IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 9, SEPTEMBER 2009 2505

Transactions Letters

Improving the Distance Properties of Turbo CodesUsing a Third Component Code: 3D Turbo Codes

Claude Berrou, Fellow, IEEE, Alexandre Graell i Amat, Member, IEEE,Youssouf Ould-Cheikh-Mouhamedou, Member, IEEE, and Yannick Saouter, Member, IEEE

AbstractThanks to the probabilistic message passing per-formed between its component decoders, a turbo decoder is ableto provide strong error correction close to the theoretical limit.However, the minimum Hamming distance (min) of a turbo codemay not be sufficiently large to ensure large asymptotic gains atvery low error rates (the so-called flattening effect). Increasingthe min of a turbo code may involve using component encoderswith a large number of states, devising more sophisticatedinternal permutations, or increasing the number of componentencoders. This paper addresses the latter option and proposes amodified turbo code in which a fraction of the parity bits areencoded by a rate-1, third encoder. The result is a noticeablyincreased min, which improves turbo decoder performance atlow error rates. Performance comparisons with turbo codes andserially concatenated convolutional codes are given.

Index TermsChannel coding, iterative decoding, parallelconcatenation, serial concatenation, turbo codes.

I. INTRODUCTION

TURBO codes (TCs) are today mainly used in AutomaticReQuest (ARQ) systems, which do not usually requirevery low error rates. Target Frame Error Rates (FER) from102 to 105 are typical for this kind of communication sys-tems. However, in future communication system generations,lower FER, down to 108, may be necessary to open the wayto real-time and more demanding applications, such as TVbroadcasting or videoconferencing. The minimum Hammingdistance (min) of a TC may not be sufficiently large to offersuch error correction at the required signal to noise ratio. Forthe current commercial applications of TCs (3G, DVB-RCS,WiMax...), commonly based on 8-state component encoders,there are several ways to increase min and thereby improvingthe performance at very low error rates. For instance, one

Paper approved by T. M. Duman, the Editor for Coding Theory andApplications of the IEEE Communications Society. Manuscript receivedOctober 10, 2007; revised August 13, 2008.

C. Berrou, A. Graell i Amat, and Y. Saouter are with Institut TELECOM,TELECOM Bretagne, UMR CNRS Lab-STICC, Universite Europeennede Bretagne, CS 83819 29238 Brest Cedex 3, France (e-mail: e-mail:{claude.berrou, alexandre.graell, yannick.saouter}@telecom-bretagne.eu).

Y. Ould-Cheikh-Mouhamedou is with the Prince Sultan Advanced Tech-nologies Research Institute (PSATRI), King Saud University, Riyadh, SaudiArabia (e-mail: ycheikh@ksu.edu.sa).

The material in this paper was presented in part at the IEEE InformationTheory Workshop (ITW), Lake Tahoe, CA, US, Sept. 2007.

A. Graell i Amat was supported by a Marie Curie Intra-European Fellow-ship within the 6th European Community Framework Programme. Y. Ould-Cheikh-Mouhamedou was supported by Region Bretagne.

Digital Object Identifier 10.1109/TCOMM.2009.09.070521

might use stronger component codes, e.g. 16-state codes,at the price of doubling the decoding complexity. Devisingmore appropriate internal permutations [1, 2] is an appealingalternative to improve min, since it does not incur anycomplexity increase. Unfortunately, designing such powerfulpermutations is not an easy task, and there are limits to themin and multiplicity values, and thus to the performanceimprovements, that can be achieved. Another way to improvemin which has been widely explored in the literature, is toconcatenate the component encoders in series rather than inparallel [3]. Serially concatenated codes (SCCs) yield higherminimum distances compared than parallel concatenation, butshows a penalty in convergence threshold, which might beunacceptable for several applications. On the other hand,irregular SCCs have been shown to achieve near-capacityconvergence thresholds, but at expense of a poorer error floor[4]. Mixed structures, like those proposed in [57], are alsopossible, combining the features of the two concatenations.Finally, multiple concatenation using an increased number ofcomponent encoders, can be used to eliminate low-weightcodewords and so improve the distance properties of the code.

This paper addresses the latter alternative to improve theminimum distance of TCs by proposing a three-dimensionTC (3D-TC). The 3D-TC we describe here is inspired by thecontributions in [6] and [7] and calls for both parallel andserial concatenation in an original approach: the proposed 3D-TC is simply derived from the classical TC by adding a partialrate-1 third dimension. A rate-1 post-encoder is concatenatedat the output of the parent turbo code, encoding only a fraction of the parity bits. The 3D-TC is a very versatile code andprovides very low error rates for a wide range of block lengthsand coding rates. It significantly improves performance in theso-called flattening region with respect to the 8-state classicalTCs, at the expense of very small increase (less than 10%)in complexity. It compares favorably to more complex codes,such as 16-state TCs.

II. THE ENCODING STRUCTURE

A block diagram of the proposed 3D-TC () is depicted inFig. 1. The information data sequence u of length bits isencoded by a TC consisting of the parallel concatenation oftwo codes, and . The corresponding code sequences aredenoted by y and y, respectively. We call this turbo codethe parent TC. In this paper, we consider the double-binary

0090-6778/09$25.00 c 2009 IEEE

2506 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 9, SEPTEMBER 2009

Cu

Cl

u

P/SPost-

encoder'

puncturing

u

yu

yl

w

1-

1-

yuch,p

ylch,p

patch

ylp

yup

yuch

ylch

Fig. 1. Block Diagram of the 3D-TC.

TC of the DVB-RCS standard [8] as the parent TC. Therefore, and are rate-2, 8-state, convolutional codes (CCs) withgenerator polynomials 15 (recursivity), 13 (redundancy), and7 (second input).

A fraction (0 1) of the parity bits y = {y,y}stemming from and are post-encoded by a rate-1,third encoder . In the sequel, we shall refer to as thepermeability rate. The bits to be post-encoded are chosen ona regular basis. For instance, if = 1/4 the permeabilitypattern is [1000] for both the upper and the lower encoders,i.e., every fourth bit in y and y is post-encoded. The inputsequence of the post-encoder is made of alternate y and y(surviving) parity bits. The number of parity bits which arefed to the post-encoder is given by:

= . (1)

The fraction 1 of parity bits which is not post-encodedis sent directly to the channel or punctured to achieve thedesired code rate. In Fig. 1 we denote by ych and y

ch the

sub-codewords of y and y, respectively, directly sent tothe channel, and by ych,p and y

ch,p their punctured versions.

Similarly, we denote by yp and yp the sub-codewords of y

and y which are post-encoded. The output of the post-encoderis denoted by w. The coded sequence y of length bits isobtained by multiplexing u, w, ych,p and y

ch,p . The overall

code rate is given by:

=1

1 + + (1 ) , (2)

where (0 1) is the fraction of surviving bits in ych andych after puncturing. Note that given the highest achievablecode rate is max = 11+ . Higher code rates may be achievedby puncturing systematic bits. On the other hand, the lowestcode rate is given by the rate of the double-binary TC, i.e., 1/2. However, the proposed 3D-TC is intended for awide range of code rates, from high ones to low ones, downto = 1/4. For a global coding rate lower than 1/2, thecomponent double-binary encoders in Fig. 1 need to generateextra parity bits. In particular, three parity bits are neededfor the lowest rate, = 1/4. For such cases, we devised anappropriate 8-state component code with three binary ouputs.

In this paper, very simple regular or quasi-regular punctur-ing patterns are applied. For example, if rate-1/2 is soughtfor and = 1/4, then, according to (2) = 1/3, and thepuncturing pattern [100] will be applied to ych and y

ch .

The material added to a standard turbo encoder, which weshall refer to as patch because it is placed just behind a pre-existing turbo encoder, is composed of:

a parallel to serial (P/S) multiplexer which takes alter-nately the parity bits y and y to be post-encoded andgroups them into a single block of bits,

a permutation denoted by which permutes the paritybits before feeding them to the post-encoder,

a rate-1 post-encoder, working on a fraction of theparity bits of each component encoder.

The proposed structure combines the features of parallel andserially concatenated codes. Increasing turns the code intomore serial, while the case = 0 corresponds to the standardparallel TC. Parameter can be tuned to tradeoff betweenconvergence and minimum distance. The proposed encodingprinciple can be extended in a straightforward manner to anyparent TC.

III. THE CHOICE OF THE POST-ENCODER

Given the parent TC and the interleaving laws for and ,the performance of the 3D-TC depends on the post-encoderand the permeability rate , which must be properly optimized.The choice of the post-encoder has to meet the followingrequirements:

1) Its decoder must be simple, adding little complexity tothe classical turbo decoder, while being able to handlesoft-input and soft-output information,

2) in order to prevent the decoder suffering from any sideeffects, while searching for very low error rates, thepost-code has to be a homogeneous block code,

3) at the first iteration (without any redundant input infor-mation), the pre-decoder associated with the rate-1 post-encoder must not exhibit too much error amplification,to prevent from a high loss in convergence.

Memory 1 and memory 2 CCs satisfy requirement (1). Onthe other hand requirement (2) is compatible with the useof Circular CCs having memory 2 (the memory 1 accumulatecode cannot be made circular using standard circular encoding,and has to be discarded). Circular convolutional codes (alsocalled tail-biting codes), are such that any state of the encoderregister is allowed as the initial state and that the encodingalways starts and ends in the same state. This makes the con-volutional code a perfect block code and prevents it from anyside effects. Moreover, no rate loss is induced by terminatingthe code.

We consider the 4-state CC with polynomials (4/5) (in octalform) for the post-encoder, instead of the classical (4/7) code.At the first step of the iterative process, the decoder of thelatter will (roughly) triple the number of errors of its input.On the other hand, the decoder of the (4/5) code only doublesthe number of errors at the first step. Therefore, from theconvergence point of view, it will be preferable to the (4/7)code. Unfortunately, the (4/5) code cannot be made circulardirectly. However, circularity can be achieved using a state-mapping encoding [9, 10].

For short block sizes ( < 1000 bits) and for medium sizes(1000 < < 5000) with low rates, the linear post-encoder(4/5) was chosen. However, for large blocks (5000 > )and for medium sizes with high rates, the 3D-TC with post-encoder (4/5) shows a flattening around FER= 106, due

C. BERROU et al.: IMPROVING THE DISTANCE PROPERTIES OF TURBO CODES USING A THIRD COMPONENT CODE: 3D TURBO CODES 2507

to poor distance properties. In these cases, we used a non-linear encoder, with recursion polynomial gr = 1 +2 (i.e.,the same recursion as the linear encoder) and feedforwardpolynomial gf = 1 + ( 2). The trellis of the proposednon-linear encoder is simply obtained from the trellis of theencoder (4/5) by swapping two branch metrics. The non-linear encoder contributes to improve the distance propertiesof the 3D-TC, with favorable consequences in the cases citedabove.

IV. THE CHOICE OF THE PERMEABILITY RATE

The choice of is a matter of trade-off between the con-vergence loss and the required min. Convergence designatesthe zone of the error rate versus signal to noise ratio /0curve where the error rate begins to decrease noticeably.Choosing a large value of will turn the code into more serial,hence leading to a higher min. However, performance willbe penalized from the convergence point of view. This resultsfrom the decoder associated with the post-encoder, which doesnot benefit from any redundant information at the first iterationand therefore multiplies the errors during the first processing.

A. Convergence Properties

Let us assume for instance that the post-encoder is the(4/5) CC. At the first iteration the associated decoder (the pre-decoder), without any extra information, roughly doubles theerrors at its input. From (1), the fraction of the codewordbits that are post-encoded is:

=

= . (3)

The fraction of the data processed by the componentdecoder of each code ( = , ) that is processed by thepre-decoder is:

=

1 +. (4)

Then, if is the probability of error at the channel output,the average probability of error at each decoder intrinsicinput is:

= 2+ (1 ) = (1 + ) . (5)Using (4) we get

=(1 + (1 + )

1 +

) , (6)

i.e., the probability of error at each decoder intrinsic inputis risen by a factor

(1+(1+)

1+

). Therefore, the addition of

the post-encoder induces a convergence loss, no matter thevalue of . This observation is in agreement with the results in[11], where it is shown by using extrinsic information transfer(EXIT) charts analysis that a threefold concatenation cannotachieve any performance gain in terms of convergence withrespect to a twofold concatenation unless the decoders can bechosen freely.

We used EXIT charts to estimate the convergence thresholdof the 3D-TC for several values of . The convergencethreshold is 0.55 dB, 0.63 dB, 0.68 dB, 0.86 dB, and 1.32dB for = 0, = 1/8, = 1/4, = 1/2 and = 1,

respectively. The convergence loss with respect to the parallelturbo code ( = 0) is very small for = 1/8 and = 1/4,and increases with the value of . For = 1 the loss is verysignificant.

B. Union Bound to the Error Probability

An upper bound on the frame error probability over mem-oryless binary-input channels with coherent detection of code can be obtained as

(/0, ) (7)

where is the output weight enumerating function (OWEF)of , i.e., the number of code sequences of weight , and() represents the pair-wise error probability. Denote by 1,2 and 3 the number of quaternary symbols 10, 01 and11, respectively, at the input of the double-binary TC. Denotealso by and the binary weight of vectors yp and y

p ,

respectively, and its sum by , = + . Similarly, we denoteby and the binary weight of vectors ych,p and y

ch,p ,

respectively, and its sum by , = +. Following [12]the OWEF of the 3D-TC with double-binary components canbe written (using random permutations) as

=

TC,,(

) (8)

where TC, is the number of codewords of the outer double-binary TC with output weights and , respectively, and, is the input-output weight enumerating function

(IOWEF) of the post-encoder, i.e., the number of codewordsof weight generated by an input weight . TC, can beexpressed as

TC,+

=

{1,2,3,,}

1,2,3,,2,1,3,,(/2

1, 2, 3

)

(9)

where

(/2

1, 2, 3

)is the multinomial coefficient.

In Fig. 2 we plot the union bound to the error probabilityfor the 3D-TC with = 1/2 for several values of on theAWGN channel for a block length of 1504 bits. Lower errorfloors are obtained for increasing values of . A slight lowerfloor with respect to the to the parent TC ( = 0) is observedfor = 1/8. The gain is already significant for = 1/2,while for higher s very significant low floors are observed.However, as shown above, this gain is obtained at a expenseof a significant convergence loss.

In this paper, = 1/4 and = 1/8 (for very long blocks)are considered, since they represent a good trade-off betweenconvergence and error floor performance.

V. PERMUTATIONS AND AND DISTANCE PROPERTIES

The proposed 3D-TC is characterized by two permutations,denoted by and , respectively. A first permutation ,

2508 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 9, SEPTEMBER 2009

0 1 2 3 4 5 6 7 8 9 10Eb/N0 (dB)

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

10-10

FE

R=0=1/8=1/4=1/2

Fig. 2. Union bound on the frame error probability of the 3D-TC (withrandom permutations) for several values of . = 1504 bits.

internal of the double binary TC at the basis of the proposed3D-TC, deals with messages of = /2 quaternary symbols.Here, is built from the combination of the Almost RegularPermutation (ARP) [1] and an intrasymbol permutation, assuggested in [13]. The intrasymbol permutation permits toeliminate a large proportion of composite return to zero(RTZ) sequences for both codes. A second permutation

is used to spread = parity bits at the output ofthe double binary TC before post-encoding. We assume

to be the simplest one, defined by the congruence relation = () = 0 + 0 (mod ) , where 0 is the startingindex, and 0 is an integer relatively prime with .

For each block length, the parameters of permutations and have been carefully chosen to guarantee a large min,even for high rates. For 57-bytes packets, the use of 3D-TCresulted in an increase in min by at least 15% and 20% forrate 4/5 and 1/2, respectively, compared to the standardizedDVB-RCS TCs. For 188-byte packets, the increase in min issignificant (i.e., more than 40%) for rate 1/2. It is also worthmentioning that the achieved min values have reasonably lowmultiplicities.

VI. DECODING THE 3D TURBO CODE

The decoding of the 3D-TC calls for the classical turboprocedure in the logarithmic domain. The decoder consistsof three SISO decoders: two 8-state SISO decoders matchedto and , denoted by 1 and 1 , respectively, and a4-state SISO decoder (the pre-decoder) to decode the post-encoder, 1 . As for standard TCs, 1 and 1 exchangeextrinsic information on the systematic symbols of the re-ceived codeword. Also, they must provide 1 with extrinsicinformation on the post-encoded parity bits. In turn, the pre-decoder feeds 1 and 1 with extrinsic information onthese parity bits. A decoding iteration consists of a singleactivation of 1 , 1 and 1 , in this order. No differencesin terms of performance were observed for other activationorders. However, the activation order might have an impacton convergence speed.

Because 1 and 1 are quaternary 8-state decodersprocessing = /2 couples of bits and the pre-decoder is abinary 4-state decoder processing = data, the relative

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Eb/N0 (dB)

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

FE

R

3D-TCDVB-RCS TC16-state DB TCSCCCTheoretical Limit

k=188 bytes

R=1/4

R=1/2R=4/5

Fig. 3. FER performance of the 3D-TC with = 1/4 for = 188 bytes, = 1/4, 1/2 and 4/5. 8 iterations.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Eb/N0 (dB)

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

FE

R

3D-TCDVB-RCS TC16-state DB TCSCCCTheoretical Limit

k=57 bytes

R=1/4

R=1/2

R=4/5

Fig. 4. FER performance of the 3D-TC with = 1/4 for = 57 bytes, = 1/4, 1/2 and 4/5. 8 iterations.

computational complexity added by the latter is very small.For instance, with = 1/4 (the largest value consideredin this paper), the additional complexity is roughly 6% interms of number of trellis transitions per decoded bit. Tothis, however, some extra-functions must be added to theclassical turbo decoder, the main one being the calculationof the extrinsic information on parity bits to be fed to thepre-decoder. Overall, the additional complexity per iterationcompared to the classical turbo decoder is less than 10% for = 1/4.

In this paper, we consider the simple Max-Log-MAP de-coding algorithm, which does not require the knowledge ofthe channel noise variance.

VII. SIMULATION RESULTS

The performance of the 3D-TC was assessed by means ofsimulation. In Figs. 3 and 4, we report frame error rate (FER)results for block sizes 188 and 57 bytes, respectively, andcoding rates 1/4, 1/2 and 4/5. In both figures = 1/4 anda maximum of 8 iterations were assumed.

The proposed code shows excellent performance for bothshort and medium block sizes. In particular, for informationblock size 188 bytes (see Fig. 3) only 0.8 dB loss is ob-served with respect to the Gallagers random coding boundat FER 107 for all code rates. For comparison purpose, the

C. BERROU et al.: IMPROVING THE DISTANCE PROPERTIES OF TURBO CODES USING A THIRD COMPONENT CODE: 3D TURBO CODES 2509

0 0.5 1 1.5 2 2.5 3 3.5 4Eb/N0 (dB)

100

10-1

10-2

10-3

10-4

10-5

10-6

10-7

10-8

10-9

BE

R,F

ER

3D-TC3GPP2 8-state TCSCCC

k=12288 bits, R=1/2

FER

BER

Fig. 5. BER and FER performance of the 3D-TC with = 1/8 for =12288 bits, = 1/2. 8 iterations.

performance of the original DVB-RCS TC is also reportedfor rates 1/2 and 4/5. For rate 1/2 the 3D-TC shows a smallconvergence loss with respect to the DVB-RCS TC. On theother hand, the error floor is significantly lowered. The largestgain is obtained for 188 bytes and = 1/2 (about 1.4 dB atFER = 107). For rate 4/5, the convergence loss is reducedwhile a significant improvement for low error rates is alsoobserved. We can also notice that the performance obtainedwith optimized permutations is clearly better than the onepredicted by the union bound based on random permutations(Fig. 2).

We also report in Figs. 3 and 4 the performance of the 16-state double binary TC described in [13] and the performanceof the SCCC proposed in [14] for rates 1/2 and 4/5. Theperformance of the proposed 3D-TC is comparable to that ofthe more complex 16-state TC. For a block length of 188bytes, the 3D-TC loses 0.1 dB in convergence with respect tothe 16-state double-binary TC. However, the proposed codeoutperforms the 16-state TC in the error floor. A similarbehavior is observed in Fig. 4 for a block length of 57 bytes.On the other hand the 3D-TC shows a convergence gain of0.1 dB and 0.3 dB with respect to the SCCC for = 1/2 and = 4/5, respectively, and = 188 bytes. However, the codein [14] is much simpler, since it is built from 4-state CCs.

In Fig. 5, performance comparison is given with respectto the 8-state TC adopted in the 3GPP2 standard and theSCCC in [14]. An information block length of 12288 bitsand 8 iterations are assumed for the three codes. Very similarperformance are observed in the waterfall region with respectto the 3GPP2. However, the proposed 3D-TC significantlyimproves the 3GPP2 code in the floor. On the other hand,the 3D-TC shows slightly better convergence than the SCCC.

VIII. CONCLUSIONS

In this paper, we presented a modified turbo code combiningthe features of parallel and serial concatenation in order toobtain increased Hamming minimum distances with respectto classical turbo codes. The simulation results corroboratethe interest of this approach. Frame error rates down to 107

are obtained near the theoretical limits without the use ofany outer block code, such as BCH or Reed-Solomon codes.

This characteristic makes this new code, called 3D turbo code(3D-TC), very versatile from the standpoint of block size andcoding rate. Finally, the internal permutations of the 3D-TCare based on very simple models enabling large degrees ofparallelism, if needed.

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