Performance Limits and Practical Decoding of Interleaved Reed-Solomon Polar Concatenated Codes

Transcription

1 Performace Limits ad Practical Decodig of Iterleaved Reed-Solomo Polar Cocateated Codes Hessam Mahdavifar, Mostafa El-Khamy, Jugwo Lee, Iyup Kag 1 arxiv: v1 [cs.it] 6 Aug 2013 Abstract A scheme for cocateatig the recetly iveted polar codes with o-biary MDS codes, as Reed-Solomo codes, is cosidered. By cocateatig biary polar codes with iterleaved Reed-Solomo codes, we prove that the proposed cocateatio scheme captures the capacity-achievig property of polar codes, while havig a sigificatly better error-decay rate. We show that for ay ɛ > 0, ad total frame legth N, the parameters of the scheme ca be set such that the frame error probability is less tha 2 N 1 ɛ, while the scheme is still capacity achievig. This improves upo 2 N 0.5 ɛ, the frame error probability of Arika s polar codes. The proposed cocateated polar codes ad Arika s polar codes are also compared for trasmissio over chaels with erasure bursts. We provide a sufficiet coditio o the legth of erasure burst which guaratees failure of the polar decoder. O the other had, it is show that the parameters of the cocateated polar code ca be set i such a way that the capacity-achievig properties of polar codes are preserved. We also propose decodig algorithms for cocateated polar codes, which sigificatly improve the error-rate performace at fiite block legths while preservig the low decodig complexity. I. INTRODUCTION Polar codes, itroduced by Arika [1], [2], are the most recet breakthrough i codig theory. Polar codes are the first ad, curretly, the oly family of codes with explicit costructio (o esemble to pick from) to achieve the capacity of a certai family of chaels (biary iput symmetric discrete memory-less chaels) as the block legth goes to ifiity. They have ecodig ad decodig algorithms with very low complexity. Their ecodig complexity is log ad their successive cacellatio (SC) decodig complexity is O(log), where is the legth of the code. However, at moderate block legths, their performace does ot compete with world s best kow codes, which prevets them from beig implemeted i practice. Also, their error expoet decreases slowly as the block legth icreases, where the error-decay rate of polar codes uder successive cacellatio decodig is asymptotically O(2 0.5 ɛ ). I this paper, we aim at providig techiques to make polar codes more practical, by providig schemes that improve their fiite legth performace, while preservig their low decodig complexity. Cocateatig ier polar codes with outer liear codes (or other variatios of cocateatio like parallel cocateatio) is a promisig path towards makig them more practical [3], [4]. By carefully costructig such codes, the cocateated Hessam Mahdavifar, Mostafa El-Khamy, Juwo Lee ad Iyup Kag are with the Mobile Solutios Lab, Samsug Research America, Sa Diego, CA 92121, U.S.A. ( {h.mahdavifar, mostafa.e, jugwo2.lee, iyup.kag}@samsug.com). costructio ca iherit the low ecodig ad decodig complexities of the ier polar code, while havig sigificatly improved error-rate performace, i compariso with the ier polar code. The performace ad decodig complexity of the cocateated code will also deped o the outer code used, the cocateatio scheme ad the decodig algorithms used for decodig the compoet codes. We chose Reed-Solomo (RS) codes as outer codes as they are maximal distace separable (MDS) codes, ad hece have the largest bouded-distace error-correctio capability at a specified code rate. RS codes also have excellet burst error-correctio capability. Recet ivestigatios have show the possibility of improvig the boud o the error-decay rate of polar codes by cocateatig them with RS codes [3]. However, this work assumed a covetioal method of cocateatio, which required the cardiality of the outer RS code alphabet to be expoetial i the block legth of the ier polar code, which makes it ifeasible for implemetatio i practical systems. I this paper, we propose a scheme for improvig the error-decay rate of polar codes by cocateatig them with iterleaved block codes. Whe deployig our proposed scheme with outer iterleaved RS block codes, the RS alphabet cardiality is o loger expoetial ad, i fact, is a desig parameter which ca be chose arbitrarily. Furthermore, we show that the code parameters ca be set such that the total scheme still achieves the capacity 1 ɛ N while the error-decay rate is asymptotically 2 for ay ɛ > 0, where N is the total block legth of the cocateated scheme. This boud provides cosiderable improvemets upo 2 N 0.5 ɛ, the error-decay rate of Arika s polar codes, ad upo the boud of [3]. Notice that 2 N is the iformatio theoretic upper boud o the probability of error of ay capacity achievig code. This ca be derived from the mai result of [5]. Therefore, our cocateated code fills the gap with the ultimate boud o the error decay rate of ay capacity achievig code. Cosiderig trasmissio over chaels with erasure bursts we show that the proposed cocateated polar codes perform well while we prove that Arika s polar codes are very weak i this regard. For theoretical aalysis, trasmissio over a arbitrary biary symmetric chael chael is cosidered while the received symbols ecouter a sigle erasure burst. We prove that if the legth of erasure burst is at least 2 1, where is the legth of polar code, the the successive cacellatio decoder fails to recover the trasmitted message. O the other had, we show that the parameters of the cocateated polar code ca be set i such a way that the cocateated polar code approaches the capacity of the chael with the same error-decay rate as i polar codes. For simulatios, Gilbert- Elliot model is cosidered wherei a chael with two states

2 2 is assumed. The good chael state is a biary erasure chael ad the bad chael state is always erasure. For some specific parameters, we show that the cocateated polar code perform well as expected, while the o-cocateated polar code fails with a very high probability of error. To costruct the cocateated polar code at fiite block legth, we propose a rate-adaptive method to miimize the rateloss resultig from the outer block code. It is kow from the theory of polar codes that ot all of the selected good bitchaels have the same performace. Some of the iformatio bits observe very strog ad almost oiseless chaels, while some other iformatio bits observe weaker chaels. This implies that a uequal protectio by the outer code is eeded, i.e. the strogest iformatio bit-chaels do ot eed aother level of protectio by the outer code, while the rest are protected by certai codes whose rates are determied by the error probability of the correspodig bit-chaels. Hece, a criterio is established for determiig the proper rates of the outer iterleaved block codes. We propose a successive method for decodig the RS-polar cocateated scheme, which is possible by the proposed iterleaved cocateatio scheme, where the symbols of each RS codeword are distributed over the same coordiates of multiple polar codewords. I the successive cacellatio (SC) decodig of the ier polar codes, the very first bits of each polar codeword that are protected by the first outer RS code are decoded first. The these bits are passed as symbols to the first outer decoder. RS decodig is doe o the first RS word to correct ay residual errors from the polar decoders, ad pass the updated iformatio back to the SC polar decoders. The the SC decoders of all polar codes update their first decoded bits, ad use that updated iformatio to cotiue successive decodig for the followig bits correspodig to the symbols of the subsequet outer words. Therefore, the errors from SC decodig do ot propagate through the whole polar codeword, which sigificatly improves the performace of our scheme. Aother mai advatage of this proposed scheme, is that all polar codes are decoded i parallel which sigificatly reduces the decodig latecy. Depedig o the chose outer code ad its chose decodig algorithm, the iformatio exchaged betwee the ier SC decoders ad the outer decoder ca be soft iformatio, as log-likelihood ratios (LLRs), or hard decisioed bits. We take advatage of the soft iformatio geerated by the successive cacellatio decoder of polar codes to perform geeralized miimum distace (GMD) list decodig [6], [7] for the outer RS code, which ehaces the error performace. For further improvemets, the SC decoder is modified so that it geerates the likelihoods of all the possible RS symbols for GMD decodig. After GMD decodig of each compoet RS word, the most likely cadidate codeword relative to the received word is picked from the list, ad SC decoder utilizes the updated RS soft ad hard outputs i further decodig of the compoet polar codes. I the case that outer codes are RS codes, more complex ad better soft decodig algorithms exist, e.g. [8], [9], however GMD was chose to preserve the boud o the decodig complexity of the polar codes. The rest of this paper is orgaized as follows: I Sectio II, some ecessary backgroud o Arika s polar codes is provided. I Sectio III, the proposed scheme is explaied i more details. We also explai how to modify the scheme to get a rate-adaptive costructio, which sigificatly improves the rate of fiite legth costructios. I Sectio IV, we describe our proposed decodig algorithms for the cocateated polar code scheme that improve the performace at fiite block legth. Asymptotic aalysis of error correctio performace alog with the proof of the bouds o the error-decay rate are provided i Sectio V. Also, the performace of the proposed cocateated polar code over chaels with erasure burst is discussed. Simulated results are show i Sectio VI. Fially, we coclude the paper i Sectio VII. II. ARIKAN S POLAR CODES A brief overview of the groudbreakig work of Arika [1] ad others [2], [10], [11] o polar codes ad chael polarizatio is provided i this sectio. The costructio of polar codes is based o a pheomeo called chael polarizatio discovered by Arika [1]. Let [ ] 1 0 G = (1) 1 1 The Kroecker powers of G are defied by iductio. G 1 = G ad for ay i > 1: [ ] G (i) G (i 1) 0 = G (i 1) G (i 1) Observe that G (i) is a 2 i 2 i matrix. Let = 2 s. The the def polarizatio trasform matrix is defied as G = R G s, where R is the bit-reversal permutatio matrix defied i [1, Sectio VII-B]. Let (U 1, U 2,..., U ), deoted by U1, be a block of idepedet ad uiform biary radom variables. Let also X1 = U1 G. X i s are trasmitted through idepedet copies of a biary-iput discrete memoryless chael (B-DMC) W. The output is deoted by Y1. This trasformatio with iput U1 ad output Y1 is called the polar trasformatio. I this trasformatio, idepedet uses of W is split ito bit-chaels assumig that a successive cacellatio decoder is deployed at the output. Uder this decodig method, all the bits U1 i 1 are already decoded ad are available at the time that U i is beig decoded. This chael is called the i-th bit chael ad is deoted by W (i). The chael polarizatio theorem states that as goes to ifiity, the bit-chaels start polarizig meaig that they either become a oise-less chael or a pureoise chael. The defiitio of bit-chaels ad the chael polarizatio theorem are discussed more precisely ext. For ay discrete memory-less chael W : X Y, let W (y x) deote the probability of receivig y Y give that x X was set, for ay x X ad y Y. Let W : X Y deote the chael that results from idepedet copies of W i the polar trasformatio i.e. W (y 1 x 1 ) def = W (y i x i ) (2) i=1

3 3 The the combied chael W is defied with trasitio probabilities give by W (y1 u 1 ) def = W ( y1 u ) 1 G = W ( y1 u 1 R G s) (3) This is the chael that the radom vector (U 1, U 2,..., U ) observes through the polar trasformatio. The trasitio probabilities for the bit-chael W (i) is give as follows: W (i) ( y 1, u i 1 1 u i ) def = 1 ( ) W 2 1 y1 u 1 (4) u i+1 {0,1} i For ay B-DMC W, the Bhattacharyya parameter of W is Z(W ) def = y Y W (y 0)W (y 1) It is easy to show that the Bhattacharyya parameter Z(W ) is always betwee 0 ad 1. Bhattacharyya parameter ca be regarded as a measure to determie how good the chael W is. Chaels with Z(W ) close to zero are almost oiseless, while chaels with Z(W ) close to oe are almost pure-oise chaels. More precisely, it ca be proved that the probability of error of a biary symmetric memoryless chael (BSM) is upperbouded by its Bhattacharyya parameter. The followig recursive formulas hold for Bhattacharyya parameters of idividual bit-chaels i the polar trasformatio: Z(W (2i 1) 2 Z(W (2i) 2 ) 2Z(W (i) ) Z(W (i) ) 2 (5) (i) ) = Z(W ) 2 (6) The equality happes i (5) if W is a biary erasure chael. The set of good bit-chaels is defied based o their Bhattacharyya parameters [2], [10]. Let [] def = {1, 2,..., } ad let β < 1 / 2 be a fixed positive costat. The the idex sets of the good bit-chaels are give by { } G (W, β) def = i [] : Z(W (i) ) < 2 β / (7) Theorem 1: [1], [2] For ay BSM chael W ad ay costat β < 1 / 2 we have G (W, β) lim = C(W ) Theorem 1 readily leads to a costructio of capacityachievig polar codes. The idea is to trasmit the iformatio bits over the good bit-chaels while freezig the iput to the other bit-chaels to a priori kow values, say zeros. The decoder for this costructed code is the successive cacellatio decoder of Arika [1]. This decoder will be described i more details later i this sectio. The key property of the ecoderdecoder pair of polar codes is summarized i the followig theorem. This theorem is (the secod part of) Propositio 2 of Arika [1]. Theorem 2: Let W be a BSM chael ad let k = G (W, β). Suppose that a message U is chose uiformly at radom from {0, 1} k, ecoded usig the polar ecoder, ad trasmitted over W. The the probability that the chael output is ot decoded to U uder successive cacellatio decodig satisfies P { Û U } Z(W (i) ) 2 β. i G (W,β) Arika proposed a low-complex implemetatio of the successive cacellatio (SC) decoder of polar codes [1]. Let = 2 s ad suppose that u 1 be the vector that is multiplied by G s ad the trasmitted over idepedet copies of W. Let y1 deote the received word. For i = 1, 2,...,, the decoder computes the likelihood ratio (LR) L (i) of u i, give the chael outputs y1 ad previously decoded û i 1 L (i) (y 1, û i 1 1 ) = W (i) 1. (y 1, û i 1 1 u i = 0) W (i) (y1, ûi 1 1 u i = 1) The likelihood fuctios L (i) ca be computed recursively as follows. Let u j 1,o ad uj 1,e deote the subvectors with odd ad eve idices, respectively. A straightforward calculatio usig the bit-chael recursio formulas for 1, gives the followig recursive formulas: L (2i 1) (y1, û1 2i 2 ) (8) = 1 + L(i) /2 (y/2 1, û 2i 2 1,e û1,o 2i 2 )L(i) /2 (y /2+1, û2i 2 1,e ) L (i) /2 (y/2 1, û 2i 2 1,e û1,o 2i 2 ) + L(i) /2 (y /2+1, û2i 2 1,e ) L (2i) (y1, û1 2i 1 ) (9) = L (i) /2 (y/2 1, û1,e 2i 2 û 2i 2 1,o L (i) )1 û2i 1 /2 (y /2+1, û2i 2 1,e ) If W (i) is ot a good bit-chael, the the decoder kows that i- th bit u i is set to zero ad therefore, û i = u i = 0. Otherwise, it makes the hard decisio based o L (i). The total umber of LRs that eed to be calculated is (1 + log ). Arika also proposed a exact order usig a (1 + log ) trellis i which the LR calculatios are carried out. III. PROPOSED RS-POLAR CONCATENATED CODE I this sectio, we describe our proposed costructio for cocateatig polar codes with outer block codes. We cosider the case whe the outer code is a RS code. We establish bouds o the error correctio performace ad discuss the rate-adaptive costructio. A. Iterleaved cocateated RS-polar codes The proposed scheme for cocateatig polar codes with outer RS codes is illustrated i Fig. 1. The symbols geerated by a certai umber of RS codewords are iterleaved ad coverted ito biary streams usig a fixed basis to provide the iput of the polar ecoders. I a specific costructio, the bits correspodig to the first symbols of all RS codewords are ecoded ito oe polar codeword. Similarly, the iformatio bits of the secod polar codeword costitutes of all bit correspodig to the secod symbol coordiates of all outer RS codewords. Hece, polar ecodig ca be doe i parallel, which reduces the ecodig latecy. More precisely, let ad m deote the legths of the ier polar code ad outer RS code, respectively. Let k deote the umber of iput bits to each polar ecoder i.e. the rate of each ier polar code is R I = k/. The symbols of the outer RS codes are draw from the fiite field F 2 t, with cardiality 2 t. It is assumed that k is divisible by t. Hece, i the proposed

4 4 Fig. 1. Proposed cocateatio scheme of Polar codes with outer iterleaved block codes. scheme, the umber of outer RS codes is r = k/t ad the umber of ier polar codes is m. The rates of the outer RS codes will be specified later. Assume that r RS codewords of legth m over F 2 t are give. For i = 1, 2,..., r, let (c i,1, c i,2,..., c i,m ) deote the i-th codeword. For j = 1, 2,..., m, the j-th polar codeword is the output of the polar ecoder of rate k/ with the iput ( I(c 1,j ), I(c 2,j ),..., I(c r,j ) ), where I(c) maps a symbol c F 2 t to its biary image with t bits. Hece, the total legth of the cocateated codeword is N = m. The iterleaver proposed here betwee the ier ad outer codes ca be viewed as a structured block iterleaver. Other polyomial iterleavers may be cosidered for further improvemets. The iterleaver plays a importat role i this proposed scheme, as it helps to elimiate the eed of large field size for the outer RS code i the previous scheme of [3]. Sice t = k/r, t ca be fixed as k grows by icreasig the umber of RS codewords. B. Rate-adaptive costructio of RS-polar cocateated code Due to the polarizatio pheomeo of polar codes, ot all bit-chaels chose to carry the iformatio bits have the same reliability. A outer RS code is ot actually eeded for the strogest bit-chaels of the ier polar code, sice the correspodig iformatio bits are already well-protected. Our costructio guaratees that all symbols of same RS code see the same set of bit-chaels, which are differet from oe RS code to aother. I other words, each RS codeword is reecoded by same bit-chael idices across the differet polar codewords. Therefore, the rate of each RS code ca be properly assiged to protect the polarized bit-chaels i such a way that all the iformatio bits are almost equally protected. Suppose that the probability of error for each of the iput bits to the polar code is give. Let k be the iformatio block legth of the ier polar code. For i = 1, 2,..., k, let P i be the probability that a error occurs whe decodig the i-th iformatio bit with the SC decoder, assumig that all the first i 1 iformatio bits were successfully recovered. Suppose that the outer RS code is over F 2 t, for some iteger t. The the total umber of RS codes is k/t. The, the first t iformatio bits of each polar codeword form oe symbol for the first RS code, the ext t iformatio bits form oe symbol for the secod RS code, etc. If we cosider oe of the ier polar codes, the probability that the first RS symbol has a error at the output of the SC polar decoder is give by 1 (1 P 1 )(1 P 2 )... (1 P t ). I geeral, for i = 1, 2,..., k/t, the probability that a symbol of the i-th RS codeword is i error assumig that symbols of precedig RS codes were decoded successfully, is give by Q i = 1 (1 P it t+1 )(1 P it t+2 )... (1 P it ). The desig criterio is as follows. Let E be the target frame error probability (FEP) of the cocateated code. The for i = 1, 2,.., k/t, let τ i be the smallest positive iteger such that ( m τ i + 1 ) Q τi+1 i < te k. (10) The, the proposed rate-adaptive (RA) cocateatio scheme deploys a τ i -error correctig RS code for the i-th outer RS code. The followig lemma shows that the FEP E is guarateed. Lemma 3: Suppose that the i-th outer RS code is a (m, m 2τ i ) code, for i = 1, 2,..., k/t, where τ i is determied by (10). The, the total frame error probability for the RS-polar cocateated code is less tha E. Proof: The i-th RS decoder is successful if the umber of its erroeous symbols is at most τ i. If some τ i + 1 of symbols out of the total m symbols are i error, the the RS decoder fails. The probability of this evet is bouded by ( ) m τ i+1 Q τ i+1 i. This is less tha te k by (10). Observe that the total FEP of the cocateated code is upper bouded by the uio boud o error probability of outer RS codes. Therefore, the probability of frame error is less tha k t te k = E. The rate-adaptive desig criterio, described above, requires kowledge of the idividual bit-chael error probability. While it ca be precisely calculated for erasure chaels, we take a umerical approach to solve this problem for a arbitrary chael, e.g. additive white Gaussia oise (AWGN) chael: Assume that all previous iput bits 1, 2,..., i 1 are provided to the SC decoder by a geie whe the i-th bit is decoded. For bit i = 1, 2...,, the decoder is ru for a sufficietly large umber of idepedet iputs to get a estimate of the probability of the evet that the i-th bit is ot successfully decoded, give that the bits idexed by 1, 2,..., i 1 were successfully decoded. A alterative way is to use the method itroduced i [13] which provides tight upper ad lower bouds o the bit-chael error probability. IV. PROPOSED DECODING METHODS FOR THE RS-POLAR CONCATENATED CODE I this sectio, we first provide a aalysis of asymptotic decodig complexity of the proposed cocateated code. The several decodig techiques are proposed to improve the fiite legth performace while the order of decodig complexity is kept the same. A. Decodig complexity To compute the decodig complexity of the cocateated RSpolar code, we take ito accout the decodig complexity of both the ier polar code ad the outer RS code. The decodig complexity of the polar code usig successive cacellatio decodig is give by O( log ), with beig the legth of the polar code [1]. Sice there are m ier polar codes i the proposed cocateated scheme, the total complexity of decodig the ier polar codes is O(m log ), which is bouded by O(N log N).

5 5 A well-kow hard-decisio bouded-distace RS decodig method is the Berlekamp-Massey (BM) algorithm. The BM algorithm is a sydrome-based method which fids the error locatios ad error magitudes separately. The decodig complexity is kow to be O(m 2 ) operatios over the field F 2 t. Oe mai advatage of the proposed cocateated code is that the RS code alphabet size is i the same order of m, whereas it is expoetial i terms of i the existig scheme of [3]. Gao proposed a sydrome-less RS decodig algorithm that uses fast Fourier trasform ad computes the message symbols directly without computig error locatios or error magitudes [12]. For RS codes over arbitrary fields, the asymptotic complexity of sydrome-less decodig based o multiplicative FFT techiques was show to be O(m log 2 m log log m). Hece by deployig sydrome-less RS decodig, the total complexity of decodig the outer RS codes ca be at most O(m log 2 m log log m) which is bouded by O(N log 2 N log log N). Therefore, the total decodig complexity of the proposed cocateated RS-polar code ca be asymptotically bouded by O(N log 2 N log log N). B. Successive cacellatio RS-polar decodig The mai drawback of the successive cacellatio decodig of polar codes is that oce a error occurs, it may propagate through the whole polar codeword. Sice the iformatio block of the polar code is protected with a outer RS code, errors i the decoded bits ca be corrected usig the outer code while the SC decoder evolves. This ca potetially mitigate the error propagatio problem ad cosequetly results i improvemet i the FEP of the proposed scheme at fiite block legths. Hece, we propose the successive cacellatio decodig algorithm for our serially cocateated RS-polar code as follows. Let k be the iformatio block legth for the ier polar code ad 2 t be the size of the alphabet for the outer RS code. Iitialize the algorithm with j = 1. The For each of the m polar codewords, decode the j-th iformatio sub-block of legth t, i.e. the iformatio bits idexed by (j 1)t + 1, (j 1)t + 2,..., jt. These operatios ca be doe i parallel for the ier polar codewords. Pass the mt hard-decisioed output bits as m symbols over F 2 t to form the j-th RS word, ad decode it with the bouded-distace RS decoder. Update the decoded bits of all m polar codewords usig the RS decoder output, ad use them to cotiue SC decodig. Icrease j by oe ad repeat, while j < k/t. Not oly the frame error probability of the RS-polar cocateated code ca be potetially improved, but also the decodig latecy ca be sigificatly reduced, thaks to the parallel processig i the first step of the decodig algorithm explaied above. C. SC Geeralized Miimum Distace decodig of RS-polar code Geeralized miimum distace (GMD) decodig was itroduced by Forey i [6], where the soft iformatio is used with algebraic bouded-distace decodig to geerate a list of codewords. I order to further improve the performace, while keepig the decodig complexity order, we propose SC decodig of the cocateated code with GMD decodig of the outer code. I the cocateated code, the likelihood of each symbol ca be computed give the LLRs of the correspodig bits geerated by the SC decoder of the ier polar code. The m symbols of each RS word are sorted with respect to their likelihoods. The α least likely symbols are declared as erasures, where the case of α = 0 is the same as a regular RS decodig. I covetioal GMD decodig, errors ad erasures decodig of RS codes is ru for α = {0, 2, 4,..., d 1}, where d is the miimum distace of the RS code. This gives a list of size at most (d + 1)/2 cadidate RS codewords at the output of the decoder. The decoder picks the closest oe to the received word, which is the passed to the polar decoders. A aive way of implemetig the GMD decodig icreases the complexity by a factor of (d + 1)/2. However, Koetter derived a fast GMD decodig algorithm which removes this factor [7], ad hece GMD ca be deployed i decodig our RS-polar code while preservig our decodig complexity boud. Sice the SC decoder actually computes the LLR s of the bits i each symbol, the likelihood of each symbol that is passed to RS decoder ca be computed. The symbol likelihoods from the differet polar SC decoders are used for GMD decodig of each RS code. D. Near ML SC-GMD decodig I the previous subsectio, at the last step of GMD RS decodig, the cadidate i the geerated list of codewords that is the closest oe to the received word is picked. Here, we further improve the performace by actually pickig the most likely codeword based o soft iformatio from the polar decoder. The first approach is to approximate the symbol probabilities usig the bit LLR s geerated by the polar decoder. This is ot precise, sice the bit LLR s i each symbol are ot idepedet. We pick the best codeword from the list geerated by GMD decoder based o its estimated probability give by the product of estimated symbol probabilities. We call this approach SC- GMD-approximate ML or SC-GMD-AML decodig. I the secod approach, the SC decoder of polar code is modified to output the soft iformatio for all the possible symbols. For a symbol costitutig of t bits, the LLR of each bit depeds o the previous bits i the symbol. I order to compute the exact symbol probabilities, the SC polar decoder computes the probabilities of all the 2 t symbols by traversig all the possible 2 t paths, for each cosecutive t bits. This icreases the complexity of polar decoder by a costat factor of t 1 i=0 2i /t. This eables ear ML decodig of outer RS code o top of the SC-GMD decodig of RS-polar code, wherei the GMD decoder picks the most likely cadidate from the geerated list. We call this method SC-GMD-ML decodig. Also, sice the SC decoder recursively calculates the LLRs, the LLRs computed alog each path are saved so whe the correct symbol is picked by the outer decoder, the LLRs computed alog the correspodig path are picked to proceed with SC decodig for the ext t bits.

6 6 V. THEORETICAL PERFORMANCE LIMITS I this sectio, upperbouds ad lowerbouds o the error correctio capability of the proposed cocateated code is derived. Also, it is show how to set the parameters of the code i order to get the upperboud 2 N 1 ɛ, for ay ɛ > 0, where N is the legth of the cocateated code, o the probability of frame error. Furthermore, the performace of the RS-polar cocateated code ad Arika s polar codes is compared assumig trasmissio over chaels with erasure bursts. A. Asymptotic aalysis of error correctio performace I our costructio, assume all outer RS codes have the same rate R o. I Lemma 4, it is show that the error probability of the cocateated code is bouded by 2 (0.5 ɛ (1 R o)/2 1)m. The i Theorem 5, we prove that m, ad R o ca be set i such a way that the error probability of the cocateated code is bouded by 2 N 1 ɛ, for ay ɛ > 0, asymptotically, while the cocateated code is still capacity achievig. This sigificatly improves the error-decay rate compared to polar codes with the same legth N. Lemma 4: I the proposed RS-polar cocateated scheme, for ay ɛ > 0 ad large eough, the probability of frame error is upper bouded by 2 ( 0.5 ɛ (1 Ro) 2 1)m, where ad m are the legths of the ier ad outer codes, respectively, ad R o is the rate of outer code. Proof: Assumig a bouded-distace RS decoder, the error correctio capability of RS codes is τ = (1 R o )m/2. The decoder for the whole cocateated scheme fails if more tha τ of the ier polar codewords are i error. Let P e deote the probability of block error of the ier polar decoder ad E be the frame error probability (FEP) of the cocateated code. E ca be derived from P e as follows: E = m i=τ+1 ( m i ) P i e(1 P e ) m i Pe τ+1 (11) τ + 1 The upper boud o the probability of error holds by the followig simple observatio. If some τ +1 of the polar codewords are i error, the a decodig error occurs. The boud is derived by coutig all the possibilities for the locatio of τ + 1 erroeous symbols. Sice some error icidets are couted multiple times, we get a upper boud i (11). By pluggig i the result of Theorem 2 o the probability of error P e of polar codes ito the upper boud i (11) we get E ɛ (τ+1) τ ɛ m(1 R o)/2 τ + 1 (12) < 2 m ɛ m(1 R o)/2 = 2 (0.5 ɛ (1 R o)/2 1)m Notice that the boud i Lemma 4 ca be tighteed usig the Stirlig s approximatio. Istead of simply boudig ( m τ+1) by 2 m, the more precise Stirlig s approximatio of 2 m(h((1 Ro)/2) 1/2 log m+o(1)), where H(x) is the biary etropy fuctio, ca be used. However, this does ot affect the result of the ext theorem i the asymptotic sese. Theorem 5: For ay ɛ > 0, the legths of the ier polar code ad outer RS code, ad the rate of outer RS code ca be set such that the frame error probability of the cocateated code of total legth N is asymptotically upper bouded by 2 N 1 ɛ, while the scheme is still capacity-achievig. Proof: The legth of the ier polar code, the legth of the outer RS code m, ad the rate of outer RS code R o ca be set as follows: = N ɛ, m = N 1 ɛ, ad R o = 1 4N ɛ(0.5 ɛ). Substitutig, m, ad R o ito the boud give by Lemma 4, oe gets E 2 (0.5 ɛ (1 R o)/2 1)m = 2 N 1 ɛ (13) as the upper boud o the FEP. With above settigs, R o 1, as N. Hece, the rate of the cocateated polar code also approaches the capacity, sice the ier polar code is prove to be capacity-achievig. A lower boud o the probability of error ca be also derived by aalyzig a optimistic case. The optimistic case is the followig: whe a polar decoder fails, oly oe sub-block of the decoded data which cotributes to oe symbol of the RS outer code is i error. Also, these erroeous symbols are distributed equally over the RS codewords i.e. each RS codeword gets a equal umber of erroeous symbols. I this case, the system ca support up to τr errors i polar decoders where r is the umber of outer RS codewords ad τ is the error-correctio capability of the outer RS code. I fact r R I /log m, where R I is the rate of ier polar code ad log m is the umber of bits i represetatio of a symbol i the RS codeword. I this optimistic case, the FEP E L is give by E L = m i=rτ+1 ( m i ) P i e(1 P e ) m i The plug i the result of [2], for large eough, ad use the approximatio discussed i Lemma 4 alog with the Stirlig s approximatio to get E L ɛ (rτ+1) rτ ɛ mr I (1 R o)/2logm (14) rτ mh(r I R o/m log m) 1.5 ɛ mr I (1 R o)/2logm B. Aalysis of erasure burst correctio performace A error burst is a commo error patter i storage systems which cosists of a cotiguous ru of erroeous symbols. A erasure burst of legth d is a ru of d cosecutive symbols that are all erased. Suppose that a polar code of legth = 2 s is costructed for trasmissio over o a B-DMC W, with respect to the set of good bit-chaels G (W, β) for a fixed β < 1 / 2. Let u 1 deote the iput message, where u i carries a iformatio bit for ay i G (W, β) ad otherwise, it is froze to zero. u 1 G is

7 7 trasmitted through idepedet copies of W. The received sequece is deoted by y1. This is the sceario cosidered i the followig two lemmas. Lemma 6: Let q s ad j 2 s q be two positive itegers. Assume that a erasure burst of legth 2 q occurs with startig idex (j 1)2 q + 1 ad edig idex j2 q. The for ay l with 0 l 2 q 1, the computed likelihood ratio of u l2 s q +1 by successive cacellatio decoder is 1. Proof: Observe that for ay i /2, the likelihood ratio L (2i 1) (y1, û (2i 1) 1 ) is 1 if oe of the terms L (i) 1, û 2i 2 û 2i 2 1,o /2 (y/2 1,e ) or L(i) /2 (y /2+1, û2i 2 1,e ) is 1. This is clear by the recursive formula for LR calculatio i (8). Calculatio of the LR of u l2 s q +1 reduces to calculatio of LRs of the form L (l+1) 2q, after s q recursios. Oe of these terms is of the form L (l+1) 2 q (yj2q (j 1)2 q +1,...), where the secod coordiate is some liear combiatio of û 1, û 2,..., û l2 s q. Sice all the output symbols y j2q (j 1)2 q +1 are erased, the later is always 1. This completes the proof of the lemma. Lemma 7: Let q s/2. The for ay β < 1 / 2 ad large is a good bit-chael. Proof: We show that l = 2 q 1 satisfies the required coditio. Let Z = Z(W ). The by (5) ad iductio o r it is easy to show that eough, there exists l 2 q 1 such that W (l2s q +1) Also, by (6) ad iductio o i we have Z(W (2q ) 2 q ) = Z2q (15) Z(W ((2q 1)2 i +1) 2 i+q ) 2 i Z(W (2r ) 2 q ) (16) (15) ad (16) together imply that Z(W ((2q 1)2 s q +1) ) 2 s q Z 2q 2 s/2 Z 2s/2 I order to coclude that (2 q 1)2 s q + 1 is a good bit-chael idex, we eed to show that for large eough s 2 s/2 Z 2s/2 < 2 s 2 2sβ Or equivaletly, by takig logarithm from both sides s 2 + 2s/2 log Z < s 2 sβ which holds for large eough s give the fact that log Z < 0 ad β < 1 / 2. This completes the proof of lemma. Theorem 8: Cosider a Arika s polar code of legth, costructed for trasmissio over a BSM chael W. If a erasure burst of legth at least 2 1 occurs, the successive cacellatio decoder always fails to recover the trasmitted message with probability at least 0.5. Proof: Observe that ay burst of legth 2 1 = 2 2 s/2 1 icludes a burst of legth 2 s/2 with startig idex (j 1)2 s/2 + 1 ad edig idex j2 s/2, for some j 2 s/2. The by Lemma 7, there exists a good bit-chael with idex of the form l2 s/ By Lemma 6, the LR of u l2 s/2 +1 is always 1 ad therefore, there is a probability of error 0.5 associated with this iformatio bit. This completes the proof of theorem. The above theorem shows that polar codes are very weak with respect to erasure bursts. I fact, a vaishig fractio of erasures lead to a failure i the successive cacellatio decoder with probability 0.5, while it does ot chage the effective capacity of chael asymptotically. Next, we show that RSpolar cocateated codes ca perform very well i this regard. Let the total legth of the RS-polar cocateated code be N = m, where is the legth of polar code ad m is the legth of the RS code. Let also d be the miimum distace of the RS code. Lemma 9: The RS-Polar cocateated code, with outer code of miimum distace d ad ier code of legth, ca recover from erasure bursts as log as the legth of the burst is at most (d 2) + 1. Proof: Observe that a burst of legth (d 2)+1 overlaps with at most d 1 polar codewords. I the worst case, assume that ay polar decoder that ecouters at least oe erasure declares failure ad output erasures to the outer decoder. At most d 1 of the ier polar decoders fail i this sceario. The miimum distace of the outer RS code is d which meas that it ca correct d 1 or less erasures. Therefore, RS-polar cocateated code ca correct all the erasures resulted from the erasure burst. Suppose that m ad the miimum distace of the RS code is at least 4. The (d 2) N + 1 Therefore, a polar code of legth N fails to recover from a erasure burst of legth (d 2) + 1 by Theorem 8, while our RS-polar code ca recover from the erasure burst. By fixig d to be ay positive iteger ad lettig to go to ifiity, the total probability of error of the cocateated code is m2 β, which goes to zero as m. The oly costrait o m is that m. Also, it is assumed that m goes to ifiity so that the rate of outer RS code approaches 1. Sice the polar code is capacity-achievig, the total cocateated code is also capacityachievig. All of these happe while a o-cocateated polar code of equal legth N fails with probability at least 0.5 i this sceario, for large eough N. VI. SIMULATED NUMERICAL PERFORMANCE A. Simulated performace over AWGN chael Trasmissio over AWGN chael is assumed. The ier polar code of legth 2 9 = 512 ad outer Reed-Solomo code of legth 15 over F 2 4 are cosidered. The rates of ier ad outer codes are desiged such that the total rate of scheme is 1/3. We use the RA method explaied i Sectio III-B to costruct the cocateated code. The actual probability of error of the bit-chaels uder SC decodig correspodig to a polar code of legth = 512 are estimated over a AWGN chael with E b /N 0 = 2 db. We take a Mote-Carlo simulatio-based approach for this estimatio as discussed i Sectio III-B. As a alterative way, oe ca use the method proposed i [13]. To desig the outer RS code, we follow the criterio proposed for the RA costructio i Sectio III-B with the differece that the total rate of the code is fixed rather tha the probability of error. To guaratee the total desig rate of the cocateated code, the costructio is optimized over differet possible rates k/ of the ier polar codes as well, ad the rate-adaptive costructio method is as follows:

8 8 For k divisible by 4 ad betwee 170 (ier rate 1/3) ad 256 (ier rate 1/2) do the followig. Pick the best k bit-chaels that have smaller probability of errors ad sort them with respect to their idex. Set target probability of error P e for each of the small sub-blocks of legth 4 (2 4 is the size of the alphabet for RS code). This replaces te/k i (10). The for i = 1, 2,.., k/4, fid the τ i -error correctig RS code accordig to the criterio proposed i (10). Calculate the total rate of the scheme ad compare it with 1/3. If it is almost 1/3, the calculate the total FEP ad move o to the ext k. Otherwise, adjust the probability of error P e accordigly ad repeat these steps. At the ed, k = 204 is picked which results i the lowest frame error rate. The performace of the proposed costructio with discussed decodig techiques is show i Figure 2. The results are compared with a polar code of the same legth 512. Sice all the 15 ier polar codes i the RS-polar cocateated code are decoded i parallel, the two schemes have the same decodig latecy. For the cocateated scheme, the codeword error rate of ier polar codes is defied as the error rate of the ier polar codeword. The aim is to have a fair compariso with a polar code of the same block size 512 whe the rates are equal, where the rate loss due to RS outer code is take ito accout. At a block error probability (BLER) of 10 4, it is observed that the proposed rate-adaptive (RA) cocateated RS-polar code with the proposed SC decodig of the cocateated code has more tha 1 db SNR gai over the o-cocateated polar code with the same decodig latecy. The GMD decodig, o top of SC, helps at higher BLERs, e.g. 0.1 db further SNR gai at BLER= Furthermore, there is about 0.2 db SNR gai usig GMD decodig with approximate ML (GMD-AML) o top of the SC decodig ad 0.3 db further SNR gai usig our proposed SC-GMD-ML decodig of the cocateated code. Also, the proposed SC-GMD-ML decodig algorithm offers more tha 2 db SNR gai over covetioal serial decodig of the RS-polar cocateated code, i which the outer RS code is a (15, 11) code ad the ier polar code is a (512, 232) code. The performace of this cocateated scheme is also compared with the outer compoet Reed-Solomo code with GMD decodig. The (63, 21) RS code over F 2 6 is picked, where its rate is equal to that of our scheme. The legth of its biary represetatio is 6 63 = 378 which is close to that of the ier polar code. It is observed that the performace of the RS code with GMD decodig is about 9 db SNR worse tha that of the proposed scheme with GMD decodig. Therefore, it is ot show i Figure 2. B. Trade-off betwee polarizatio order ad RS decodig radius It is iterestig to aalyze the trade-off betwee the legths of ier ad outer codes assumig that the total legth of the cocateated code is fixed. The loger the ier polar code is, the further polarizatio happes which results i better performace i the ier code. O the other had, loger outer RS code provides better error correctio capability ad also a Block error rate = 512, rate = 1/ E b /N 0 [db] polar code regular RS-polar RA-SC RA-SC-GMD RA-SC-GMD-AML RA-SC-GMD-ML Fig. 2. Performace of the cocateated scheme usig GMD-ML decodig techique wider rage of rates to be chose by the rate adaptive scheme, thereby makig the cocateated code more efficiet. Recall the proof of Theorem 5 wherei the code parameters are set as follows: the ier block legth is = N ɛ ad the outer block legth is m = N 1 ɛ. The the error decay rate is N 1 ɛ as bouded by 2 proved i Theorem 5. Therefore, if ɛ is decreased, the boud o the error decay rate will be improved, asymptotically. As a result, it is well-justified to say that the smaller the ier polar code is, the further improvemet i error decay rate happes i as asymptotic sese. I fiite block legth, there is o aalytical way to determie the optimum parameters for ier ad outer block legths. I this subsectio, we provide simulatio results i a attempt to fid out the optimum parameters whe the total block legth is about The total rate of the scheme is fixed ad is equal to 1/2. The rate-adaptive scheme, except for the case of = 16, i which there is oly oe outer RS code, is desiged at E b /N 0 = 2 db. The FEP varies depedig o the scheme ad is set for each of them separately i order to maitai the same rate of 1/2 for the total cocateated code. More details o each desig of the cocateated code is as follows: For the case of RS code over F 2 5 of legth 31 ad ier polar code of legth 512 (simply deoted by RS(31)-polar(512)), the optimum value of k, the iformatio block legth of polar code, is 295. There are 295/5 = 59 outer RS codes i total. For the case of RS(63)-polar(256), k = 144 ad there are 24 outer RS codes of legth 63 over F 2 6. Cosequetly, for the cocateated code RS(127)-polar(128), k = 70 ad there are 10 outer RS codes of legth 127 over F 2 7. We have also chose the extreme case of RS(1023)-polar(16) to compare with other schemes, where k = 10 ad there is oly oe RS code of legth 1023 over F The simulatio results are show i Figure 3. It ca be observed that the cocateated code RS(31)-polar(512) shows a better performace comparig to the other schemes. Notice that the extreme case of RS(1023)-polar(16) has a much sharper slope of frame error rate i terms of SNR comparig to other cases. However, its performace is much worse which may be due to the relatively very small umber of polarizatio levels.

9 9 frame error rate Fig E b /N 0 Trade-offs betwee the legths of ier ad outer codes RS(15)-polar(1024) RS(31)-polar(512) RS(63)-polar(256) RS(127)-polar(128) RS(1024)-polar(16) C. Performace gais from cocateatio over chaels with erasure bursts We have also observed the advatage of our RS-polar cocateated code comparig to o-cocateated polar codes over chaels with erasure bursts through simulatios. A Gilbert- Elliot model for producig erasure bursts is cosidered. This model is based o a Markov chai with two states for the chael. The chael is either i the good state, deoted by G, or i the bad state, deoted by B. Whe the chael is i state G, stays i this state with probability P ad chages to state B with probability 1 P. Likewise, whe the chael is i state B, stays i this state with probability Q ad chages to state G with probability 1 Q. For simulatio, state G is cosidered as biary erasure chael with probability of erasure 0.1 ad state B is always a erasure. I the RS-polar cocateated code, the legth of ier polar code is set as 2 9 = 512. The outer RS code is a (15, 11) code which ca correct up to four erasures. The total legth of the cocateated code is N = = The rate of ier polar code is set as 0.68 so that the total rate of the cocateated code is 0.5. The trasitio probabilities i the Gilbert-Elliot model are picked as P = ad Q = This way, the average legth of a ru of bad chael states is 1/(1 Q) = 100, which is close to N. Likewise, the average legth of cosecutive good chael states is 1/(1 P ) = 10000, which is close to N. This model resembles the coditios i Theorem 8. Ituitively, the ier polar codes corrects the erasures resulted from the the BEC(0.1) i good state while the outer RS code corrects the failures resulted from erasure bursts. The probability of frame error estimated as usig Mote Carlo simulatios. O the other had, the o-cocateated polar code of legth 2 13 = 8192 does ot perform well i this sceario, as expected. The probability of frame error is estimated as 0.1 usig Mote Carlo simulatios. Notice that i this sceario, the computed likelihood ratios are either 0, 1 or whe the symbols are trasmitted through biary erasure chael. If the computed likelihood ratio is either 0 or, it meas that the correspodig iput bit is recovered with o error. Otherwise, the correspodig iput bit is erased. Therefore, the successive cacellatio decoder ca ot be improved usig list decodig algorithm proposed i [14]. VII. DISCUSSIONS AND CONCLUSION I this paper, we proved that by carefully cocateatig the recetly iveted polar codes with Reed-Solomo codes, a sigificat improvemet i the error-decay rate compared to o-cocateated polar codes is possible which ultimately fills the gap with the iformatio theoretic boud. The parameters of the scheme ca be set to iherit the capacity-achievig property of polar codes while workig i the same regime of low complexity. The proposed cocateated code is show to perform well over chaels with erasure bursts, while it is proved that the origial polar codes are very weak i this regard. We developed several costructio methods ad decodig techiques to improve the performace at fiite block legths, which is a step to makig polar codes more practical. There are some directios for future work as follows. The methods described i this paper ca be i geeral applied to cocateatio of polar codes with o-biary block codes. The costructio ad decodig methods ca also be used whe the outer code is a biary code by groupig each t bits, where t is a complexity parameter to be optimized. If the outer code has a low complexity soft decodig algorithm, the the decodig techiques based o GMD decodig of RS codes ca be exteded to this case as well. REFERENCES [1] E. Arika, Chael polarizatio: A method for costructig capacityachievig codes for symmetric biary-iput memoryless chaels, IEEE Tras. Iform. Theory, vol. 55, o. 7, pp , July [2] E. Arika ad E. Telatar, O the rate of chael polarizatio, preprit of July 24, 2008, arxiv.org/abs/ [3] M. Bakshi, S. Jaggi ad M. Effros, Cocateated polar codes, Proc. IEEE Iter. Symp. Iformatio Theory, pp , Austi, TX., Jue [4] A. Eslami, H. Pishro-Nik O the aalysis ad desig of fiite-legth polar codes, IEEE Tras. Commu., vol. 61, o. 3, pp , March 2013 [5] Y. Polyaskiy, H. V. Poor ad S. Verdu, Chael codig rate i the fiite blocklegth regime," IEEE Tras. Iform. Theory, vol. 56, o. 5, pp , May [6] G. D. Forey, Geeralized miimum distace decodig, IEEE Tras. Iform. Theory, vol. 12, o. 2, pp , April [7] R. Koetter, Fast geeralized miimum distace decodig of algebraic geometry ad Reed-Solomo codes, IEEE Tras. Iform. Theory, vol. 42, o. 3, pp , May [8] R. Koetter ad A. Vardy, Algebraic soft-decisio decodig of Reed- Solomo codes, IEEE Tras. Iform. Theory, 49(11), , [9] M. El-Khamy ad R. J. McEliece. Iterative algebraic soft-decisio list decodig of Reed-Solomo codes. IEEE Jour. Selected Areas Comm, 24(3), , [10] S.B. Korada, Polar codes for chael ad source codig, Ph.D. dissertatio, EPFL, Lausae, Switzerlad, May [11] S. B. Korada, E. Şaşoğlu, ad R.L. Urbake, Polar codes: Characterizatio of expoet, bouds, ad costructios, Proc. IEEE Iter. Symp. Iformatio Theory, pp , Seoul, Korea, Jue [12] S. Gao, A ew algorithm for decodig Reed-Solomo codes, Commuicatios, Iformatio ad Network Security,V. K. Bhargava, H. V. Poor, V. Tarokh, ad S. Yoo, Eds. Norwell, MA: Kluwer, 2003, pp [13] I. Tal ad A. Vardy, How to costruct polar codes, olie at [14] I. Tal ad A. Vardy, List decodig of polar codes, Proc. IEEE Iter. Symp. Iformatio Theory, Sait Petersburg, Russia, [15] H. Mahdavifar, M. El-Khamy, J. Lee ad I. Kag, O the costructio ad decodig of cocateated polar codes", Proc. IEEE Iter. Symp. Iformatio Theory, pp , Istabul, Turkey, 2013.