A Lower Bound on the Error Exponent of Linear Block Codes over the Erasure Channel

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1 A Lowr Bod o th Error Expot of Liar Block Cods ovr th Erasr Chal Erico Paolii DEI, Uivrsity of Bologa via dll Uivrsità 50, Csa (FC Italy .paolii@ibo.it Gialigi Liva KN-SAN, Grma Arospac Ctr Müchr Strass 20, Wsslig, Grmay Gialigi.Liva@dlr.d arxiv: v cs.it] 2 Ja 209 Abstract A lowr bod o th maximm liklihood (ML dcodig rror xpot of liar block cod smbls, o th rasr chal, is dvlopd. Th lowr bod trs to b positiv, ovr a smbl spcific itrval of rasr probabilitis, wh th smbl wight spctral shap fctio tds to a gativ val as th fractioal codword wight tds to zro. For ths smbls w ca thrfor lowr bod th block-wis ML dcodig thrshold. Two xampls ar prstd, amly, liar radom parity-chck cods ad fixd-rat Raptor cods with liar radom prcodrs. Whil for th formr a fll aalytical soltio is possibl, for th lattr w ca lowr bod th ML dcodig thrshold o th rasr chal by simply solvig a 2 2 systm of oliar qatios. I. INTRODUCTION I this papr, a lowr bod o th ML dcodig rror xpot of liar cod smbls wh sd ovr rasr chals (ECs is drivd. Th calclatio of th bod rqirs th kowldg of th smbl wight spctral shap oly (dr a rlativly mild coditio, as it will b discssd latr. A gral lowr bod o th rror xpot, for ay discrt mmory-lss chal (DMC, was itrodcd ]. Its calclatio ivolvs th valatio of th maximm ratio btw th smbl avrag wight mrator (AWE ad th AWE of th radom liar cod smbl. Th tchiq of ] was sd i 2] to driv a lowr bod o th ML dcodig rror xpot of (xprgatd low-dsity paritychck (LDPC cod smbls 3]. Th bod o th rror xpot itrodcd i this papr is drivd from th tight io bod o th rror probability dr ML dcodig ovr th EC for liar block cod smbls of 4], 5]. A similar approach was followd i 6] to obtai a lowr bod o th rror xpot for xprgatd LDPC cod smbls. Or work xtds th rslt of 6] to ay liar cod smbl for which th wight spctral shap is kow, with th oly rqirmt that th logarithm of th AWE of th cod smbl (ormalizd to th block lgth covrgs i th block lgth iformly to th wight spctral shap. Th lowr bod trs ot to b positiv, ovr a smbl spcific itrval of rasr probabilitis, wh th smbl wight spctral shap fctio tds to a gativ val as th fractioal codword wight tds to zro. For A shortr vrsio of this papr, omittig som proofs, has b sbmittd to th 209 IEEE Itratioal Symposim o Iformatio Thory (ISIT. th liar radom cod smbl, w show that th bod o th rror xpot rcovrs Gallagr s radom codig rror xpot 7]. Th kowldg of th lowr bod o th rror xpot allows obtaiig a lowr bod o th smbl s ML rasr dcodig thrshold. As a xampl of applicatio, w driv a lowr bod o ML rasr dcodig thrshold for th smbl of fixd-rat Raptor cods 8] itrodcd i 9]. Rmarkably, th rslt is obtaid by simply solvig a 2 2 systm of oliar qatios. For th aalyzd smbls, th bod o th rror xpot drivd i this papr shows to b cosidrably tightr tha th gral bod of ] wh th lattr is spcializd to th biary rasr chal (BEC. II. PRELIMINARIES W cosidr trasmissio of liar block cods costrctd ovr F q, th fiit fild of ordr q, o a mmorylss q-ary rasr chal (q-ec o which ach codword symbol is corrctly rcivd with probability ǫ ad rasd with probability ǫ. A cod smbl is dfid as a st of cods alog with a probability distribtio o sch cods. W dot by C(,r,q a gric smbl of liar block cods ovr F q of lgth ad dsig rat r, ad by C C(,r,q a radom cod i th smbl. Th block-wis ML dcodig rror probability of C ovr th q-ec is idicatd as P B (C,ǫ ad its xpctatio ovr th smbl as E C(,r,q P B (C,ǫ]. W dfi th ML dcodig thrshold for th smbl C(,r,q ovr th q-ec as ǫ ML = spǫ (0, : E C(,r,qP B (C,ǫ] 0 as. Or startig poit is a ppr bod o E C(,r,q P B (C,ǫ] dvlopd i 4] for biary cods ad xtdd i 5] to obiary os. W hav ( E C(,r,q P B (C,ǫ] ǫ ( ǫ + ( r = =( r+ ( ǫ ( ǫ mi, w= ( w Aw ( w A(x = i=0 A ix i is th AWE of C. Giv th AWE A(x th growth rat of th wight distribtio, or wight spctral shap, of C(,r,q is dfid as 2 G(ω = lim loga ω. 2 I this papr all logarithms ar to th bas 2. (

2 W dot th Kllback-Liblr (KL divrgc btw two Brolli distribtios with paramtrs ad v, both i (0,, by D(,v = log v +( log v. Morovr, w dot by H b ( = log ( log(, 0, th biary tropy fctio. Throghot th papr w mak s of th lowr ad ppr bods ( + 2H b(k/ 2 H b(k/ (2 k o th biomial cofficit, valid for all ogativ itgrs k. For ay two pairs (x,y ad (x 2,y 2 of rals, w writ (x,y (x 2,y 2 wh x x 2 ad y y 2. Rcall that a sqcf of ral-vald fctios oa R covrgs iformly to th fctio f : A R o A 0 A if for ay ε > 0 thr xists 0 (ε sch that, for all 0 (ε, f (x f(x < ε x A 0. W writ f f to idicat that f covrgs to f iformly. A cssary ad sfficit coditio for iform covrgc is stablishd by th followig lmma 0, Th. 7.0]. Lmma. Ltlim f (x = f(x x A 0. Thf f o A 0 if ad oly if sp x A0 f (x f(x 0 as. Th followig rslt will also b sfl. Lmma 2. Lt f,g : A R R b bodd fctios. Th if f(x if g(x sp f(x g(x. x A x A x A III. MAIN RESULTS This sctio prsts th mai rslts of this papr. A lowr bod o th asymptotic rror xpot o liar block cod smbls ovr th rasr chal is first dvlopd i Thorm. Th, Thorm 2 shows how this bod allows lowr bodig ǫ ML for smbls for which G(ω is cotios i (0,] ad gativ for small ogh ω. Thorm. Cosidr a liar block cod smbl C(,r,q ad lt its wight spctral shap G(ω b wll-dfid i0, ]. If loga ω G(ω th lim log E C(,r,qP B (C,ǫ] E G (ǫ Th fctio f ǫ (δ is dfid as ad E G (ǫ = if δ (0,] f ǫ(δ. (3 f ǫ (δ = D(δ,ǫ+g + (δ g + (δ = max0,g(δ (4 ( ω ] g(δ = if δh b +H b (ω G(ω. (5 ω (0,δ] δ Proof: Th proof is orgaizd ito two parts. W first ppr bod th right-had sid of ( to obtai a lowr bod o log E C(,r,qP B (C,ǫ]. Th w tak th limit of th lowr bod as. Lowr bodig log E C(,r,qP B (C,ǫ]: Th ppr bod ( ca b writt i th qivalt, mor compact form E C(,r,q P B (C,ǫ] ( ǫ ( ǫ mi = Lttig w = ω ad = δ, w hav 3, w= E C(,r,q P B (C,ǫ] ( a max ǫ ( ǫ,..., ( ] Aw mi, ( w w= w ( b max ǫ ( ǫ,..., (( ] Aw mi, max ( w,..., w ( w = max ǫ δ ( ǫ ( δ δ,..., δ (( ] δ δ Aω mi, max ( ω,...,δ ω ω c max 2 (H b(δ+δlogǫ+( δlog( ǫ δ,..., mi, δ(+ (δhb(ω max 2 ω,...,δ d = max 2 D(δ,ǫ δ,..., mi, δ(+ (δhb(ω max sp mi δ Q (0,] f = sp mi 2 D(δ,ǫ, δ(+ δ (0,] 2 D(δ,ǫ, δ(+ ω,...,δ 2 sp ω Q (0,δ] sp ω (0,δ] ( w δ H b(ω+ δ H b(ω+ Aw ( w. ] log Aω ] log Aω ] 2 (δh b( ω δ H log Aω b(ω+ ] 2 (δh b( ω δ H log Aω b(ω+ I th abov dvlopmt: a ad b ar d to h l= f(l hmax l N h f(l. Morovr: c is d to applicatio of th ppr ad lowr bods i (2; d to xpadig H b (δ ad rcallig th dfiitio of KL divrgc; to th fact that th sprmm ovr Q (0,] ppr bods th maximm ovr,..., ad, similarly, th sprmm ovr Q (0,δ] ppr bods th maximm ovr,...,δ; f to th dsity of Q. I th fial xprssio, both δ ad ω ar cosidrd as ral variabls. Th bod (6 is valid for ay lgth, rat r, ad fild ordr q. 3 For otatioal simplicity hraftr w writ A ω i li of A ω. (6

3 Nxt w xploit (6 to bod log E C(,r,qP B (C,ǫ] from blow. Owig to logarithm mootoicity w obtai log E C(,r,qP B (C,ǫ] if δ (0,] f (δ. (7 f ǫ, (δ = log+d(δ,ǫ+max 0, if ω (0,δ] ( log q δ(+ ( ω δh b +H b (ω loga ω. δ 2 Takig th limit: Nxt, w tak th limit as i both sids of (7. To kp th otatio compact w dfi h δ, (ω = log ω b( δ(+ δh +H b (ω loga ω δ g (δ + = max0,g (δ. g (δ = if ω (0,δ] h δ,(ω ad W also dfi ( ω h δ (ω = δh b +H b (ω G(ω (8 δ so that g(δ dfid i (5 flfills g(δ = if ω (0,δ] h δ (ǫ. W start by showig that f ǫ, f ǫ o ay itrval a,] sch that 0 < a <. W first show that g (δ g(δ o a,]. To this prpos w writ sp g (δ g(δ = sp if h δ,(ω if h δ(ω ω (0,δ] ω (0,δ] a sp sp ω (0,δ] = sp sp ω (0,δ] b sp h δ, (ω h δ (ω log δ(+ δ(+ log + loga ω G(ω + sp loga ω G(ω ω (0,δ] a is d to Lmma 2 ad b to triagl iqality. I th last xprssio, th first addd covrgs to zro as sic q is costat ad δ a,] with a > 0. Morovr, th scod addd covrgs to zro d to th hypothsis that (/loga ω G(ω ad by Lmma. Agai by Lmma w cocld that g (δ g(δ. Uiform covrgc of g (δ to g(δ trs ito iform covrgc of g + (δ to g + (δ. I fact, w hav g + (δ g + (δ g (δ g(δ for all δ ad, which implis 0 sp g + (δ g+ (δ sp g (δ g(δ. By sqz thorm w hav sp g (δ g + + (δ 0 as, ad thrfor g + g + by Lmma. W ar ow i a positio to prov iform covrgc of f ǫ, to f ǫ. I fact, w hav sp f ǫ, (δ f ǫ (δ = sp log +g+ (δ g+ (δ log + sp g (δ g + + (δ w applid triagl iqality. Covrgc to zro of th last xprssio is garatd by g + g +. Uiform covrgc of f ǫ, (δ to f ǫ (δ lads s to th statmt, as follows. Rcall that, if f f o A 0 th lim if x A0 f (x = if x A0 lim f (x = if x A0 f(x, i.., w ca xchag limit ad ifimm. Hc w ca writ lim log E C(,r,qP B (C,ǫ] lim if f ǫ,(δ = if = if f ǫ(δ if δ (0,] f ǫ(δ. lim f ǫ,(δ I th prvios qatio array, th first iqality is jstifid by th fact that if α α, β β, ad α β for all (possibly, largr tha som 0, th α β. Morovr, th two qalitis ar jstifid by f ǫ, (δ f ǫ (δ. Rmark. Th fctio E G (ǫ giv by (3 is ogativ for all 0 < ǫ <, sic it is dfid as th ifimm of th sm of two ogativ qatitis. Morovr, sic E G (ǫ bods th rror xpot of th giv smbl from blow, it mst flfill E G (ǫ = 0 for all r ǫ. Rmark 2. Th lowr bod E G (ǫ trs ot to b slss for all smbls for which G(ω 0 as ω 0 +, as for ay sch smbl w hav E G (ǫ = 0 for all 0 < ǫ <. To s this, simply obsrv that dr this sttig w hav if ω (0,δ] h δ (ω lim ω 0 + h δ (ω = 0 for all 0 < δ, ad thrfor g + (δ = 0 for all 0 < δ. Th, E G (ǫ = if δ (0,] D(δ,ǫ = 0 for all 0 < ǫ < (simply tak δ = ǫ. Th followig lmma charactrizs th fctio g + (δ dfid i (4. Lmma 3. Th fctio g + (δ has th followig proprtis: g + (δ = 0 for all r δ ; 2 If G(ω is cotios i (0, th g + (δ is oicrasig ad cotios; 3 If G(ω is cotios i (0, ad lim ω 0 + G(ω = γ < 0 th: a lim δ 0 + g + (δ = γ ; b δ = spδ (0, r] : g + (δ > 0 is strictly positiv; c g + (δ > 0 δ (0,δ ; g + (δ = 0 δ δ,]. Proof: Tak ay r δ ad lt ǫ = δ. W mst hav E G (δ = if δ (0,] D(δ,δ + g + (δ] = 0 (Rmark. This yilds δ = δ, hc g + (δ = E G (δ = 0. 2 Th fctioh(ω,δ = h δ (ω is cotios ad drivabl with rspct to (w.r.t. δ. Sic h(ω, δ/ δ = log((δ ω/δ < 0, w hav h δ (ω > h δ2 (ω 0 < ω δ < δ 2. (9 Morovr, cotiity of G(ω trs ito cotiity of h δ (ω also w.r.t. ω. W dfi h δ (0 = lim ω 0 + h δ (ω, so that h δ (ω is cotios (w.r.t. ω o th compact 0,δ]. W lt ˆω δ = argmi ω 0,δ] h δ (ω. Takigz < y ad sig (9, w ca writ g(y = h y (ˆω y < h y (ˆω z < h z (ˆω z = g(z which shows that

4 g(δ is mootoically dcrasig ad, as a cosqc, that g + (δ is o-icrasig. Nxt, w prov cotiity of g(δ as it implis cotiity of g + (δ. W d to show that for ay θ > 0 thr xists α(θ s.t. z y < α(θ implis g(z g(y < θ. It is asy to prov that for ay θ > 0 thr xists α (θ s.t. z y < α (θ implis h z (ω h y (ω < θ/2 for all ω (0,miy,z. 4 W rfr to this proprty as Proprty. Morovr, cotiity of h δ (ω w.r.t. ω, srs that for ay θ > 0 thr xists α 2 (θ s.t. ξ ω < α 2 (θ implis h δ (ξ h δ (ω < θ/2. W rfr to this proprty as Proprty 2. Hraftr w addrss th casz < y, th argmt forz > y big vry similar. Lt y z < miα (θ,α 2 (θ ad rcall th abov dfiitio of ˆω y ad ˆω y. W d to distigish two cass. Cas : ˆω y < z. Proprty implis h z (ˆω y h y (ˆω y < θ/2. By (9 w hav h z (ˆω y > h y (ˆω y ad thrfor h z (ˆω y h y (ˆω y = h z (ˆω y h y (ˆω y = h z (ˆω y h z (ˆω z +h z (ˆω z h y (ˆω y a = h z (ˆω y h z (ˆω z + h z (ˆω z h y (ˆω y a is d to th dfiitios of ˆω z ad ˆω y ad to z < y. Ths, g(z g(y = h z (ˆω z h y (ˆω y h z (ˆω y h y (ˆω y < θ/2 < θ. Cas 2: ˆω y z. Proprty ad proprty 2 imply h z (z h y (z < θ/2 ad h y (z h y (ˆω y < θ/2, rspctivly, yildig (by triagl iqality h z (z h y (ˆω y h z (z h y (z + h y (z h y (ˆω y < θ. Howvr, w also hav h z (z h y (ˆω y a = h z (z h y (ˆω y = h z (z h z (ˆω z +h z (ˆω z h y (ˆω y b = h z (z h z (ˆω z + h z (ˆω z h y (ˆω y both a ad b ar d to h z (z h z (ˆω z h y (ˆω y. Hc, g(z g(y = h z (ˆω z h y (ˆω y h z (z h y (ˆω y < θ. 3a Lt s look at th bhavior of g + (δ as δ 0 +. Sic 0 < ω δ, w mst also hav ω 0 +, which yilds lim δ 0 + g + (δ = max0,lim (δ,ω (0 +,0 +,0<ω δh(ω,δ = γ. 3b Th fctio g + (δ tds to a positiv mbr as δ 0 + ad is zro for ay δ btw r ad. Sic th fctio is cotios, δ = spδ (0, r] : g + (δ > 0 mst b strictly positiv. 3c Sic g + (δ is also o-icrasig, it mst b positiv o th whol itrval (0,δ ad mst b ll ls (i.., o δ,]. Th xt thorm shows that, dr coditios o G(ω, thr xists a itrval of vals of ǫ ovr which E G (ǫ is positiv. For th corrspodig smbls, E G (ǫ is thrfor sfl to lowr bod ǫ ML. 4 Th proof is basd o th obsrvatio that h z(ω h y(ω icrass mootoically with ω, yildig h z(ω h y(ω h z(m h y(m = MH b ( z y /M, M = maxz,y. Cotiity of H b ( lads to th coclsio. Thorm 2. Ltδ = spδ (0, r] : g + (δ > 0 r. If G(ω is cotios i (0, ad lim ω 0 + G(ω < 0, th E G (ǫ > 0 ǫ (0,δ ad E G (ǫ = 0 ǫ δ,], ad thrfor ǫ ML δ. Proof: Tak ay ǫ s.t. δ ǫ. W hav 0 E G (ǫ = if δ (D(δ,ǫ + g + (δ D(ǫ,ǫ + g + (ǫ = 0, ad thrfor E G (ǫ = 0. Tak ow ay ǫ s.t. 0 < ǫ < δ. Sic D(δ,ǫ ad g + (δ ar both ogativ fctios, to hav E G (ǫ = if δ (D(δ,ǫ+g + (δ = 0 w d to fid δ s.t. both D(δ,ǫ ad g + (δ ar ll. To hav D(δ,ǫ = 0 w d to choos δ = ǫ; howvr, sic 0 < ǫ < r w hav g + (ǫ > 0 ad thrfor E G (ǫ > 0. I th xt sctio w prst rslts for two smbls flfillig th hypothss of Thorm 2, amly, th smbl of liar radom parity-chck cods ovr F q ad th smbl of fixd-rat biary Raptor cods with liar radom prcodrs 9]. For th first smbl th fctio E G (ǫ ca b obtaid aalytically ad coicids with Gallagr s radom codig bod ovr th q-ec. For th scod o, E G (ǫ shall b comptd mrically. Howvr, if oly th lowr bod o ǫ ML is of itrst, it may b comptd by simply solvig a 2 2 systm of qatios. IV. RESULTS FOR SPECIFIC ENSEMBLES A. Liar Radom Parity-Chck Cods Cosidr th smbl of liar radom parity-chck cods ovr F q idcd by a ( r radom parity-chck matrix whos tris ar idpdt ad idtically distribtd (i.i.d. radom variabls iformly distribtd i F q. For this smbl w hav th followig rslt. Thorm 3. For th smbl of liar radom parity-chck cods w hav δ = r ad thrfor ǫ ML = r. Morovr log ( ǫ q +ǫ rlogq 0 < ǫ < ǫ c E G (ǫ = D( r,ǫ ǫ c ǫ < r (0 0 ǫ r ǫ c = ( r/(+(r. Proof: Th xpctd wight mrator of th liar radom parity-chck smbl is A ω = ( ω (q ω q ( r ad th corrspodig wight spctral shap is G(ω = H b (ω+ωlog(q ( rlogq. Uiform covrgc of loga ω to G(ω may b provd i a vry simpl way, by obsrvig that sp ω loga ω G(ω = sp ( ω log H b (ω ω sp log(+ = log(+ ω w applid th lowr bod i (2. Sic log(+ 0 as, w cocld that loga ω G(ω. Th fctio h δ (ω dfid i (8 assms th form ( ω h δ (ω = δh b ωlog(+( rlogq. δ

5 Lt ˆω(δ = q q δ. It is asy to s that this fctio tds to ( rlogq wh ω 0 +, is mootoically dcrasig for ω (0,ˆω(δ, taks a miimm at ω = ˆω(δ, ad icrass mootoically for ω (ˆω(δ,δ]. Hc, w hav g(ω = if ω (0,δ] h δ (ω = h δ (ˆω(δ = ( r δlogq so that g + ( r δlogq if0<δ < r (δ= max0,g(δ = 0 if r δ <. Th paramtr δ is thrfor qal to r. Sic ǫ ML δ = r ad ǫ ML r, w obtai ǫ ML = r. Nxt, w dvlop E G (ǫ aalytically. Basd o th abov fidigs, w hav E G (ǫ = mi if D(δ,ǫ+( r δlogq], δ (0, r if D(δ,ǫ δ r,] that immdiatly yilds E G (ǫ = 0 for all ǫ r (it sffics to tak δ = ǫ, corrspodig to th third row of (0. For 0 < ǫ < r w d to aalyz th fctio f ǫ (δ = D(δ,ǫ+ qǫ +(q ǫ. g + (δ = D(δ,ǫ + ( r δlogq. Lt ˆδ(ǫ = Takig th drivativ with rspct to δ, it is immdiat to s that this fctio dcrass mootoically for δ < ˆδ(ǫ, taks a miimm at δ = ˆδ(ǫ, ad icrass mootoically for δ > ˆδ(ǫ. Hraftr w d to distigish th two cass ˆδ(ǫ < r ad ˆδ(ǫ r. It is immdiat to vrify that thy corrspod to0 < ǫ < ǫ c adǫ c ǫ < r, rspctivly, ǫ c = ( r/(+(r. Cas : 0 < ǫ < ǫ c. I this cas th fctio D(δ,ǫ+( r δlogq has a miimm at δ = ˆδ(ǫ. It taks th val D( r,ǫ at δ = r. Thrfor w obtai E G (ǫ = mid(ˆδ(ǫ,ǫ+( r ˆδ(ǫlogq,D( r,ǫ = D(ˆδ(ǫ,ǫ+( r ˆδ(ǫlogq ( ǫ = log +ǫ rlogq q th third xprssio follows from simpl algbraic maiplatio. This yilds th first row of (0. Cas 2: ǫ c < ǫ < r. I this cas th fctio D(δ,ǫ+ ( r δlogq is mootoically dcrasig for δ (0, r, so its ifimm is tak as δ ( r. W obtai E G (ǫ = mid( r,ǫ,d( r,ǫ = D( r,ǫ that corrspods to th scod row of (0. Rmark 3. Itrstigly, th xprssio (0 of E G (ǫ trs ot to coicid with that of Gallagr s radom codig rror xpot for th q-ec ]. B. Fixd-Rat Raptor Cods with Liar Radom Prcodrs I this sbsctio w cosidr biary fixd-rat Raptor cod smbls with liar radom prcodig. A vctor of r iformatio bits is first codd by a otr liar block cod pickd radomly i th smbl of biary liar radom parity-chck cods with dsig rat r o, providig a vctor of r/r o itrmdiat bits. Itrmdiat bits ar frthr codd by a ir fixd-rat Lby-trasform (LT cod of rat r i ad otpt dgr distribtio Ω(x = j Ω jx j, gratig codd bits. Th ovrall dsig rat is r = r o r i. Th wight spctral shap of this smbl was charactrizd i 9]. It is giv by G(ω = H b (ω r i ( r o ν ω (λ 0 ( ν ω (λ = H b (λ+ωlog(ρ(λ+( ωlog( ρ(λ ad λ 0 = λ 0 (ω = argmaxν ω (λ. (2 λ D I (2, D = 0, if Ω j = 0 for ay v j ad D = (0, othrwis. Morovr,ρ(λ = d 2 j= Ω j ( 2λ j ], big d th maximm LT otpt dgr. Agai from 9]: G(ω i ( is cotios. 2 lim ω 0 + G(ω < 0 iff (r i,r o P, P = (r i,r o (0,0 : r i ( r o 3 Th drivativ of G(ω is > max λ D r ih b (λ+log( ρ(λ]. (3 G (ω = log ω ω +log ρ(λ 0 ρ(λ 0. (4 4 G (ω > 0 for 0 < ω < 2 ad lim ω 0 + G (ω = +. Uiform covrgc of loga ω to G(ω ca b provd sig argmts from 9, Sc. III]. Morovr, th hypothss of Thorm 2 ar satisfid wh(r i,r o P, P is giv by (3. As opposd to liar radom parity-chck smbls, i this cas E G (ǫ shall b comptd mrically. Howvr, if oly th lowr bod δ o th ML dcodig thrshold ǫ ML is of itrst, it may b comptd fficitly, as show xt. Thorm 4. Cosidr a biary Raptor smbl with a liar radom prcodr ad lt (r i,r o P. Th ǫ ML δ δ is th smallst ˆδ s.t. (ˆδ,ˆλ 0 is a soltio of th 2 2 systm r i ( r o r i H b (ˆλ 0 ( ˆδlog( ρ(ˆλ 0 = 0 (5 r i log ˆλ 0 ˆλ 0 ˆδ ρ(ˆλ 0 ρ (ˆλ 0 log = 0. (6 Proof: If (r i,r o P th Thorm 2 applis. W hav ǫ ML δ, δ = spδ (0, r] : g + (δ > 0 ad δ > 0. Owig to cotiity of h δ (ω w ca writ g + (δ = max0,h δ (ˆω, ˆω = ˆω(δ = argmax ω (0,δ] h δ (ω. From (8 ad (4 w obtai dh δ (ω dω = log ω δ ω log ρ(λ 0 ρ(λ 0 (7 which rvals how dh δ (ω/dω + as ω δ. 5 Ths, th maximm caot b tak at ω = δ ad ˆω mst b a soltio of dh δ (ω/dω = 0. Dfiig ˆλ 0 = argmax λ D νˆω (λ ad 5 ρ(λ 0 (ω caot covrg to as ω δ for ay 0 < δ r.

6 rcallig (7, aftr som algbraic maiplatio this traslats to ˆω = δρ(ˆλ 0. (8 Th paramtr ˆλ 0 mst b a soltio of dνˆω (λ/dλ = 0. Dvlopig th drivativ w obtai r i log ˆλ 0 + ˆω ρ (ˆλ 0 ˆλ 0 ρ(ˆλ 0 log ( ˆω ρ (ˆλ 0 log = 0. ρ(ˆλ 0 (9 So far w hav show that g + (δ = max0,h δ (ˆω (ˆω,ˆλ 0 is a soltio of th systm of simltaos qatios (8 ad (9. Rcall ow from Thorm 2 that g + (δ > 0 for all 0 < δ < δ ad g + (δ = 0 for all δ δ. This cssarily implis h δ (ˆω > 0 for all 0 < δ < δ ad h δ (ˆω = 0, i.., δ is th smallst ˆδ sch that hˆδ(ˆω = 0, i.., aftr simpl maiplatio, th smallst ˆδ sch that ( r i ( r o r i H b (ˆλ 0 ( ˆδ ( ˆδlog ˆωˆδ = 0. Sbstittig (8 (with δ = ˆδ ito this lattr qatio yilds (5, whil sbstittig it ito (9 yilds (6. Exampl. Lt r o = 0.99, r i = 0.8, ad Ω(x b th LT otpt distribtio of 3GPP Raptor cods, i.., Ω(x = x x x x x x x 40. By dirct calclatio o ca vrify that (r i,r o P. Solvig (5 ad (6 w obtai th iq soltio (ˆδ,ˆλ 0 = ( , from which w cocld that ǫ ML δ = This bod is rlativly tightr that th o obtaid by mployig th gral bod i ], which rtrs ǫ ML V. CONCLUSIONS A lowr bod o th ML rror xpot of liar cod smbls ovr rasr chals has b drivd. Th lowr bod rqirs, dr mild coditios, jst th kowldg of th smbl wight spctral shap. Th applicatio to som liar block cod smbls has b dmostratd. For th spcific cas of fixd-rat Raptor cod smbls, th bod allows to compt a lowr bod o th ML dcodig thrshold, that is rmarkably tightr with rspct to th lowr bod obtaid with stablishd tchiqs. REFERENCES ] N. Shlma ad M. Fdr, Radom codig tchiqs for oradom cods, IEEE Tras. If. Thory, vol. 45, o. 6, pp , Sp ] G. Millr ad D. Brshti, Bods o th maximm-liklihood dcodig rror probability of low-dsity parity-chck cods, IEEE Tras. If. Thory, vol. 47, o. 7, pp , ] R. G. Gallagr, Low-Dsity Parity-Chck Cods. Cambridg, MA: M.I.T. Prss, ] C. Di, D. Proitti, T. Richardso, E. Tlatar, ad R. Urbak, Fiit lgth aalysis of low-dsity parity-chck cods o th biary rasr chal, IEEE Tras. If. Thory, vol. 48, pp , ] G. Liva, E. Paolii, ad M. Chiai, Bods o th rror probability of block cods ovr th q-ary rasr chal, IEEE Tras. Comm., vol. 6, o. 6, pp , J ] D. Brshti ad G. Millr, Asymptotic mratio mthods for aalyzig LDPC cods, IEEE Tras. If. Thory, vol. 50, o. 6, pp. 5 3, J ] R. G. Gallagr, Iformatio Thory ad Rliabl Commicatio. Nw York: Wily, ] M. Shokrollahi, Raptor cods, IEEE Tras. If. Thory, vol. 52, o. 6, pp , J ] F. Lázaro, E. Paolii, G. Liva, ad G. Bach, Distac spctrm of fixd-rat Raptor cods with liar radom prcodrs, IEEE J. Sl. Aras Comm., vol. 34, o. 2, pp , Fb ] W. Rdi, Pricipls of Mathmatical Aalysis, 3rd d. Nw York: McGraw-Hill, 976. ] S. Fashadi, S. O. Ghara, ad A. K. Khadai, Codig ovr a rasr chal with a larg alphabt siz, i Proc IEEE It. Symp. If. Thory, Toroto, Caada, Jl. 2008, pp