design of rate-compatible block turbo code with a low-degree generating polynomial

Wireless Pers CommunDOI 10.1007/s11277-014-1853-5

Design of Rate-Compatible Block Turbo Codewith a Low-Degree Generating Polynomial

Kyunghoon Kwon · Jun Heo

© Springer Science+Business Media New York 2014

Abstract This paper presents a Rate-Compatible Block Turbo Code (RC-BTC) withincreased transmission capacity. The proposed RC-BTC uses a new algorithm called infor-mation augmenting scheme, and can achieve a higher code rate than conventional one. Anew Error Location Finding (ELF) Decoding Algorithm is introduced to solve the decodingproblem of the information augmenting scheme. As a result, a novel error-correcting capabil-ity of the BTC-applied Chase-ELF Hybrid Decoder, combining the previous Chase–Pyndiahalgorithm and the proposed ELF algorithm, is demonstrated via Monte-Carlo simulation. Inaddition, the proposed code maximally increases the transmission capacity with the Chase-ELF Hybrid decoder using an information augmenting scheme.

Keywords Block Turbo Code · Rate-Compatible code · IEEE 802.16e · WiMAX

1 Introduction

Block Turbo Code (BTC), a serially concatenated code, was first introduced in 1954 by Eliasas a product code [1,2]. However, due to the use of a Hard-Input Hard-Output (HIHO) algo-rithm to decode the data with low error correction capability and high decoding complexity,it had a poor performance, and was not attractive to many users. Recently, Pyndiah proposeda novel decoding algorithm of BTC with high error-correction capability and reasonabledecoding complexity using the Chase algorithm [3,4]. The Pyndiah–Chase algorithm imple-ments iterative decoding which is performed using several Soft-Input Soft-Output (SISO)decoders with low complexity. The Chase algorithm is a suboptimum algorithm showingnear-ML performance for linear block codes at a high code rate [5]. This algorithm offers

K. Kwon · J. Heo (B)School of Electrical Engineering, Korea University, 5-1 Anam-dong,Sungbuk-gu, Seoul, Republic of Koreae-mail: [email protected]

K. Kwone-mail: [email protected]

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a good compromise between performance and complexity and is very attractive for imple-mentation. Because of that, BTC has already been adopted in the IEEE 802.16 standard forwireless wide-area networks (WiMAX) [6], and is now studied in the IEEE 802.22 standardfor wireless regional-area networks using TV bands [7]. Moreover, it has also been widelyused in satellite communication systems and digital storage systems [8].

Despite its near optimum performance and reasonable complexity, the algorithm has somelimitations in code rate or codeword length due to the use of the conventional BCH code withfixed codeword and information length. In the case of other error correction codes, i.e. Low-Density Parity Check (LDPC) Code, the code rates are controlled by using parity puncturingor information shortening schemes for time varying channel [9,10]. But the BTC, which isconstructed by the regular BCH code for each row and column, can achieve its code rateby using information shortening scheme or constructing asymmetry forms without othersolutions for a fixed codeword length. Moreover, the channel of the WiMAX standard canbe expressed by a time varying channel because users can easily approach the system, andthis channel may vary depending on the number of users. Due to the reason, the transmis-sion system has to select a forward error correction (FEC) code with a suitable code rateusing shortening or puncturing schemes for channel conditions when data is transmitted.Then, BTC can use the information shortening scheme or lower rate code to maximize itserror correcting capability for a bad channels. However, even if the channel conditions arevery good, the conventional BTC uses a single error-correcting BCH code in the WiMAXsystem because there is no other solution for achieving higher code rate than conventionalone.

The outline of the paper is as follows. In Sect. 2, we review the information-shorteningscheme and present the information-augmenting scheme. In Sect. 3, the conventional Chasealgorithm is concisely summarized to understand the decoding procedure, and the encodingand decoding procedure of the proposed RC-BTC is mentioned. The performances of thissystem is analyzed and evaluated in Sect. 4. A conclusion is drawn in Sect. 5.

2 Preliminaries

2.1 Notation Explanation

In this section, the proposed scheme called by information augmenting scheme, and its prop-erties are presented. Before describing the scheme, some quantities are defined as follows:

• n, k, and dmin are codeword length, information length, and minimum hamming distanceof linear block codes, respectively.

• s and a are the number of shortening and augmenting bits, respectively.• ei (X) and Si (X) are the error and syndrome polynomials with single error in i th position,

respectively.• σ(X) is the error location polynomial obtained from Berlekamp–Messy algorithm with

the syndrome polynomial S(X).• R and Y are the observations and the hard decision values of the received signal, respec-

tively.• T and Z are the test patterns and the test sequences, respectively. The test sequence Z is

the modulo sum of Y and T.• C and D are the decoded codeword using Z and the optimum decision vector with minimum

Euclidean distance between R and C, respectively.

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Design of Rate-Compatible Block Turbo Code

• W is the extrinsic information for the next iterative decoding.• α and β are the scaling factor and the reliability factor, respectively.

2.2 Information Shortening and Augmenting Scheme

The main limitation of the conventional BTC is the weakness in the rate and length com-patibility, because (n, k, dmin) BCH codes constructing BTC have to use regular parameters,where n, k, and dmin designate codeword length, information length, and minimum hammingdistance, respectively [11]. Since the BCH generating polynomial is decided by these fixedparameters, BCH code only controls its code rate using an information-shortening schemeas a rate-compatible code. Given the (n, k) BCH code of length n and dimension k, con-sider the set of codewords where s information bits are equal to zero. If the s informationbits are deleted from each of the codewords, obtaining (n − s, k − s, dmin) shortened BCHcode, where n − s and k − s are the shortened codeword length and shortened informationlength, respectively. This procedure is called information-shortening, and rate-compatibleBCH codes achieve lower code rate. However, only the code rate is decreased.

In general, the conventional linear block codes, i.e. BCH , can use rate-compatible schemesthrough information-shortening scheme in order to obtain lower code rate than conventionalone due to the fixed values of information and codeword length. This is why an informationaugmenting scheme is proposed to achieve higher rate and longer codeword length in rate-compatible codes over conventional codes. Given (n, k) BCH code, the information which arethe redundant length n −k, the number of error correction “t”, and its generating polynomial,are obtained. If the a information bits are added from each of these codewords, the (n +a, k + a) augmented BCH code is obtained where n + a and k + a are the augmentedcodeword and information length, respectively. k+a information bits are encoded by using the(n, k) BCH generating polynomial, then an n +a codeword is generated. Since an equivalentgenerating polynomial is used, the redundant (parity) length is still n − k bits.

2.3 Properties of BCH Codes with Low-Degree Generating Polynomials(Information Augmenting Scheme)

As mentioned above, conventional BTC achieves rate-compatible codes that have low rate.If the rate-compatible BTC is constructed, the structure of BCH codes constructing BTCshould adjust the usable rate-compatible scheme for higher code rate. BCH codes generatecodewords using standardized generating polynomials with fixed codeword length, infor-mation length, and minimum hamming distance. However, if the information augmentingscheme is applied to BCH code, as components of BTC, rate-compatible code for highercode rate than conventional one can be generated. The conventional algebraic decoder, i.e.the Berlekamp-Messy algorithm, is applied to BCH code with an information-augmentingscheme. A syndrome polynomial based on the received polynomial using Berlekamp-Messyalgorithm can be achieved. In the case of the conventional linear block code, there is aone-to-one correspondence between syndrome polynomials and error polynomials, and noinformation-augmenting scheme. Using an information-augmenting scheme, the error pat-terns with uniform cycle correspond to a syndrome polynomial. The length of a cycle canbe estimated using a standardized (n, k) BCH generating polynomial. When the informationbits are encoded using (7, 4) BCH g-poly with an information-augmenting scheme, errorpolynomials ei (X) = Xi , Xi+7, Xi+14, . . . correspond to a syndrome polynomial Si (X).The proposed scheme has an equivalent syndrome polynomial for various error polynomialswith n-cycles of the (n, k) BCH generating polynomial. The generating polynomials of the

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BCH code for each code rate and its coefficients, g0, . . . , gn−k are listed in Table 1. Figure 1shows the encoder of RC-BTC with the information augmenting scheme and Fig. 2 showsthe block diagram of its encoder structure with α, the number of information augmented bits.

The information-augmenting codeword encoded by a conventional (n, k) BCH generatingpolynomial still has n − k parity bits. After the decoding procedure is activated, a syndromepolynomial of the information-augmenting scheme is generated. If the codeword length islonger than n-bits, it is only estimated via the syndrome polynomial whether an error occurred.Assuming high signal to noise ratio (SNR) and single error correction (n, k) BCH code, it

Table 1 The coefficients of a Polynomial Division Circuit for each generating polynomial

BCH g(x) g0 g1 g2 g3 g4 g5 g6 g7 g8

(255, 247) 1 + x2 + x3 + x4 + x8 1 0 1 1 1 0 0 0 1

(127, 120) 1 + x3 + x7 1 0 0 1 0 0 0 1 –

(63, 57) 1 + x + x6 1 1 0 0 0 0 1 – –

(31, 26) 1 + x2 + x5 1 0 1 0 0 1 – – –

(15, 11) 1 + x + x4 1 1 0 0 1 – – – –

(7, 4) 1 + x + x3 1 1 0 1 – – – – –

Fig. 1 Encoder of the RC-BTC with information-augmenting scheme (Polynomial Division Circuit)

Fig. 2 Construction of BlockTurbo Code withinformation-augmenting scheme

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can be estimated that the single error in any place is one of the predictable error locationpolynomials, σ(X) = Xl , Xl+n, Xl+2n, . . ., and 0 ≤ l ≤ n − 1, where σ(X) is obtainedfrom the syndrome polynomial S(X).

3 Rate-Compatible Block Turbo Codes with Low-Degree Generating Polynomial

The encoding procedure of the proposed system is equivalent to conventional BTC. Butbefore the encoding procedure, a target rate must be determined and defined the number ofbits added to each row, and column as ‘a’ to increase the number of information bits for ratecompatibility.

Rtarget = (KBCH + a)2

(NBCH + a)2 , a is the added bits.

a =⌈√

Rtarget · NBCH − KBCH

1 − √Rtarget

⌉. (1)

After determining the value of ‘a’, conventional encoding procedure is performed, and theinformation bits are initially arranged in a (K +a)-by-(K +a) array due to the characteristicof BTC consisting of two-dimensional product codes. Then, each row is encoded using BCHcodes. Afterwards, the resulting (N + a) columns are encoded using the same BCH codes,and the product code is obtained, which consist of (N + a) rows and columns. The receiverdecodes the information using the signal received via the transmission channel. In this paper,a novel decoding algorithm is proposed to decode the proposed system with an information-augmenting scheme. First, the conventional Chase algorithm used in BTC is introduced, andthen, a suitable decoding algorithm for the information-augmenting scheme is proposed.Finally, a Hybrid algorithm of the Chase and proposed algorithm, the most suitable decodingalgorithm for Rate-Compatible BTC, will be presented.

3.1 Chase Decoding Algorithm

In this section, we briefly explain the decoding procedure of Block Turbo Codes. In [3,4], theobservations are defined as R. The Chase algorithm using observation R can be described asfollows:

1. Obtain the hard decision sequence Y=(y1, . . . , yl , . . . , yn) from R, where yl = 1 ifrl ≥ 0 and yl = 0 if rl ≤ 0. Then, determine the position of the p = �dmin/2� leastreliable binary elements of Y using R.

2. Form test patterns Tq (q = 2p), which define all combinations of binary sequences inthe p least reliable positions. The length of test pattern sequences is n and the positionsexcept the p least reliable positions are all zeros.

3. Form test sequences Zq where zql = yl ⊕ tq

l and decode Zq using an algebraic (or hard)decoder and add the codeword Cq to subset Ω .

4. Determine optimum decision D, which defines the minimum Euclidean distance vectorin subset Ω from R. We operate the maximum likelihood (ML) decision for subset Ω

based on the following decision rule:

D = Ci i f |R − Ci |2 ≤ |R − Cl |2 ∀l ∈ [1, 2k], l = i, (2)

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Fig. 3 Block diagram of Block Turbo Decoder

where Ci = (c1i , · · · , cl

i , · · · , cni ) is the i th codeword of C and

|R − Ci |2 =n∑

l=1

(rl − ci

l

)2(3)

is the square of the Euclidean distance between R and Ci .5. Calculate the extrinsic information based on soft output computation for the j th bit of the

codeword. If C j exists such that C ∈ Ω and c j = d j , calculate the extrinsic informationfor every bit of the decision sequence. The extrinsic information is computed as

w j =( |R − C|2 − |R − D|2

4

)· d j . (4)

If codeword C is not found, another method for computing the soft output should be found.The solution is to use the following equation:

w j = β × d j wi th β ≥ 0. (5)

The value of reliability factor β increases with the number of iterations. During iterativedecoding, β starts with a low value, then gradually becomes a higher value. Figure 3 showsthe Chase decoder of the Block Turbo Code.

The received signal is denoted as a matrix [R]. The first decoder performs the soft decodingof the rows of [R] and the second one performs on the columns of [R]. The subsequentiteration performs similarly. The data input in the i-th iteration is given by

[R(i)] = [R] + α(i)[W(i)], (6)

where W(i) is the extrinsic information matrix and α(i) is a scaling factor to eliminate thedifference between the standard deviation of the matrix [R] and matrix [W(i)]. This scalingfactor is also used to reduce the effect of the extrinsic information in the soft decoder to thefirst decoding steps when the bit error rate (BER) is relatively high. The value of this factorincreases with the iteration increasing [3]. The Chase decoder outputs the updated extrinsicmatrix used for the next iteration.

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Table 2 Error polynomial E(x)

and syndrome polynomial S(x)S(x) E(x)

1 1 X7 X14 X21 · · ·X X X8 X15 X22 · · ·X2 X2 X9 X16 X23 · · ·1 + X X3 X10 X17 X24 · · ·X + X2 X4 X11 X18 X25 · · ·1 + X + X2 X5 X12 X19 X26 · · ·1 + X2 X6 X13 X20 X27 · · ·

3.2 Error Location Finding (ELF) Algorithm

The syndrome polynomial for errors is required to decode the RC-BTC codewords. BCH codegenerated by a low-degree generating polynomial has an equivalent syndrome polynomialfor many other errors with periodic location in a high-SNR regions. When the generatingpolynomial of (7, 4) BCH code is applied to RC-BTC, the error and syndrome polynomialsare as shown in Table 2.

An Error Location Finding (ELF) Algorithm is derived by using this property. The ELFalgorithm can be described as follows:

1. Obtain the hard decision sequence Y′ as mentioned in step 1 of the Chase algorithm.Then, temporarily decode Y′ using an algebraic (or hard) decoder and find the syndromepolynomial.

2. Determine the position of the p′ = �NBT C/(2m − 1)� predictable error positions wherem and NBT C are the degree of the generating polynomial and the codeword length ofeach row and column with information-augmenting scheme, respectively. Then form testpatterns Tp′

, which define all cases of one bit-flip binary sequences which provide anerror location polynomial using the Berlekamp-Messy algorithm. Test patterns Tp′

areNBT C -dimensional binary vectors with a single “1” in the error location polynomial.Then, a second test pattern is generated with a single “1” in the error location polynomialmultiple X2m−1, and · · · , p′th test pattern is produced with a single “1” in the (X2m−1)p′

thlocation.

3. Form test sequences Zp′where z p′

l = y′l ⊕ t p′

l and add the codeword Cp′where Cp′

is

equivalent Zp′to subset Ω ′.

4. Determine optimum decision D′, which defines minimum Euclidean distance vector insubset Ω ′ from R. Decoder operates the maximum likelihood (ML) decision for subsetΩ ′ based on the following decision rule as mentioned Eqs. (2) and (3) in step 4 of Chasealgorithm.

5. Calculate extrinsic information based on soft output computation for the j th bit of thecodeword. Next, the process is entirely the same as described earlier. The conventionalalgorithm uses a soft value to update the information for the next iterative decodingprocedure in the conventional Chase algorithm, but the proposed ELF algorithm needs are-defined value of scaling and reliability factor not equivalent to previous factors due tothe use of hard information updates. Therefore the scaling and reliability factor shouldbe redefined as follows:

α(m) = [0.0, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.6]β(m) = [0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.6, 0.6]. (7)

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3.3 CHASE-ELF Hybrid Decoding Algorithm

When the decoding procedure is performed with the proposed information-augmentingscheme, the iterative coding gain should be maximized with the soft and hard informa-tion values. To do this, the decoding algorithm uses a combination of two algorithms. TheChase-ELF Hybrid algorithm can be described as follows:

1. Obtain the hard decision sequence Y from received signal R, and then determine thenumber of the p′ test patterns. Also, form test pattern Tp′

by implementation of the ELFalgorithm.

2. After generating Tp′test patterns, determine the position of the p = �dmin/2� least

reliable binary elements of Y using R. Then, form test patterns Tq (q = 2p), whichdefine all combinations of binary sequences in the p least reliable positions via theChase algorithm as mentioned above. The number of all test patterns is q + p′. This canbe mathematically expressed as:

q + p′ = 2p + �NBCH/(2m − 1)�, p = (dmin,BTC − 1)/2. (8)

3. Form test sequences Zq where zql = yl ⊕ tq

l and decode Zq using an algebraic (or hard)

decoder and add the codeword Cq to subset Ω . Also form test sequences Zp′where

z p′l = y′

l ⊕ t p′l and add the codeword Cp′

where Cp′is equivalent Zp′

to subset Ω ′.4. Determine optimum decision D′, which defines the minimum Euclidean distance vector

in subset Ω +Ω ′ from R. We operate the maximum likelihood (ML) decision for subsetΩ + Ω ′ based on the following decision rule as mentioned in Eqs. (2) and (3) in step 4of the Chase algorithm.

5. Calculate extrinsic information based on soft output computation for j th bit of codewordvia the same process as described earlier. The conventional algorithm uses soft values toupdate the information for the next iterative decoding procedure in the conventional Chasealgorithm, and the ELF algorithm uses hard values for iterative decoding. Therefore, thescaling and reliability factors should be redefined to reflect the Chase and proposed ELFalgorithms. The redefined scaling and reliability factors are the optimized values frommany simulations, and are the reflected values to update soft and hard information. In theHybrid decoder, the β reliability factor values are equal to the values of the conventionalChase algorithm because the Chase algorithm assumes a main role due to the T q testpatterns. However, the error correcting capability of the ELF algorithm is also powerfulin high SNR regions. Therefore, the value of the scaling factor α similar to the ELFalgorithm is used.

α(m) = [0.0, 0.5, 0.5, 0.5, 0.5, 0.7, 0.7, 0.7]β(m) = [0.2, 0.4, 0.6, 0.8, 1.0, 1.0, 1.0, 1.0]. (9)

In Fig. 3, Block Turbo Decoder uses only Chase decoder. If the proposed Chase-ELFHybrid decoder is applied to the RC-Hybrid BTC, chase decoder has to be changed theproposed hybrid decoder. Figure 4 shows a block diagram of the Chase-ELF Hybrid decoder.

4 Numerical Results

In the previous section, the proposed decoding algorithm, Chase-ELF, is studied with theconventional Chase algorithm. For all systems, an additive white Gaussian noise (AWGN)channel, BPSK modulation, and up to six iterations are assumed. (31, 26)2 BTC and (63, 57)2

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Fig. 4 Block diagram of Chase-ELF Hybrid decoder

2 2.5 3 3.5 4 4.5 510

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

Uncoded

(31,26)2 conv BTC itr 6 (R=0.7034)

(31,26)2 Hybrid BTC itr 6 (R=0.7034)

(63,57)2 conv BTC itr 6 (R=0.8186)

(63,57)2 Hybrid BTC itr 6 (R=0.8186)

(255,247)2 conv BTC itr 6 (R=0.9382)

(255,247)2 Hybrid BTC itr 6 (R=0.9382)

Fig. 5 Performance comparison between the conventional BTC (Pyndiah–Chase algorithm) and the proposedBTC (Chase-ELF) at iteration 6 for various code rates

BTC adopted in the IEEE 802.16 (WiMAX) standard are considered, as well as (255, 247)2

BTC. Figure 5 shows the bit error rate (BER) performance of the proposed and conventionalalgorithms. The proposed systems with (31, 26)2 BTC, (63, 57)2 BTC and (255, 247)2 BTCshow consistent performance gain over the conventional BTC. In the region of BER of 10−5,an approximate gain of 0.15 dB is observed. The reason for this coding gain is that thenumber of test patterns of the conventional BTC and the proposed BTC are set to 16 and 17,respectively. The test pattern Tq is equivalent to each BTC, but the proposed system needs anadditional test pattern Tp′

due to the use of the ELF algorithm. In other words, if an occurederror position is not equivalent to the position of the least reliable binary elements of Y,an added test pattern Tp′

can correct the error that is not corrected by test patterns Tq . The

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K. Kwon, J. Heo

−4 −2 0 2 4 6 8 10 120

200

400

600

800

Mean

The

Num

ber

of L

LR

val

ues

(a) (255,247)2 conventional BTC at 4.6dB

itr = 1 , avg = 1.028090itr = 3 , avg = 1.640033itr = 6 , avg = 1.493337

itr 1

itr 6

itr 3

−4 −2 0 2 4 6 8 10 120

200

400

600

800

Mean

The

Num

ber

of L

LR

val

ues

(b) (255,247)2 Hybrid BTC at 4.6dB

itr = 1 , avg = 1.053350itr = 3 , avg = 1.454557itr = 6 , avg = 5.346564itr 1 itr 6itr 3

Fig. 6 The results of tracking probability density function (PDF) based on the extrinsic information forconventional and proposed BTC

added test pattern crucially functions in enabling BTCs to find the more reliable codewordby iterative decoding.

Figure 6 shows the probability density functions (PDFs) based on the extrinsic informa-tion of the conventional BTC and the proposed Hybrid BTC with (255,247) BCH code. Inthe 4.6 dB SNR region, the BER performance of the conventional BTC and Hybrid BTC arearound 10−3 and 10−5, respectively, because the proposed scheme has a powerful iterativedecoding effect. In the iterative decoding process, the soft output tends to have a Gaussiandistribution for any identically distributed input data [3]. Figure 6a shows the PDFs of(255, 247)2 conventional BTC output for the first, third, and sixth iterations. The PDFsof three different iterations are almost the same. This suggests that the PDF does not evolveas the iterations progress. In other words, performance cannot be enhanced by iterations.Figure 6b depicts that the extrinsic information of the proposed system, (255, 247)2 HybridBTC, is a Gaussian distribution at each of the iterations, and the average log likelihood ratio(LLR) values are increased by increasing the number of iterations for the first, third, andsixth iterations. The PDF is evolved as the iterations progress, and this confirms that theperformance can be enhanced by iterations.

Also, the proposed BTC can control its code rate according to the channel conditionsas a rate-compatible code. Rate-Compatible Hybrid BTC can be constructed using othergenerating polynomials with low degree as mentioned in Fig. 1. The transmitter encodesusing a low-degree generating polynomial to change the code rate, then the receiver decodesthe data using the Chase-ELF Hybrid decoder. The proposed BTC with the single errorcorrection BCH is simulated because the conventional BTC with single error correctionBCH code is used in the WiMAX system. Single error correction BCH can maximize the

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2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 710

−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

Uncoded

(31,26)2 conv BTC itr 6 (R=0.7034)

(31,26)2 Hybrid BTC itr 6 (R=0.7034)

(30,26)2 RC hybrid BTC itr 6 (R=0.7511)


Fig. 7 (31, 26)2 RC-Hybrid BTC with three different code rates; the circle, square and diamond correspondto code rates of 0.7034, 0.7511, and 0.8038, respectively, and all of the codes have 676 information bits

code rate of BTC because the proposed system needs minimal parity bits. Figures 7, 8 and 9show the performance of the RC-Hybrid BTC. The RC-Hybrid BTC obtained from the mothercodes, (31, 26)2 BTC, (63, 57)2 BTC and (255, 247)2 BTC, respectively, are encoded withthe low-degree generating polynomial in Table 1.

The performances of conventional and RC-Hybrid (31, 26)2 BTCs are depicted for variouscode rates in Fig. 7. (30, 26)2 and (29, 26)2 RC-Hybrid BTCs are generated with (15,11) and(7,4) BCH generating polynomials, respectively. Both Chase-algorithms have 4 p values and16 test patterns, and the number of ELF test patterns p′ are 2 and 5, respectively. As a result,the number of total test patterns are 18 and 21, respectively. The SNR value at 10−5 of theconventional and RC-Hybrid BTCs with various code rates are 3.6, 5, and 5.5 dB, respectively.The SNR value at 10−5 of the proposed BTC is 3.6 dB, which is equivalent to the conventionalBTC. The RC-Hybrid BTC achieved a transmission capacity improved by a maximum of12.5 % with the (7,4) BCH generating polynomial, but it has a performance degradationof about 1.9 dB. Also, using (30, 26)2 RC-Hybrid BTC with a (15,11) BCH generatingpolynomial can improve the transmission capacity by around 6.35 % with a performancedegradation of 1.4 dB. Assuming that the RC-Hybrid BTC is used in good conditions of atime varying channel in the WiMAX system, the performance degradation converges to zerobecause the good conditions of a time varying channel allow relatively high SNR values orlow noise variances.

In Fig. 8, the performances of (63, 57)2 RC-Hybrid BTCs are depicted for various coderates. As mentioned above (63, 57)2 BTC is used in the IEEE 802.16 WiMAX standard with(31, 26)2 BTC. The RC-Hybrid BTCs are generated by (63,57), (31,26), (15,11), and (7,4)BCH generating polynomials, respectively, and have 17, 18, 21, and 25 total test patterns,

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K. Kwon, J. Heo

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 710

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

Uncoded

(63,57)2 conv BTC itr 6 (R=0.8186)

(63,57)2 Hybrid BTC itr 6 (R=0.8186)




Fig. 8 (63, 57)2 RC-Hybrid BTC with four different code rates; the circle, square, diamond and trianglecorrespond to code rates of 0.8186, 0.8452, 0.8732 and 0.8038, respectively, and all of the codes have 3249information bits

respectively. In these cases, the (60, 57)2 RC-Hybrid BTC with (7,4) BCH code achievedtransmission capacity improved by a maximum of 9.3 % with performance degradation of2 dB.

Figure 9 shows the performance comparison between (255, 247)2 conventional BTC and(255, 247)2 RC-Hybrid BTC at various code rates. The (255, 247)2 conventional BTC has avery high code rate, and the RC-Hybrid BTC has an even higher code rate, approaching thecode rate of one. Surprisingly, this code with a code rate of 0.9761 approaches the region of10−5 BER with an SNR value of only 6.2 dB.

5 Conclusion

We proposed an information-augmenting scheme to design rate-compatible BTC for achiev-ing very higher code rates approaching one than conventional one. A new decodingalgorithm, the ELF algorithm was introduced, which can correct errors by using theerror locations. In addition, the RC-BTC with the Chase-ELF Hybrid algorithm wasproposed as well, combining the conventional Chase algorithm with the proposed ELFalgorithm.

The proposed algorithms were verified by comparing the Chase-ELF Hybrid BTC withthe conventional BTC. The Chase-ELF Hybrid BTC showed consistent performance gainaround 0.15dB at various code rates. Moreover, by tracking its PDF of extrinsic information,the proposed algorithm was verified as having improved iteration gains. Another importantcontribution was that this code can control its code rates. Compared to the conventional

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3.5 4 4.5 5 5.5 6 6.5 7 7.5 810

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

Uncoded

(255,247)2 conv BTC itr 6 (R=0.9382)

(255,247)2 Hybrid BTC itr 6 (R=0.9382)




Fig. 9 (255, 247)2 RC-Hybrid BTC with four different code rates; the circle, square, diamond and trianglecorrespond to code rates of 0.9382, 0.9456, 0.9531 and 0.9761, respectively, and all of the codes have 61009information bits

algorithm in IEEE 802.16 and 802.22 systems, this algorithm has an advantage of higher coderate and improved transmission capacity than conventional one. Particularly, the (255, 247)2

RC-Hybrid BTC with very higher code rate approaching one can be implemented.Therefore, considering an improved performance of the proposed scheme, the RC-Hybrid

BTC will perform a key role in the next generation high capacity mobile communicationsystems which require the use of error correction code with higher code rate.

Acknowledgments This research was supported by the Ministry of Science, ICT and Future Planning(MSIP), Korea, under the IT/SW Creative research program supervised by the National IT Industry PromotionAgency (NIPA) (NIPA-2013-H0502-13-1022).

References

1. Forney, G. D. (1966). Concatenated codes. Cambridge, MA: MIT Press.2. Elias, P. (1954). Error-free coding. IRE Professional Group on Information Theory, 4(4), 29–37.3. Pyndiah, R. (1998). Near optimum decoding of product codes: Block turbo codes. IEEE Transactions on

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Proceeding of GLOBECOM.5. Chase, D. (1972). A class of algorithms for decoding block codes with channel measurement information.

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trol (MAC) and Physical Layer (PHY) specifications: Policies and procedures for operation in the TVbands. IEEE std 802.22–2011 (pp. 1–680).

8. Song, H., & Cruz, J. R. (2000). Block turbo codes for magnetic recording channels. In Proceedings ofICC.

9. Ha, J., Kim, J., & McLaughlin, S. W. (2004). Rate-compatible puncturing of low-density parity-checkcodes. IEEE Transactions on Information Theory, 50(11), 2824–2836.

10. Tian, T., & Jones, C. R. (2005). Construction of rate-compatible LDPC codes utilizing informationshortening and parity puncturing. EURASIP Journal on Wireless Communications and Networking, 5,789–795.

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Kyunghoon Kwon received the B.S., degree in electronics engineer-ing from Sejong University, Seoul, Korea, in 2010. He received theM.S., degree in electronics engineering from Korea University, Seoul,Korea, in 2012. He is presently a Ph.D. course in the School of Electri-cal Engineering at Korea University, Seoul, Korea. His research inter-ests include wireless communications, error-correct coding, and LDPCcode.

Jun Heo received the B.S., and M.S., degrees in electronics engineer-ing from Seoul National University, Seoul, Korea in 1989 and 1991,respectively and the Ph.D. degree in electrical engineering from theUniversity of Southern California, Los Angeles, USA in 2002. During1991–1997, he was a senior research engineer at LG Electronics Co.,Inc. During the 2003–2006, he was an assistant professor in the elec-tronics engineering department, Konkuk University, Seoul, Korea. Heis presently a professor in the School of Electrical Engineering at KoreaUniversity, Seoul, Korea. His research interests include channel codingtheory and Digital communication systems.

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