last row f

17 7 γ ( k) + S S = an no connecton f the value s 0. Then, the coeffcents of the generator k polynomal g ( x ) can be represente as δ( γ() γ( )){ ( ) δ( γ( ) ) ( )}, g = + k ut S k + S S ut S k k k = where we efne the unt step functon as follows:, t 0 ut () = 0, t < 0. For the above Example 3 when M =7, g0 = u( t 3) u( t 6), g = u( t 5), g = u( t 6). As mentone earler, the propose slng wnow encong metho can be apple to other block coes f the nonsystematc part of ther party-check matrx has lower-trangular form as shown n Fg. 4. In fact, the wnow sze can be lowere f the lower-trangular form n Fg. 4 has lower-trangular 0 s n t, whch can be attempte by column an row permutaton for a gven party-check matrx. Another way to mplement the encoer of the propose E RC coes s by usng a smple teratve erasure ecoer. Recall that all the noes n k-sr matrx can be recovere n k teratons of erasure econg snce they are all k-sr noes. For the propose coes, even f all the party bts are erase, we can obtan the exact party bts wthn (+) teratons usng a smple erasure ecoer or a general message-passng LDPC ecoer as long as the systematc bts are known exactly (ths s the case at the encoer). In a transcever system, ths can be a bg avantage n terms of complexty. We only nee to prove an LDPC ecoer for both encong an econg, an o not nee any extra encoer. Even though we may not be able to use the shft-regster mplementaton of slng-wnow metho for the encoer when N () < M, we can easly apply the effcent encong metho v propose n []. Followng the notaton n [], let the party-check matrx H be represente as H A B C = D E F. Then, A D s the systematc part of the B E RC coes, = L E s the submatrx of the nonsystematc part consstng of noes wth egree hgher than two an C = T F s the

18 8 submatrx of the nonsystematc part consstng of egree two noes. For E RC coes, we know the exact sequence of the matrx T. Furthermore, the matrx C s a lower trangular wth ones on the agonal. Thus preprocessng s not requre for puttng the matrx n the form use n []. Ths makes t easy to apply the effcent encong technques n [] to E RC coes. V. SIMULATIONS In ths secton we present smulaton results of E RC coes an the compare ther performance wth eira coes an general rregular LDPC coes. We conser rate-/ mother coes wth block length of 00. For a far comparson we use the egree strbutons presente n [5] for rate-/ coes: λ 6 ( x) = x x x 5 6 ρ( x) = 0.4x x. However, the actual egree strbutons for the E RC coes are slghtly fferent to compensate for the rght egree of H. These are gven below. λ 6 ( x) = x x x ρ( x) = x x x x x x x. The progressve ege growth (PEG) algorthm n [] s apple to esgn the systematc matrx H to mprove grth characterstcs for both eira coes an E RC coes. The PEG algorthm was also use to generate the general rregular LDPC coes. Frst, we conser ranom puncturng for the puncturng strategy for eira coes an general rregular LDPC coes. The puncturng performance comparsons between eira coes an E RC coes are shown n Fg. 6. From Fg. 6, the E RC coes show more powerful puncturng performance at hgher coe rates. At a rate of 0.8 an BER = 0-5, E RC coes outperform eira coes by over 0.8B. A smlar performance gap was observe n the comparsons wth general rregular LDPC coes (the curve s omtte ue to lack of space). Next, we apply the puncturng algorthm propose n [8-9] to eira coes an general rregular LDPC coes. As mentone earler, ths puncturng algorthm has a lmt on the number of low-sr

19 9 noes that t can fn. In fact, the puncturng algorthm n [8-9] assgns 300 noes as -SR noes, an cannot fn further k-sr noes (k ) f we try to maxmze the number of -SR noes. To get a hgh rate (R = 0.7, 0.8, 0.9) n eira coes, we puncture ranomly after the puncturng lmtaton of rate 0.67, whch estroys the prevous tree structure of -SR noes resultng n poor performance. To ncrease the number of varable noes that can be puncture for eira coes, one can mpose a lmtaton on the number of the lower-sr noes when the puncturng algorthm n [8-9] s apple, thus trang off fewer -SR noes for more number of -SR an 3-SR noes. In ths case, however, the puncturng performance for lower rate s worse than the case when -SR noes are maxmze. For general rregular LDPC coes, we can fn 389 -SR noes, 45 -SR noes, 3-SR noes, so the maxmum puncturng rate s Above the puncturng lmt, we apply ranom puncturng to get hgher rates. The puncturng performance of eira coes an general rregular LDPC coes wth the puncturng algorthm n [8-9] are shown n Fg. 7 an 8. Even wth the best effort ntentonal puncturng algorthm n [8-9], the E RC coes show better puncturng performance across the entre range of rates, especally at hgher rates. For coe rate of 0.9 an BER of 0-5 the E RC coes outperform the eira coes an general rregular LDPC coes by 0.7B an.5 B respectvely. For practcal purposes, esgnng a low rate E RC coe an provng a we range of rates by puncturng are useful. There are other methos to lower the rates such as extenng an shortenng. However, these methos often ncrease harware complexty or the performance of lower rate coe may not be goo enough. On the other han, puncture low rate (R < 0.5) stanar rregular mother coes have ba performance at hgh puncturng rates. The E RC coes show relatvely less performance egraaton when puncture as compare to other LDPC coes. For E RC coes, all the egree- noes n the partes can be puncture. As an example, we conser a rate-0.4 mother coe of whch egree strbutons are optmze n AWGN channel (ege perspectve): λ 9 ( x) = 0.947x x x 5 ρ( x) = x. Snce we assgn all the egree- noes to partes an hgher egree noes to messages, 88.4% of the partes are egree- noes an the remanng.6% of the partes are egree-3 noes from the

20 0 above egree strbutons. Thus, the structure of E RC coes s change from the orgnal one, an the E RC coes can acheve rate of 0.85 snce all the egree- noes can be puncture. For rate-0.4 mother coe wth N = 000, K = 800, an N () = 06, we have the epth = 4, an γ() = 600, γ() = 300, γ(3) = 50, γ(4) =. In aton, the E RC coes can have perfect rght egree concentraton at egree 6. We apply the PEG algorthm to generate matrx other than egree- partes. To compare the puncturng performance, the general rregular LDPC coes wth the same egree strbutons as above are generate by usng the PEG algorthm. The best-effort puncturng algorthm n [8-9] s apple to the general rregular LDPC coes. The maxmum achevable rate of ths general rregular LDPC coe s 0.69 wth puncturng algorthm n [8-9]. So, after the lmt we apply ranom puncturng. The puncturng performance comparson between E RC coes an general rregular LDPC coes s epcte n Fg. 9 an 0. In Fg. 9 an 0, the E RC coes show goo performance over a we range of rates 0.4~0.85. At a BER of 0-5 n Fg. 9, the E RC coes outperform the general rregular LDPC coes by.0b an.7b at rate 0.8 an 0.85, respectvely. The same tren can be observe n FER performance n Fg. 0. v VI. CONCLUSIONS We have propose a new class of coes calle E RC coes that have several esrable features. Frst, the coes are effcently encoable. We have presente a shft-regster base mplementaton of the encoer whch has low-complexty an emonstrate that a smple erasure ecoer can be use for the encong of these coes. Thus, we can share a message-passng ecoer for both encong an econg f t s apple to transcever systems whch requre an encoer/ecoer par. Secon, we have shown that the nonsystematc part of the party-check matrx are cycle-free, whch ensures goo coe characterstcs. From smulatons, the performance of the E RC coes (mother coes) s as goo as that of eira coes an other rregular LDPC coes. Thr, the E RC coes show better performance uner puncturng than other rregular LDPC coes an eira coes n all ranges of coe rates an are partcularly goo at hgh rates. Fnally, the E RC coes can prove goo performance over a we range of rates even when they are esgne for rates lower

21 than 0.5. We beleve that these characterstcs of E RC coes are valuable when they are apple to IR Hybr-ARQ systems. APPENDIX Proof of Fact : From the efnton, M shoul be M <. By efnton, M can be represente by M = γ () + R, where R s the remaner when M s ve by,.e., R = 0 or. Then, we have M γ () = γ() + R = γ() + R M γ () γ() = γ() + R = γ(3) + R 3 M γ () γ() γ( ) = γ( ) + R = γ( ) + R (a) In the above equatons, the remaners can be R, R,..., R = 0 or. From the equatons above, we also have ( γ ) ( γ ) M + R = () + R = () + R ( γ ) ( γ ) M + R + R = () + R = (3) + R 3 ( γ ) M + R + R + R = ( ) + R = γ( ) + R (b) ( γ ) M + R + R + R = + R (c) ( ) In equaton (b), the LHS s strctly greater than - from the range of M. So, γ ( ) n RHS snce R = 0 or. On the other han, γ ( ) + R n equaton (c) has to be snce the sum of the LHS of (c) s at most + -. Thus, we conclue that γ ( ) = an R = 0. Then, from (a), we have γ () + γ() + + γ( ) = M. Now, note that γ () γ() γ( ) an snce γ ( ) =, therefore γ ( k) for k. Proof of Lemma : Note that H s lower-trangular wth ones on the agonal. Therefore n the case when j () k =, snce h, j D ( D γ ) = + for 0 j γ () we have a set of columns whose

22 frst entry s on the agonal. Therefore the frst entry of these columns s connecte to a row of egree- an the lemma hols for k =. Now conser k. The j k -th column n the k-sr matrx has a sequence s gven by γ ( ) ( ) h D D jk+ Sk k k, j = + k jk+ Sk jk+ Sk We shall emonstrate that the frst entry of = D + D, where 0 j γ ( k). k h k, j s connecte to a column n the l-sr matrx for k l < k. An mmeate consequence of the lower-trangular nature of H s that connecte to the secon entry of h k, j k can only be h l, j l, the j l -th column n the l-sr matrx. Suppose that the secon entry of the j l -th column n the l-sr matrx s connecte to the frst entry of the j k -th column n the k-sr matrx. Ths mples jl + Sl = jk + Sk. Clearly jl = jk + Sk Sl 0 snce k > l an 0 j γ ( k). We shall now show that j = j + S S γ () l. Ths means that for a k l k k l gven j k, t s possble to fn a unque column j l belongng to the l-sr matrx to whch t s connecte. From the proof of Fact, we have S = M γ ( ) R, where R = 0 or. Snce S + k γ ( k) = S an k Sk S, we have jl = jk + Sk Sl γ ( k) + Sk Sl = Sk Sl S Sl = M γ( ) R ( M γ( l) Rl) = γ () l Rl γ () l. Therefore, for a gven j k, we can fn a corresponng j l n the l-sr matrx for l < k. Note that the frst entry of j k s connecte to the corresponng j l. Snce the matrx s lower-trangular, ths entry cannot have any connecton wth a m-sr matrx where m > k. Therefore ths partcular row has egree exactly k. Ths conclues the proof. REFERENCES [] T. Rcharson an R. Urbanke, Effcent Encong of Low-Densty Party-Check Coes, IEEE Trans. Inform. Theory, vol. 47, no., pp , Feb. 00.

23 [] S. Ln, L. Chen, J. Xu an I. Djurjevc, Near Shannon lmt quascyclc low-ensty party-check coes, n Proc. 003 IEEE GLOBECOM Conf. San Francsco, CA, Dec [3] M. Tanner, A recursve approach to low complexty coes, IEEE Trans. Inform. Theory, vol. IT-7, pp , Sept. 98. [4] H. Jn, A. Khanekar, an R. McElece, Irregular repeat-accumulate coes, n Proc. n. Int. Symp. Turbo Coes an Relate Topcs, Brest, France, pp. -8, Sept [5] M. Yang, W. E. Ryan, an Y. L, Desgn of Effcently Encoable Moerate-Length Hgh-Rate Irregular LDPC Coes, IEEE Trans. Comm., vol. 5, no. 4, pp , Apr [6] J. Hagenauer, Rate-compatble puncture convolutonal coes (RCPC coes) an ther applcatons, IEEE Trans. Comm., vol. 36, pp , Apr [7] J. Ha, J. Km, S. W. McLaughln, Rate-Compatble Puncturng of Low-Densty Party-Check Coes, IEEE Trans. Inform Theory, vol.50, no., Nov [8] J. Ha, J. Km, D. Klnc, an S. W. McLaughln, "Rate-Compatble Puncture Low-Densty Party-Check Coes wth Short Block Lengths," IEEE Trans. Inform. Theory, vol. 5, no., Feb [9] J. Ha, J. Km, an S. W. McLaughln, Puncturng for Fnte Length Low-Densty Party-Check Coes, n Proc. Int. Symp. Inform. Theory, Chcago, 004. [0] M. R. Yazan an A. H. Banhashem, On Constructon of Rate-Compatble Low-Densty Party-Check Coes, IEEE Comm. Letters, vol. 8, no. 3, pp. 59-6, Mar [] T. Rcharson, A. Shokrollah, an R. Urbanke, Desgn of capacty- approachng rregular low-ensty party-check coes, IEEE Trans. Inform. Theory, vol. 47, pp , Feb. 00. [] X. Hu, E. Eleftherou, an D. M. Arnol, Progressve ege-growth Tanner graphs, n Proc. IEEE GLOBECOM, San Antono, Texas, Nov. 00, pp [3] T. Tan, C. Jones, J. D. Vllasenor, an R. D. Wesel, Selectve Avoance of Cycles n Irregular LDPC Coe Constructon, IEEE Trans. On Comm., vol. 5, no. 8, pp. 4-47, 004 [4] A. Ramamoorthy an R. D. Wesel, Constructon of Short Block Length Irregular Low-Densty Party-Check Coes, n Proc. IEEE Int. Conf. on Comm., Pars, June 004. [5] W. Weng, A. Ramamoorthy, an R. D. Wesel, Lowerng the Error Floors of Hgh-Rate LDPC Coes by Graph Contonng, VTC 004, Los Angeles, Calforna. 3

24 Fg.. k-sr noe n a graph 4

25 Fg.. Constructon of the Party-check matrx of the propose coes. 5

26 Fg. 3. An example of shft-regster mplementaton of E RC coes. 6

27 Fg. 4. Nonsystematc part of a party-check matrx for applyng slng wnow encong metho. 7

28 Fg. 5. Tmng agram of coeffcents of slng wnow encoer. 8

29 9 0 - E RC coes (K=600) eira coes (K=600) BER E b /N o [B] Fg. 6. The puncturng BER performance comparson between E RC coes an eira coes wth ranom puncturng. Flle crcles are for E RC coes an unflle crcles are for eira coes. Rates are 0.5 (mother coes), 0.6, 0.7, 0.8, an 0.9 from left to rght.

30 E RC coes (K=600) eira coes (K=600) BER E b /N o [B] Fg. 7. The puncturng BER performance comparson between E RC coes an eira coes wth puncturng algorthm n [8-9]. Flle crcles are for E RC coes an unflle crcles are for eira coes. Rates are 0.5 (mother coes), 0.6, 0.7, 0.8, an 0.9 from left to rght

31 3 0 - E RC coes (K=600) general LDPC coes (K=600) BER E b /N o [B] Fg. 8. The puncturng BER performance comparson between E RC coes an general rregular LDPC coes wth puncturng algorthm n [8-9]. Flle crcles are for E RC coes an unflle crcles are for general rregular LDPC coes. Rates are 0.5 (mother coes), 0.6, 0.7, 0.8, an 0.9 from left to rght.

32 3 0 - E RC coes (K=800) General rregular LDPC coes (K=800) BER E b /N o [B] Fg. 9. The puncturng BER performance comparson between E RC coes an general rregular LDPC coes wth puncturng algorthm n [8-9]. Flle crcles are for E RC coes an unflle crcles are for general rregular LDPC coes. Rates are 0.4 (mother coes), 0.5, 0.6, 0.7, 0.8, an 0.85 from left to rght.

33 E RC coe (K=800) General rregular LDPC coes (K=800) FER E b /N o [B] Fg. 0. The puncturng FER performance comparson between E RC coes an general rregular LDPC coes wth puncturng algorthm n [8-9]. Flle crcles are for E RC coes an unflle crcles are for general rregular LDPC coes. Rates are 0.4 (mother coes), 0.5, 0.6, 0.7, 0.8, an 0.85 from left to rght.