POLAR codes proposed by Arıkan [1] have been proved. Does Gaussian Approximation Work Well for The Long-Length Polar Code Construction?

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1 1 Does Gaussia Appoximatio Wok Well fo The Log-Legth Pola Coe Costuctio? Jicheg Dai, Stuet Membe, IEEE, Kai iu, Membe, IEEE, Zhogwei Si, Membe, IEEE, Chao Dog, Membe, IEEE a Jiau Li, Membe, IEEE axiv: v3 [cs.it] 16 Ma 017 Abstact Gaussia appoximatio GA is wiely use to costuct pola coes. Howeve whe the coe legth is log, the subchael selectio iaccuacy ue to the calculatio eo of covetioal appoximate GA AGA, which uses a two-segmet appoximatio fuctio, esults i a catastophic pefomace loss. I this pape, ew piciples to esig the GA appoximatio fuctios fo pola coes ae popose. Fist, we itouce the cocepts of polaizatio violatio set PVS a polaizatio evesal set PRS to explai the essetial easos that the covetioal AGA scheme caot wok well fo the log-legth pola coe costuctio. I fact, these two sets will lea to the ak eo of subsequet subchaels, which meas the oes of subchaels ae misalige, which is a sevee poblem fo pola coe costuctio. Seco, we popose a ew metic, ame cumulative-logaithmic eo CLE, to quatitatively evaluate the emaie appoximatio eo of AGA i logaithm. We eive the uppe bou of CLE to simplify its calculatio. Fially, guie by PVS, PRS a CLE bou aalysis, we popose ew costuctio ules base o a multi-segmet appoximatio fuctio, which obviously impove the calculatio accuacy of AGA so as to esue the excellet pefomace of pola coes especially fo the log coe legths. umeical a simulatio esults iicate that the popose AGA schemes ae citical to costuct the high-pefomace pola coes. Iex Tems Pola coes, Gaussia appoximatio GA, polaizatio violatio set PVS, polaizatio evesal set PRS, cumulative-logaithmic eo CLE. I. ITRODUCTIO POLAR coes popose by Aıka [1] have bee pove to achieve the capacity of ay symmetic biay iput symmetic iscete memoyless chaels B-DMCs ue a successive cacellatio SC ecoe as the coe legth goes to ifiity. Recetly, pola coes have bee ietifie as oe of the chael coig schemes i the 5G wieless commuicatio system ue to its excellet pefomace []. To costuct pola coes, the chael eliabilities ae calculate efficietly usig the symmetic capacities of subchaels o the Bhattachayya paametes fo the biay-iput easue chaels BECs. As a heuistic metho, Aıka has suggeste to use the ecusio which is optimal oly fo BECs also fo othe B-DMCs [3]. Moi et al. egae the costuctio poblem as a istace of esity evolutio DE [4], which theoetically has the highest This wok is suppote by the atioal atual Sciece Fouatio of Chia o & o , BUPT-SICE Excellet Gauate Stuets Iovatio Fu a Huawei HIRP poject. The mateial i this pape was pesete i pat at the Recet Results Sessio of IEEE Iteatioal Symposium o Ifomatio Theoy ISIT, Hog Kog, Jue 015. The authos ae with the Key Laboatoy of Uivesal Wieless Commuicatios, Miisty of Eucatio, Beijig Uivesity of Posts a Telecommuicatios, Beijig , Chia aijicheg, iukai, sizhogwei, ogchao, jli}@bupt.eu.c. accuacy. Cosieig its high computatioal complexity, Tal a Vay evise two appoximatio methos to simplify the calculatio of DE, by which oe ca get the uppe a lowe bous o the eo pobability of each subchael. Tal a Vay s metho has almost o pefomace loss compae with DE [5] [8]. Aftewas, Gaussia appoximatio GA was popose to futhe euce the computatioal complexity of DE [9] without much sacifice i accuacy, which became popula i the costuctio of pola coes thaks to its goo taeoff betwee the complexity a pefomace. I the GA costuctio of pola coes, the bit log-likelihoo atio LLR of each subchael is assume to obey a costait Gaussia istibutio i which the mea is half of the vaiace. Hece, the iteative evaluatio of each subchael eliability is oly ivolve with the mea upate of LLRs. Howeve, the LLR mea upates i check oes still epe o complex itegatio. Cosequetly, fo costuctio of pola coes, the computatioal complexity of exact GA eote by EGA gows expoetially with the polaizatio levels. This makes EGA too complicate to be pactically employe. Theefoe, i pactical implemetatio, like GA utilize i LDPC coes, the well-kow appoximate vesio of GA eote by AGA give by Chug et al. base o a two-segmet appoximatio fuctio is use to spee up the calculatios [10] [1]. Iitially, the appoximatio fuctio i covetioal AGA chose by Chug is suitable fo LDPC coes. Howeve, i piciple, we o t kow whethe this appoximatio metho ca be also goo fo the pola coes. I fact, the calculatio eo of AGA vesus EGA will be accumulate a amplifie i the ecusio pocess of pola coes costuctio. Cosequetly, this pheomeo causes the iaccuate subchael selectio a esults i a catastophic block eo atio BLER pefomace loss fo the log coe legths. Takig the pola coe stuctue chaacteistics ito cosieatio, it is lack of the compehesive famewok to esig the AGA schemes fo the costuctio of pola coes. I aitio, the pefomace evaluatio metho fo iffeet AGA is also abset. O the pola coe costuctio, we fi that the ak eo of the subchaels a the calculatio eo of each subchael s eliability ae two citical factos to affect the accuacy of AGA appoximatio fuctio. Hee, akig eo meas the oes of subchaels ae misalige. Fo vaious AGA schemes, the two factos will esult i iffeet evaluatio eo of each subchael. Followe by above two istotios, we eveal the essetial easo that the covetioal AGA scheme leas to a catastophic pefomace loss. Ou ultimate goal is to popose systematic esig ules of the AGA appoximatio fuctio fo pola coes, which achieves the excellet pefomace as

2 well as euces the GA computatioal complexity. Ou aim i this pape is to povie the ew piciples to esig the multi-segmet GA appoximatio fuctios so as to impove the calculatio accuacy of AGA a guaatee the excellet pefomace of pola coes. The mai cotibutios ca be summaize i the followig thee aspects: Fist, we take a close ivestigatio at the easo behi the poo pefomace of the log-legth pola coes whe the covetioal vesios of AGA e.g. Chug s scheme ae use. To this e, we itouce the cocepts of polaizatio violatio set PVS a polaizatio evesal set PRS. I the AGA pocess, whe the subchael s LLR mea belogs to the two sets, it will big i the ak eo a the polaizatio is violate o evete amog the subsequet subchaels. This pheomeo is ot cosistet with Aıka s fuametal polaizatio elatioship. The two sets eveal the essetial easo that pola coes costucte by the covetioal AGA peset poo pefomace at log coe legths. Seco, afte elimiatig PVS a PRS, we futhe popose a ew metic, ame cumulative-logaithmic eo CLE of chael polaizatio, to quatitatively evaluate the emaie calculatio eo betwee AGA a EGA i the costuctio of pola coes. We also eive the uppe bou of CLE to simplify its calculatio. With this bou, the pefomace of iffeet vesios of AGA ca be easily evaluate by aalytic calculatio athe tha euat the Mote-Calo simulatio. Fially, guie by PVS, PRS a the CLE bou, we popose ew esig ules fo the impove AGA techiques which is tailoe fo the pola coe costuctio. I this way, a systematic famewok is establishe to esig the high accuacy a low complexity AGA scheme fo pola coes at ay coe legth. Followe by the popose ules, thee ew AGA schemes ae give to guaatee the excellet pefomace of pola coes. The emaie of the pape is ogaize as follows. The pelimiaies of pola coig ae escibe i Sectio II. The the covetioal GA is itouce i Sectio III. Sectio IV makes etaile eo aalysis of GA, i which the cocepts of PVS, PRS a CLE ae popose. The the ew esig ules of AGA appoximatio fuctios ae give i Sectio V, a ew AGA schemes with complexity compaiso ae also give i this pat. Diffeet vesios of AGA ae compae with the help of CLE bou i Sectio VI, whee the simulatio esults ae also aalyze. Fially, Sectio VII coclues this pape. A. otatio Covetios II. PRELIMIARIES I this pape, we use calligaphic chaactes, such as X, to eote sets. Let X eote caiality of X. We wite lowecase lettes e.g., x to eote scalas. We use otatio v 1 to eote a vecto v 1, v,, v a v j i to eote a subvecto v i, v i+1,, v j. The sets of biay a eal fiel ae eote by B a R, espectively. Specially, let a, b eote Gaussia istibutio, whee a a b epeset the mea a the vaiace espectively. Fo pola coig, oly squae matices ae ivolve i this pape, a they ae eote by bol lettes. The subscipt of a matix iicates its size, e.g. F epesets a matix F. The Koecke pouct of two matices F a G is expesse as F G, a the -fol Koecke powe of F is eote by F. Thoughout this pape, log meas logaithm to base, a l stas fo the atual logaithm. B. Pola Coes a SC Decoig Let W : X Y eote a B-DMC with iput alphabet X a output alphabet Y. The chael tasitio pobabilities ae give by W y x, x X a y Y. Give the coe legth =, = 1,,, the ifomatio legth K, a the coe ate R = K/, the pola coig is escibe as [1]. Afte the chael combiig a splittig opeatios o iepeet uplicates of W, we obtai successive uses of sythesize biay iput chaels W j, j = 1,,,, with tasitio pobabilities W j y 1, u j 1 1 u j. The ifomatio bits ca be assige to the chaels with iices i the ifomatio set A, which ae the moe eliable subchaels. The complemetay set A c eotes the foze bit set a the foze bits u A c ca be set as the fixe bit values, such as all zeos, fo the symmetic chaels. To put it i aothe way [1], pola coig is pefome o the costait x 1 = u 1 G, whee G is the geeato matix a u 1, x 1 0, 1} ae the souce a coe block espectively. The souce block u 1 cosists of ifomatio bits u A a foze bits u A c. The geeato matix ca be efie as G = B F, whee B is the bit-evesal pemutatio matix a F = [ ]. As metioe i [1], pola coes ca be ecoe by successive cacellatio SC ecoig algoithm. Let û 1 eote a estimate of souce block u 1. Afte eceivig y1, the bits û j ae successively etemie with iex fom 1 to i the followig way: hj y û j = 1, û j 1 1 j A, u j j A c 1, whee h j y1, û j 1 1 = 0 if W j y 1,ûj W j y1,ûj 1 1 othewise , Give a pola coe with coe legth, ifomatio legth K a selecte chaels iices A, the BLER ue SC ecoig algoithm is uppe boue by P e, K, A P e W j, 3 j A whee P e W j is the eo pobability of the j-th subchael. This BLER uppe bou is ame as the SC bou. III. GAUSSIA APPROXIMATIO FOR POLAR CODES I this sectio, we use the coe tee to escibe the pocess of chael polaizatio. Base o the tee stuctue, we peset a aalyze the basic poceue of GA.

3 3 A. Coe Tee The chael polaizatio pocess ca be expesse o a coe tee. Fo a pola coe with legth =, the coespoig coe tee T is a pefect biay tee 1. Specifically, T ca be epesete as a -tuple V, B, whee V a B eote the set of oes a the set of eges, espectively. Depth of a oe is the legth of the path fom the oot to this oe. The set of all the oes at a give epth i is eote by V i, i = 0, 1,,,. The oot oe has a epth of zeo. Let v j i, j = 1,,, i, eote the j-th oe fom left to ight i V i. As a illustatio, Fig. 1 shows a toy example of coe tee with = 16, which iclues 4 levels. I the oes set V, the oe fom left to ight is eote by v. Except fo the oes at the -th epth, each vj i V i has two esceats i V i+1, a the two coespoig eges ae labele as 0 a 1, espectively. The oes v j V ae calle leaf oes. Let T v j i eote a subtee with a oot oe v j i. The epth of this subtee ca be efie as i which iicates the iffeece betwee the epth of the leaf oe a that of the oot oe. I aitio, the oe v j i has two subtees, that is, the left subtee T left = T v j 1 i+1 a the ight subtee T ight = T v j i+1. All the eges i the set B ae patitioe ito levels B l, l = 1,,,. Each ege i the l-th level B l is iciet to two oes: oe at epth l 1 a the othe at epth l. A i-epth oe is coespoig to a path b 1, b,, b i which cosists of i eges, with b l B l, l = 1,,, i. A vecto b i 1 = b 1, b,, b i is use to epict the above path. B. Gaussia Appoximatio fo Pola Coes Tifoov [9] suggests a pola coe costuctio metho fo the biay iput AWG BI-AWG chaels base o a Gaussia assumptio i evey ecusio step. Fo the BI-AWG chaels with oise vaiace σ, the coe bits ae moulate usig biay phase shift keyig BPSK. The tasitio pobability W y x is witte as W y x = 1 y 1 x e σ, 4 πσ whee x B a y R. The LLR of each eceive symbol y is eote by L y = l W y 0 W y 1 = y σ. 5 Without loss of geeality, we assume that all-zeo coewo is tasmitte. Oe ca check L y σ, 4 σ. GA assumptio: The LLR of each subchael obeys a costait Gaussia istibutio i which the mea is half of the vaiace [9] [1]. Accoig to the GA assumptio, the oly issue eee to be ealt with is the LLR mea. Theefoe, i the costuctio of pola coes, to obtai the eliability of each subchael, we tace thei LLR mea. This ecusive calculatio pocess ca be pefome o the coe tee. The set of LLRs coespoig 1 A pefect biay tee is a biay tee i which all iteio oes have two chile a all leaves have the same epth o same level. Fig. 1. A example of coe tee fo = 16, = 4. The e bol ege shows the ecusive calculatio pocess of m 8 4. to the oes at epth i is eote by L i, i = 0, 1,,,. Let L j i, j = 1,,, i, eote the j-th elemet i L i. We wite m j i as the mea of L j i. So the mea of LLR fom the chael ifomatio ca be witte as m 1 0 = σ, a ue the GA assumptio we have L j i m j i, m j i. 6 Hee, m j i ca be compute ecusively as m j 1 i+1 = f c m j i, m j i+1 = f v m j i, 7 whee the fuctios f c t a f v t ae use fo check oes left bach a vaiable oes ight bach, espectively. The physical meaig of the fuctio vaiable t stas fo subchael s LLR mea i GA costuctio. We have f c t = φ φ t, 8 f v t = t. I EGA, φ t is witte as 1 φ t = 1 4πt R tah z e z t 4t z t > 0, 1 t = 0, whee tah eotes hypebolic taget fuctio. It is easy to check that φ t is cotiuous a mootoically eceasig o [0, +, with φ 0 = 1 a φ + = 0 [10]. As a illustatio, the e bol ege i Fig. 1 epicts the ecusive, whose coespoig path is eote by b 1, b, b 3, b 4 = 0, 1, 1, 1. Obviously, the exact calculatio of LLR mea i check oes equies complex itegatio, which esults i a high computatioal complexity. Theefoe, Chug et al. give the well kow two-segmet appoximatio fuctio of φ t, eote by ϕ t, fo the aalysis of LDPC coes i [10], calculatio pocess of m 8 4 ϕ t = e 0.457t < t < 10, π t e t t t is also wiely use i the costuctio of pola coes [11]. Its coespoig AGA algoithm is eote by Chug.

4 4 By the GA assumptio of 6, the eo pobabilities of polaize subchael P e W j ca be witte as P e W j = Q mj = Q m j, 11 m j whee Q ς = 1 witte as + π ς P e, K, A Q j A e z z. Thus, the SC bou ca be m j. 1 Sice Q ς is a mootoe eceasig fuctio, the subchael W j with a lage mea mj has highe eliability. The costuctio of pola coes coespos to the selectio of best K subchaels amog as ifomatio set A i tems of the LLR meas m j, whee j = 1,,,. IV. ERROR AALYSIS OF GAUSSIA APPROXIMATIO I this sectio, we itouce the cocepts of polaizatio violatio set PVS a polaizatio evesal set PRS. By calculatig the two sets, we emostate the itisic easo that pola coes costucte by covetioal AGA suffe fom catastophic pefomace loss at log coe legths. I oe to quatitatively evaluate the emaie calculatio eo betwee AGA a EGA, we futhe popose the cocept of cumulative-logaithmic eo CLE of chael polaizatio a give a bou to simplify its calculatio. Base o the CLE bou, we ca efficietly evaluate the pefomace of iffeet appoximatio fuctios i AGA. A. PVS a PRS Popositio 1. Ue the GA assumptio, each subchael s symmetic capacity mootoically iceases with its LLR mea. It is woth oticig that a BI-AWG chael s capacity I W is witte as I W = h σ = 1 W y x W y x log y, x B R W y 0 + W y 1 13 whee the tasitio pobability W y x is give as 4. I tems of the GA piciple, each subchael is appoximate by a BI-AWG chael W with LLR mea m. I aitio, the vaiace of coespoig aitive white Gaussia oise is σ = m ue GA assumptio. Sice the fuctio h σ mootoically eceases with σ, the symmetic capacity I W mootoically iceases with its LLR mea m. I aitio, we have lim m 0 I W = 0, lim m + I W = Popositio. Fo the size-two chael polaizatio, suppose W, W W 1, W. Ue the GA assumptio, the LLR meas coespoig to W, W 1 a W ae epesete as m, m 1 a m, espectively. The, the LLR meas shoul satisfy m 1 m m 15 with equality if a oly if m = 0 o m = +. This esult follows the [1, Popositio 4]. Combiig with Popositio 1, the poof of 15 is immeiate. It ca be see fom 15 that the eliability of the oigial chael W is eistibute. Base o this itepetatio, we may say that afte oe step polaizatio, a ba chael W 1 a a goo chael W have bee ceate. Popositio 3. I the AGA costuctio of pola coes, the appoximatio fuctio Ω t of φ t shoul satisfy 0 < Ω t < 1, 16 which is the ecessay a sufficiet coitio fo Ω t to satisfy Popositio. Poof. Diffeet fom that i LDPC coes, the appoximatio fuctio Ω t fo pola coes shoul guaatee that the elatioship of 15 hols. Theefoe, i the size-two chael polaizatio, fo t 0, +, Ω t shoul satisfy Ω Ω t < t < t, 17 which follows Popositio. Recall that φ t is cotiuous a mootoically eceasig o [0, + [10], we theefoe assume its appoximate fom Ω t mootoically eceases o 0, +. Cosequetly, the left iequality i 17 ca be simplifie as 1 1 Ω t > Ω t 0 < Ω t < 1, 18 I tu, if Ω t satisfies 0 < Ω t < 1, we have 1 1 Ω t > 1 1 Ω t Ω Ω t < Ω Ω t }} =t Ω Ω t < t < t 19 The above aalysis iicates that 16 is the ecessay a sufficiet coitio fo Ω t to satisfy Popositio. If Ω t caot meet 16, its appoximatio eo with espect to the exact φ t will esult i the followig two types of eliability ak eo: Type 1 I the size-two chael polaizatio, if Ω t leas to m m 1 < m, this eo iicates that the eliabilities of subchaels ae patially violate, which is ame as the polaizatio violatio pheomeo. Type Futhemoe, whe Ω t leas to m < m m 1. This eo iicates the eliabilities of subchaels have bee wogly evese, which is ame as the polaizatio evesal pheomeo. I this pape, we aalyze the eo betwee AGA a EGA, athe tha the eo of GA itself.

5 5 Defiitio 1. Give the appoximatio fuctio Ω t, the polaizatio violatio set PVS S PVS is efie as S PVS = t t Ω Ω t } < t, 0 whee t 0, +. Obviously, i the size-two chael polaizatio, fo ay LLR mea m belogig to S PVS, Ω t will cetaily lea to m m 1 < m, which violates the basic oe i Popositio. Theefoe, fo the AGA algoithm with Ω t, if S PVS, ay subchael whose LLR mea belogs to S PVS will iaccuately ceate two goo chael i the size-two chael polaizatio, which will lea to obvious appoximatio eo i the subsequet AGA pocess. Defiitio. Give the appoximatio fuctio Ω t, the polaizatio evesal set PRS S PRS is efie as S PRS = t Ω Ω t } t, 1 whee t 0, +. Iteestigly, i the size-two chael polaizatio, fo ay LLR mea m belogig to S PRS, Ω t will esult i m < m m 1. I othe wos, the split goo chael a ba chael swap thei oles ue to the calculatio eo of Ω t, which yiels sevee eo i the size-two polaizatio. This pheomeo the leas to the substatial eo i subchaels positio ak 3. Popositio 4. The elatioship betwee PVS a PRS ca be expesse as S PRS S PVS. Poof. ote that whe t 0, +, fo ay t S PRS, it makes Ω Ω t t by Defiitio, which will ievitably esult i Ω Ω t t. Theefoe, accoig to Defiitio 1, the poof of Popositio 4 is immeiate. I othe wos, S PRS is the sufficiet coitio of S PVS. O the cotay, if S PVS =, we ca eive S PRS =. Suppose Ω t mootoically eceases o 0, +, the left iequality i 0 ca theefoe be simplifie as Ω t 1 1 Ω t Ω t 1. 3 Aalogously, the iequality i 1 will also be simplifie as 1 1 Ω t Ω t Ω t Ωt Ω t. 4 Example: ecall that i Chug s covetioal AGA scheme, the two-segmet appoximatio fuctio ϕ t is a specific fom of Ω t. Sice ϕ 0 = e > 1, ϕ t caot satisfy 16. Fo ϕ t, its coespoig PVS a PRS ae eote by S PVS = a 1, a ] a S PRS = 0, a 1 ], espectively. The bouay poits a 1 a a ae give i the followig equatios ϕ a1 ϕa 1 = ϕ a 1, a1 = , ϕ a = 1, a = , 5 3 Oe shoul otice that we caot have m < m sice m = m. Thee just exist thee oes, which coespo to 15, PVS a PRS. Fig.. Schematic plot of ϕ t, whee S PVS = , ] a S PRS = 0, ]. which follows 3 a 4. Hece fo ϕ t we have S PVS = , ], S PRS = 0, ], 6 which ae eote i Fig.. Theoem 1. Fo the -chael tasfom, whee =, 1, suppose that the oigial chael s LLR mea has two cofiguatios, which ae eote by m 1 0 a m 1 0. If they satisfy m 1 0 m 1 0, the ue the GA assumptio, fo j = 1,,,, we have m j m j. 7 Poof. This esult will be pove by mathematical iuctio. Ue the GA assumptio, ecall that φ t is cotiuous a mootoically eceasig o [0, + [10], oe ca easily check that f c t a f v t i 8 mootoously icease o [0, +. Suppose = k, k = 1, if the two LLR mea cofiguatios of the oigial chael satisfy m 1 0 m 1 0, the f c m 1 0 m 1 0 f v f c m 1 0 m 1 0 f v,, m 1 1 m 1 1, m 1 m 1. 8 With the icease LLR mea of the oigial chael W, it ca be see that the LLR meas of two polaize subchaels will stictly icease. ext, suppose = k, if we have m 1 0 m 1 0, the fo j = 1,,, k, m j k oe ca check f c m j k m j k f v f c m j k hols. Thus, whe = k+1, m j k m j k f v,, m j 1 k+1 m j 1 k+1, m j k+1 m j k+1. 9 I othe wos, fo j = 1,,, k+1, m j k+1 m j k+1. Fom above aalysis, the poof of 7 is fiishe. Theoem 1 iicates that ue the GA assumptio, if the LLR mea of the oigial chael W iceases, the polaize

6 6 Fig. 3. A coe tee epesetatio of polaizatio violatio a polaizatio evesal, whee = 16 a = 4. m 1 1 S PRS a m 5 3 S PVS. subchael s LLR mea will icease togethe. Combiig PVS a PRS, Theoem 1 seves to aalyze the subchael s ak eo a the poo pefomace of the log-legth pola coes whe Chug s covetioal AGA is use. Fo the AGA costuctio of pola coes with Ω t, suppose S PRS, the fo ay LLR mea m j i S PRS, we have m j 1 i+1 m j i+1 by the efiitio of S PRS i the ecusive calculatio of GA. Howeve, φ t i EGA ca guaatee S PRS = a S PVS =, which claims m j 1 i+1 < m j i+1 fo ay m j i > 0. The above aalysis iicates that i the AGA pocess, as pesete i the coe tee, ue to S PRS, the coitio m j i S PRS will lea to the ak eo amog the leaf oes which belog to the left subtee T left = T v j 1 i+1 a the ight subtee T ight = T v j i+1. I othe wos, we have m j 1 i +s m j 1 i +s+ i 1, 30 whee s = 1,,, i 1. This esult follows fom Theoem 1 by mechaically applyig m j 1 i+1 m 1 0 a m j i+1 m 1 0. The we have m s i 1 m s i 1. ote that i these two subtees, m s i 1 a m s i 1 coespo to the left a ight sie i 30, espectively. Howeve, i EGA, the i 30 shoul be <. This polaizatio evesal pheomeo leas to the ak eo of subchael s positio, which iectly affects the selectio of ifomatio set A. As a example, Fig. 3 shows the polaizatio violatio a polaizatio evesal i the coe tee, whee = 16 a = 4. I the AGA computatio pocess, m 1 1 S PRS. The we have m 1 m, fo the leaf oes with T left = T v 1 a T ight = T v, followig Theoem, oe ca check m 1 4 m 5 4, m 4 m 6 4, m 3 4 m 7 4, m4 4 m Similaly, sice m 5 3 S PVS, we have m 9 4 m 5 3, wheeas i EGA, it shoul be m 9 4 < m 5 3, a all the i 31 shoul be <. Thus, compae to EGA, the appoximatio eo of AGA esults i obvious ak eo withi the 16 leaf oes. The above aalysis iicates that i the AGA costuctio of pola coes, if S PRS, its appoximatio eo leas to polaizatio evesal. Cosieig its efiitio, S PRS lies i the viciity of 0. As state i [1], whe tes to ifiity, the symmetic capacity tems IW j } cluste aou 0 a 1, a the coespoig LLR meas cluste aou 0 a +. So whe the coe legth becomes loge, thee ae moe LLR meas fallig i S PRS uig the AGA ecusive computatio pocess. This is the essetial easo that pola coes costucte by some AGA suffe fom catastophic pefomace loss with the log coe legths. Duig the ecusive pocess of AGA, assumig the umbe of coe tee oes belogig to the two sets 4 ae eote by µ PVS a µ PRS, the coespoig atios with espect to all oes ae witte as θ PVS = µ PVS 1 k=0 k, θ PRS = µ PRS 1. 3 k k=0 Fo Chug s covetioal AGA, Table I gives the istibutio of µ PVS a µ PRS with iffeet polaizatio levels. The coespoig θ PVS a θ PRS ae also liste i Table I, whee E b / 0 = 1B σ = Fom Table I, we obseve that with the icease of coe legth, thee ae moe a moe oes fallig i PRS, a its coespoig atio also becomes lage. Theefoe, the pola coes costucte by Chug s covetioal AGA suffe fom catastophic jitte i pefomace whe thei coe legths ae log. TABLE I THE UMBER OF ODES AD ITS PERCETAGE WHOSE LLR MEA BELOGS TO S PVS AD S PRS DURIG THE RECURSIVE PROCESS OF CHUG S COVETIOAL AGA, WHERE E b / 0 = 1B. µ PVS θ PVS µ PRS θ PRS % % % % % % % % % % % % % % % % % % % % % % Fig. 4 emostates the BLER pefomace of pola coes costucte by Chug s covetioal AGA a Tal&Vay s metho ue the BI-AWG chaels. I Fig. 4, pola coes ae costucte epeig o the sigal-to-oise atio SR oe by oe, a all the schemes have coe ate R = 1/3 with SC ecoig. The coe legth is set to be 1, 14 a 18. We obseve that fo log coe legths, Chug s scheme obviously pesets catastophic pefomace loss. It is cosistet with the aalysis of Table I. Remak 1. Fo the AWG chaels, compae with the accuate DE o Tal&Vay s algoithm, EGA is also fou 4 The leaf oes whose LLR mea falls ito the two sets ae ot coute i, because it will ot lea to ak eo amog the esceats.

7 7 Fig. 4. BLER pefomace compaiso of pola coes with the coe legth = = 1, 14, 18 a coe ate R = 1/3 i the AWG chael. to well appoximate the actual polaize subchaels. ote that EGA has stict oe pesevig popety, which follows Popositio. This oe pesevig popety of EGA iee gives easoable itepetatio about its goo esults vesus DE. Theefoe, i geeal, the eo betwee EGA a DE is so small that it ca be igoe. Remak. Recall that Aıka i [3] suggeste a heuistic BEC appoximatio metho to costuct the pola coes fo abitay biay-iput chaels, which has also yiele goo esults i expeimets. The above PVS a PRS aalyses also give itepetatio about this goo esults. Oe ca check BEC appoximatio has stict oe pesevig popety i size-two polaizatio tasfom, which shows IW 1 < I W < IW. Heuistic BEC appoximatio will ot lea to polaizatio violatio a polaizatio evesal i this sese. Theefoe, this heuistic metho will just big some moeate pefomace loss athe tha catastophic jitte. B. CLE of Chael Polaizatio Polaizatio violatio a polaizatio evesal eveal the essetial easo that AGA caot wok well fo log coe legths. Besies these two sets, i this subsectio, we futhe popose a ew metic, ame cumulative-logaithmic eo CLE of chael polaizatio, to quatitatively evaluate the emaie appoximatio eo betwee AGA a EGA. The CLE is utilize to guie the esig of GA appoximatio fuctio Ω t. Fo Ω t i AGA, suppose its S PVS = a S PRS =, CLE will play a cucial ole i evaluatig the pefomace of AGA. We coce about the subchael s capacity, which is a fuctio of LLR mea ue GA assumptio. Hece, the iffeece betwee Ω t a φ t bigs i calculatio eo i subchael capacity evaluatio. The oigial absolute eo of capacity calculatio betwee AGA a EGA is eote by t, which is a fuctio of LLR mea t. Without ambiguity, t will be abbeviate to i this pape. Assume occus afte ecusios, eote by. Thus is accumulate as fial eo afte polaizatio levels, a this pocess ca be epesete o a subtee with a epth. To evaluate the calculatio eo of AGA, we focus o the iffeece of subchael s capacity calculate by AGA a EGA i logaithmic omai. The capacities calculate by EGA ca fom a set I efie o this coe subtee with the followig popeties: O this subtee, the set of capacities coespoig to the oes at a give epth is eote by I, =, +1,,. Let I k, k = 1,,,, eote the k-th elemet i I. Fo each I k I, I k takes value o [0, 1]. Fo >, I k is a fuctio of path b +1, which is actually the biay expasio of k 1 i.e., k 1 = i=1 b +i i. Theefoe, at the oot oe v j, ca be witte as = Ĩ1 I 1 = h h, 33 whee Ĩ1, I 1 a m j, m j m j m j sta fo the iitial symmetic capacities a LLR meas calculate by AGA a EGA espectively, a the fomula of h is witte i 13. As state i Remak, without much sacifice i accuacy, BEC appoximatio will act as faithful suogate fo GA i eo aalysis. Accoig to the iteatio stuctue i chael polaizatio tasfom i [1], we ca eive as follows I k 1 +1 = I k I k +1 = Ik whe b +1 = 0, I k whe b +1 = 1. Futhemoe, whe b +1 = 1, we have I k i logaithmic omai, we ca get logi k 1 +1 = logi k whe b +1 = 0, +1 Ik logi k +1 logik + 1 whe b +1 = Thus, 35 Defie Ĩk = I k + k, whee Ĩk eotes the capacity coespoig to AGA, k epesets the absolute eo of capacity calculatio betwee AGA a EGA. Fo <, Ĩ k a I k epeset the capacities, which ae calculate by BEC appoximatio, of AGA a EGA espectively. I this pape, We oly aalyze the eo betwee AGA a EGA, athe tha the eo of GA itself o the eo bought i by heuistic BEC appoximatio. Theefoe, 1 =. Let ρ k = k /Ik eote the elative eo, a e k = logĩk logi k = log 1 + ρ k eote the capacity calculatio eo i logaithmic omai. Hece, the patial cumulative-logaithmic eo PCLE ca be witte as C : = e k k=1. 36 The cumulative-logaithmic eo CLE will be C = C. CLE Bou C :. The pecise calculatio of CLE is too complicate to be aalyze usig ecusive elatio 34. I this subsectio, we popose a uppe bou o CLE to simplify its calculatio.

8 8 Defiitio 3. Fo the k-th leaf oe coespoig to a path b +1, we efie + 1 i : b i = 0} = α a + 1 i : b i = 1} = α. Theoem. The logaithm eo ca be boue by e k e k α log 1 + ρ 1 = α log I 1. Poof. Let ẽ k eote the bou of e k. The, ẽk is etemie by specifyig ẽ 1 = e 1 = log 1 + ρ 1 a ẽ k 1 +1 = D ẽ k whe b +1 = 0, ẽ k +1 = E ẽ k 37 whe b +1 = 1, whee E : R R, E z = z eotes equality, D : R R, D z = z eotes oublig. If 1 0, it claims that k 0 hols i tems of Theoem 1. ote that uig the iteatio, whe b +1 = 0, 0 e k 1 +1 = e k = ẽ k Whe b +1 = 1, it ca be pove that e k +1 ẽk +1. Accoig to the fist equatio of 34, it is easy to get that Ĩ k +1 = Ĩk I k Ĩk = + k = I k I k }} =I k +1 I k Theefoe, e k +1 ca be witte as e k +1 = log 1 + ρ k +1 = log 1 + k I aitio, we have ẽ k +1 = log 1 + ρ k +1 The we ca check ρ k +1 ρk + k k + k I k k }} = k +1 Ik k k I k I k = e k k +1 = I k = log I k 1 + k I k k 0. 4 I k It ca be ifee that ẽ k e k 0 hols. Recall Defiitio 3, uig the iteative calculatio of ẽ k, we cout oublig α times a equality α times. Hece, we have 0 e k Aalgously, if 1 4, we ca get 0 > e k ẽ k = E α D α e 1 < 0, we have k > ẽ k = E α D α e 1 = α e < 0. Fom 38 a = α e Combig 43 a 44, we pove the theoem. Theoem 3. PCLE C : ca be uppe boue by C : 3 log 1 + ρ 1 = 3 log I 1. Poof. Fo ay k 1,,, }, the umbe of vectos b +1 satisfyig Defiitio 3 is α, whee b +1 is the biay expasio of k 1. Combie with efiitio 36 a Theoem, C : satisfies the followig costait C : ẽ k k=1 = α=0 α α e 1 = 3 e The last equatio i 45 uses biomial theoem. Theefoe, CLE C ca be uppe boue by 3 log 1 + ρ 1, a the expoet stas fo polaizatio levels. V. IMPROVED GAUSSIA APPROXIMATIO I this sectio, guie by the pevious PVS, PRS a CLE aalyses, we popose ew ules to esig AGA fo the costuctio of pola coes. The we give thee specific foms of the appoximatio fuctio i AGA, which have avatages i both complexity a pefomace. A. Desig Rules of AGA Fo chaels othe tha BEC e.g. AWG chael, AGA is wiely use to costuct pola coes. Howeve, i pactical implemetatio, the accuacy of appoximatio fuctio Ω t will geatly affect the costuctio of pola coes especially whe the coe legth is log. Accoig to Theoem 3, the iitial eo will be expoetially amplifie with the icease of polaizatio levels. ote that PCLE bou is maily affecte by two factos: the fist tem 3 is elevat to polaizatio levels, a the seco tem is epeet o the oigial elative is tiy. Theefoe, fo the goo chaels whose capacities I 1 appoach 1, thei oigial elative eos ρ 1 ae so small that eo ρ 1 = 1 /I 1. I geeal, the absolute eo 1 they ca be igoe. Howeve, fo the ba chaels whose capacities I 1 appoach 0, thei oigial elative eos ae ot egligible. Subsequetly, give a fixe absolute eo 1, the moe I 1 is close to 0, the lage the oigial elative eo ρ 1 becomes. Above aalysis iicates that CLE bou is maily affecte by the tems C : with the small iitial capacity I 1, which coespos to the ba chaels. Due to the eo, some foze subchaels will be wogly ietifie as ifomatio-cayig oes ole flippig, which esults i the pefomace egaatio. ote that the capacity I 1 mootoically iceases with LLR mea t ue the GA assumptio. I aitio, accoig to 14, we have lim t 0 I1 = 0, lim t + I1 = Guie by above aalyses, the AGA appoximatio fuctio esig is compose of thee ules: Rule 1 PVS a PRS elimiatig: Ω t shoul guaatee S PVS = a S PRS =. Accoig to Popositio

9 9 4 a its covese-egative popositio, if S PVS =, we have S PRS =. Hece, i oe to empty PVS a PRS, we shoul esue 0 < Ω t < 1 fo ay t 0, +. Rule Low SR esig: Whe t comes close to 0, we must guaatee lim Ω t = 1 to euce appoximatio t 0 eo. Sice CLE bou is amplifie expoetially with the gowth of polaizatio levels, the oly way to euce CLE bou is to lowe the iitial elative eo ρ 1. Theefoe, Ω t ees to be ivie ito moe segmets whe t appoaches 0. This ule ca euce the oigial absolute eo 1 i the viciity of 0 so as to lowe ρ 1. Rule 3 High SR esig: Whe t stays away fom 0, thaks to the elatively lage I 1, CLE bou ca toleate a moe obvious absolute eo 1. Theefoe, Ω t ca be selecte with some simple foms to euce the computatioal complexity. Besies, Ω t shoul keep cotiuity betwee the ajacet two segmets, which ca mitigate the jitte of CLE bou by keepig smooth of iitial eo. Amog these thee ules, Rule 1 is the most cucial, which ca pevet the coespoig AGA pola coe costuctio scheme fom sevee pefomace loss. The the impotace of Rule is seco, which euces the emaie appoximatio eo betwee AGA a EGA. Rule 3 plays a less impotat ole, which helps to futhe euce the computatioal complexity of AGA. B. Impove AGA Appoximatio Fuctio Recall the aalysis i Sectio IV, Chug s two-segmet AGA scheme leas to S PVS a S PRS, which violates Rule 1. Theefoe, guie by above ules, we esig the followig ew two-segmet appoximatio fuctio by employig cuve-fittig algoithm with miimum mea squae eo citeio, eote by Ω t, e t 0.41t 0 < t a, Ω t = e 0.944t a < t, whee the bouay poit a = The coespoig ew AGA algoithm is eote by AGA-. I the costuctio of pola coes, compae with Chug s scheme, AGA- ca elimiate ak eo a theefoe will ot lea to catastophic pefomace loss at the log coe legths. I aitio, sice the ivese fuctio of Ω t ca be aalytically obtaie, it has lowe calculatig complexity. Futhemoe, guie by Rule 3, fo f c t i 8, whe t leaves away fom 0, 1 Ω t tes to 1, which iicates 1 Ω t 1 Ω t f c t t. 48 Followe by Popositio, f c t shoul satisfy f c t < t. Thus, whe t stays away fom 0, the complex f c t ca be futhe appoximate as f c t = t ε, 49 whee ε eotes the offset. The accoig to 8, whe Ω t tes to 0, oe ca check Ω t ε = Ω t Ω t Ω t. 50 Theefoe, fo Ω t, i tems of 47, whe t 0 we have e 0.944t ε = e 0.944t l ε =.3544, 51 whee t ε shoul locate i the seco segmet, amely t ε > a, which claims t > Followig Rule 3, fo the etie AGA- scheme, its f c t is futhe simplifie as f c t = Ω Ω t 0 t τ, t.3544 t > τ, 5 whee the bouay poit is τ = Fom the subsequet CLE aalysis, we fi that although AGA- ca satisfy Rule 1 so as to emove the ak eo, it will still big obvious appoximatio eo i the viciity of 0. Theefoe, whe the coe legth is elatively log, AGA- will also big some moeate pefomace loss. I oe to futhe impove the GA costuctio pefomace, we popose a ew piecewise fuctio Ω 3 t with thee segmets by usig cuve-fittig algoithm, that is Ω 3 t = e t t 0 < t a, e 0.457t a < t b, e 0.83t b < t, 53 whee the bouay poits a = a b = Its coespoig AGA algoithm is eote by AGA-3. It is specially esige fo pola coes, which follows the popose ules. AGA-3 ca bette satisfy Rule 1 a Rule. Especially i the thi segmet, amely the high SR egio, AGA-3 has lowe computatioal complexity compae to Chug s scheme which obeys Rule 3. I aitio, the ivese fuctio of Ω 3 t ca be easily obtaie. Simila with AGA-, fo the whole AGA-3 scheme, its f c t ca be futhe simplifie as f c t = Ω Ω 3 t 0 t τ, t.4476 t > τ, 54 whee the bouay poit is τ = Thee is o oubt that, if the coe legth becomes extemely log, the thee-segmet appoximatio fuctio i AGA-3 will also big calculatio eo, which obeys Theoem 3 a is state i Rule. Hece, we esig the followig fou-segmet impove AGA scheme fo extemely log pola coe, that is Ω 4 t = e t 0.499t 0 < t a, e t t a < t b, e 0.457t b < t c, e 0.83t c < t, 55 whee the bouay poits a = , b = a c = The appoximatio accuacy i the viciity of 0 is futhe impove. Its coespoig AGA algoithm is eote by AGA-4. The ivese fuctio of Ω 4 t ca also be aalytically obtaie. Sice the last two segmets i Ω 4 t ae the same as that i Ω 3 t, the f c t fuctio i AGA-4 also has the same fom as the equatio 54. The computatioal complexity of f c t has bee emakably euce thaks to Rule 3 i the above thee popose AGA schemes. Moeove, Rule 1 a Rule help AGA to achieve

10 10 excellet pefomace. Thus, fo the geeal costuctio of pola coes, the popose AGA- scheme is a goo alteate to impove the Chug s covetioal AGA scheme. Whe the coe legth becomes loge, the popose AGA-3 scheme will achieve bette pefomace tha AGA- scheme. If the coe legth becomes extemely log, AGA-4 will peset bette pefomace. Whe the coe legth cotiues to gow, AGA-4 will ievitably big calculatio eo, which follows Theoem 3 a Rule. evetheless, followe by Rule 1 to 3, we ca still esig AGA with moe complex multi-segmet appoximatio fuctio to keep the calculatio accuacy. I this way, we povie a systematic famewok to esig high accuacy AGA scheme fo pola coes at ay coe legth. As fo the specific foms of the ew AGA appoximatio fuctios, it is heuistically obtaie fom Chug s oigial ϕ t while takig the Rule 1 to 3 ito cosieatio. The fist segmet shoul esue lim Ω t = 1 a use some complex t 0 fuctios to euce the elative eo. Whe t stays away fom 0, Ω t ca be selecte with some simple foms to euce the computatioal complexity. Afte choosig the appoximatio fom, its paametes ae acquie by cuve-fittig. Cetaily, oe ca choose iffeet appoximatio fuctios accoig to the popose ules. Fially, the pefomace will be evaluate by calculatig its CLE bou. Fo othe chaels W athe tha the AWG e.g. the Rayleigh faig chael, oe may expect bette pefomace at the expese of moe complexity i the coe costuctio by usig DE. Howeve, we ca appoximate the chael W usig a AWG chael W with σ, whee I W = I W = h σ. 56 The coe costuctio is the the pefome ove each of the equivalet AWG chaels i the same GA way as that i the covetioal AWG case. As that will be show i Sectio VI, the popose AGA base costuctio of pola coes woks well fo othe chaels as well. This is sigificat i that it shows the obustess of the AGA costuctio agaist ucetaity a vaiatio i chael paametes. C. Complexity Compaiso I this pat, we compae the oe of complexities with fou typical pola coe costuctio methos, which ae Aıka s heuistic BEC appoximatio metho, DE algoithm, Tal a Vay s metho a ou popose AGA schemes. ote that the complexity oe povie oly iclues the omiat tems, a the etaile umbe of calculatio epes heavily o the specific hawae implemetatio, which is beyo the scope of this pape. Oe ca check the piciple of these fou methos is to ecusively calculate each subchael s eliability o the coe tee. Theefoe, thei the total umbe of oe visitig complexity ca be witte as O + O / + O /4 + +O +O 1 = O. It seems that these fou methos have the same umbe of oe visitig. Futhemoe, we aalyze the computatioal complexity of each visitig. Fo the heuistic BEC appoximatio, the computatioal complexity of each visitig opeatio is O 1, which ca be igoe. Hece, it has the lowest complexity amog the fou schemes. DE ees a fast Fouie tasfom FFT a a ivese fast Fouie tasfom IFFT opeatio whe calculatig the pobability esity fuctio PDF of LLR of each bit. The coespoig complexity is O ξ log ξ, whee ξ eotes the umbe of samples. Howeve, a typical value of ξ is about 10 5 [13], implyig a huge computatioal bue i pactical applicatio. The ifficulty is futhe aggavate by the quatizatio eos, which will be accumulate ove multiple polaizatio levels. Tal a Vay s metho ca be viewe as a appoximate vesio of DE [5] a has a complexity of Oµ log µ, whee µ is a fixe, eve a positive itege iepeet of coe legth. I geeal, a typical value of µ is 56 which is much less tha ξ. Hece, this metho has much lowe complexity tha DE. But whe the coe legth becomes log, Tal a Vay s metho still ivolves elatively high computatioal complexity. Regaig the popose AGA metho, the complexity of calculatig f v t ca be igoe sice the esult ca be obtaie easily. Fo f c t, whe t > τ, its calculatig complexity ca also be igoe. Whe 0 t τ, sice the ivese fuctio i f c t ca be aalytically obtaie, the computatioal complexity is O 1. Fom these compaiso, it ca be coclue that ou popose AGA schemes have simila calculatig complexity oe with the heuistic BEC appoximatio, which ae much lowe tha DE o Tal a Vay s algoithms. VI. PERFORMACE EVALUATIO The pecise esults a coespoig uppe bous of CLE fo vaious AGAs ae epicte i Fig. 5, whee the polaizatio level = 8 a = 0 CLE C = C 0:8. It ca be fou that the CLE bou a the exact esult coicie well. Theefoe, the CLE bou may be use as a effective tool to evaluate the pefomace of vaious AGA schemes. Fom Fig. 5, we ca see that the CLE bou of Chug s covetioal AGA scheme Fig. 5. The pecise esults a coespoig uppe bous of CLE fo vaious AGAs, whee t stas fo LLR mea a polaizatio level = 8.

11 11 is obviously highe tha that of AGA- a AGA-3, which iicates the poo pefomace of pola coes whe Chug s AGA scheme is use. Compae with AGA-, the CLE bou of AGA-3 also shows some pefomace gai. Futhemoe, AGA-4 ca achieve the best pefomace amog these AGA schemes. We otice that thee exists a o-mootoic behavio fo the CLE bou. This pheomeo is cause by fuctio fittig because thee is iffeet elative eo ρ 1 fo iffeet t, which esults i the jitte of CLE bou. ext we compae the BLER pefomace amog iffeet costuctio schemes ue the BI-AWG chael, which is show i Fig. 6. All the schemes have coe ate R = 1/3 with the SC ecoig. The coe legths ae set to 1, 14 a 18, espectively. Fom these esults, it ca be fou that the BEC appoximatio scheme emostates some moeate pefomace loss compae with othe avace costuctio methos. Fo the extemely log coe legths, sice DE falls ito a huge computatioal bue i the pactical applicatio, we use Tal a Vay s metho as a alteate with almost o pefomace loss. Theefoe, Tal a Vay s costuctio possesses the highest accuacy i Fig. 6. Howeve, it still equies high computatioal complexity at log coe legths. Amog the AGA schemes, we ca see that Chug s scheme suffes fom a amatic pefomace loss with the icease of coe legth sice it violates Rule 1. O the cotast, ou popose AGA- achieves goo pefomace. Hece, AGA- ca be employe as a goo alteate fo Chug s two-segmet metho at some moeate coe legths. It ca also be obseve that the popose AGA-3 scheme achieve bette pefomace which follows Rule 1 a Rule. Futhemoe, we ca see that AGA-4 scheme appoaches the pefomace of Tal a Vay s metho, which ca be use fo some extemely log coe legth. I aitio, with the icease of coe legth, the pefomace gap betwee AGA-3 a AGA-4 becomes lage. Theefoe, i tems of the pevious CLE bou aalyses a simulatio esults, it ca be peicte that the pefomace gap will become moe a moe obvious with the icease of coe legth. The compehesive BLER pefomace compaisos with the SC ecoig i tems of iffeet costuctio schemes ue the Rayleigh faig chaels ae give i Fig. 7. The coe legth is = 14, a the coe ates R ae set to 1/3 a /5, espectively. Simila with Fig. 6, we ca see that DE algoithm achieves the best pefomace. Chug s scheme also pesets poo pefomace sice it caot satisfy Rule 1. O the cotay, AGA-4 scheme suffes fom a igoable loss of pefomace compae with DE algoithm. Futhemoe, it becomes moe computatioally efficiet a implemetable i pactical use tha the fome. I aitio, AGA-4 ca stably achieve B gai fo iffeet coe ates compae with AGA-3 scheme. These esults shows the obustess of the AGA costuctio agaist ucetaity a vaiatio i chael paametes, which is valuable fo pola coig. I Fig. 8, we povie the BLER pefomace of pola coes by usig the aaptive cyclic euacy check CRC aie SC list ecoig A-CASCL [14], [15]. The maximum list size i the A-CASCL ecoe is eote by L max. The 16-bit CRC i LTE staa [16] is use. The pefomace of LTE Fig. 6. The SC ecoig BLER pefomace compaiso of pola coes with the coe legth = = 1, 14, 18 a the coe ate R = 1/3 ove the BI-AWG chael. Fig. 7. The SC ecoig BLER pefomace compaiso of pola coes with coe legth = 14 = a coe ates R = 1/3 a R = /5 ove the Rayleigh faig chael. Fig. 8. The A-CASCL ecoig BLER pefomace compaiso of pola coe a LTE tubo coe with the coe legth = 14 = a the coe ates R = 1/3. The chael is cofigue as the BI-AWG.

12 1 tubo coe is also give as a compaiso, whee the Log-MAP ecoig is applie i the tubo ecoe with 8 iteatios [17]. We ca see that Chug s covetioal AGA scheme shows poo pefomace. It iicates that this taitioal two-segmet AGA scheme is ot suitable fo the pola coe costuctio. O the cotay, the pola coes costucte by the popose AGA-4 scheme pefom well. Whe L max = 3, pola coe ca outpefom LTE tubo coe i the low SR egio. Fo the high SR egio, although tubo coe sometimes pefoms bette tha pola coe, it suffes fom the eo floo. Whe L max is set to 18, pola coe ca outpefom LTE tubo coe fo ay SR. Aitioally, this pola coe costucte by AGA-4 scheme with L max = 18 A-CASCL ecoig ca achieve BLER 10 3 at E b / 0 = 0.51B. We compae this pefomace to the Shao limit at the same fiite block legth, which is povie i [18]. The maximum ate that ca be achieve at block legth a BLER ɛ ca be well appoximate by V R max = C Q 1 ɛ, 57 whee C is the chael capacity a V is a quatity calle the chael ispesio that ca be compute fom the chael statistics, usig the fomula: V = Va [ log ] W Y X. 58 W Y Fo the BI-AWG chael, the tasitio pobability W y x is witte i 4. The chael ispesio V is witte as V = 1 [ ] W y x W y x log y W y 0 + W y 1 x B R I W. 59 By usig 57, we ca calculate the Shao limit fo the, K = 16384, 5461 coe which is ame as the ispesio bou i Fig. 8. To achieve a ate R = 5461/16384 = 1/3, the miimum E b / 0 equie is 0.186B. Hece, pola coe costucte by AGA-4 with L max = 18 is 0.696B fom the Shao limit. Whe L max iceases, this SR gap will become smalle. REFERECES [1] E. Aıka, Chael polaizatio: A metho fo costuctig capacity achievig coes fo symmetic biay-iput memoyless chaels, i IEEE Tas. If. Theoy, vol. 55, o. 7, pp , Jul [] Chaimas otes RA1#87, Reo, USA. [3] E. Aıka, A pefomace compaiso of pola coes a Ree-Mulle coes, i IEEE Commu. Lett., vol. 1, o. 6, pp , Ju [4] R. Moi a T. Taaka, Pefomace of pola coes with the costuctio usig esity evolutio, i IEEE Commu. Lett., vol. 13, o. 7, pp , Jul [5] I. Tal a A. Vay, How to Costuct Pola Coes, i IEEE Tas. If. Theoy, vol. 59, o. 10, pp , Oct [6] R. Peasai, S. H. Hassai, I. Tal a E. Telata, O the costuctio of pola coes, i Poc. IEEE It. Symp. Ifom. Theoy ISIT, pp , St. Petesbug, 011. [7] I. Tal, O the Costuctio of Pola Coes fo Chaels with Moeate Iput Alphabet Sizes, i Poc. IEEE It. Symp. Ifom. Theoy ISIT, pp , Hog Kog, 015. [8] C. Schellig a A. Schmeik, Costuctio of Pola Coes Exploitig Chael Tasfomatio Stuctue, i IEEE Commu. Lett., vol. 19, o. 1, pp , Dec [9] P. Tifoov, Efficiet Desig a Decoig of Pola Coes, i IEEE Tas. Commu., vol. 60, o. 11, pp , ov. 01. [10] S.-Y. Chug, T. J. Richaso a R. L. Ubake, Aalysis of sumpouct ecoig of low-esity paity-check coes usig a Gaussia appoximatio, i IEEE Tas. If. Theoy, vol. 47, o., pp , Feb [11] D. Wu, Y. Li a Y. Su, Costuctio a block eo ate aalysis of pola coes ove AWG chael base o Gaussia appoximatio, i IEEE Commu. Lett., vol. 18, o. 7, pp , Jul [1] H. Li a J. Yua, A pactical costuctio metho fo pola coes i AWG chaels, i Poc. IEEE TECO Spig Cof., pp. 3-6, 013. [13] K. iu, K. Che, J. Li a Q.T. Zhag, Pola coes: Pimay cocepts a pactical ecoig algoithms, i IEEE Commu. Mag., vol. 5, o. 7, pp , Jul [14] K. iu a K. Che, CRC-aie ecoig of pola coes, i IEEE Commu. Lett., vol. 16, o. 10, pp , Oct. 01. [15] B. Li, H. She, a D. Tse, A aaptive successive cacellatio list ecoe fo pola coes with cyclic euacy check, i IEEE Commu. Lett., vol. 16, o. 1, pp , 01. [16] 3GPP TS 36.1: Multiplexig a chael coig, Release 1, 014. [17] S. Li a D. J. Costello, Eo Cotol Coig e., Petice- Hall, Ic., 004. [18] Y. Polyaskiy, H. Vicet Poo, a S. Veu, Chael coig ate i the fiite blocklegth egime, i IEEE Tas. If. Theoy, vol. 56, o. 5, pp , May 010. VII. COCLUSIO I this pape, we itouce the cocepts of PVS a PRS which explai the essetial easo that pola coes costucte by covetioal AGA expesses poo pefomace at log coe legths. The we popose a ew metic, ame CLE, to quatitatively evaluate the emaie eo of AGA. We futhe eive the uppe bou of CLE to simplify its calculatio. Guie by PVS, PRS a CLE bou aalysis, we popose ew ules to esig AGA fo pola coes. Simulatio esults show that the pefomace of all AGA schemes is cosistet with CLE aalysis. Whe the polaizatio levels icease, Chug s covetioal AGA scheme suffes fom a catastophic pefomace jitte. O the cotay, the popose AGA methos guie by the popose ules stably guaatee the excellet pefomace of pola coes fo both the AWG chaels a the Rayleigh faig chaels.