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670IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

Design of Low-Density Parity-Check

Codes for Modulation and Detection

Stephan ten Brink, Gerhard Kramer, Member, IEEE, and Alexei Ashikhmin, Member, IEEE

Abstract—A coding and modulation technique is studied where

the coded bits of an irregular low-density parity-check (LDPC)

codearepasseddirectlytoamodulator.Atthereceiver,thevariable

nodes of the LDPC decoder graph are connected to detector nodes,

and iterativedecoding is accomplishedby viewing thevariable and

detectornodesasonedecoder.Thecodeisoptimizedbyperforming

a curve fitting on extrinsic information transfer charts. Design ex-

amples are given for additive white Gaussian noise channels, as

well as multiple-input, multiple-output (MIMO) fading channels

where the receiver, but not the transmitter, knows the channel.

For the MIMO channels, the technique operates within 1.25 dB of

capacity for various antenna configurations, and thereby outper-

forms a scheme employing a parallel concatenated (turbo) code by

wide marginswhen there are more transmitthan receiveantennas.

IndexTerms—Fading,iterative

parity-check (LDPC) codes, multiple-input, multiple-output

(MIMO) detection, mutual information.

decoding,low-density

I. INTRODUCTION

I

noisy channels [1]–[5]. We consider two problems associated

with LDPC codes. The first is how to combine a code with a

modulator and detector. The second is how to design the code

for iterative decoding, i.e., how to choose good degree distribu-

tions for the modulator, channel, and detector.

We approach the first problem by mapping the coded bits of

an irregular LDPC code directly onto a modulation signal set.

The mapping is arranged to facilitate code design. At the re-

ceiver, we consider thegraphicalrepresentation ofan LDPCde-

coder [2]–[4] and connect the LDPC variable nodes to detector

nodes.

We deal with the second problem by using a curve-fitting

procedure on extrinsic information transfer (EXIT) charts

[6]. The design methodology is illustrated for two types of

channels: additive white Gaussian noise (AWGN) channels

with binary phase-shift keying (BPSK), and multiple-input,

multiple-output (MIMO) fading channels with quadrature

phase-shift keying (QPSK). The MIMO code design can be

be extended in a straightforward way to other modulators,

channels, and detectors. We remark that the curve fitting might

TERATIVE decoding of low-density parity-check (LDPC)

codes is a powerful method for approaching capacity on

Paper approved by H. El Gamal, the Editor for Space–Time Coding and

Spread Spectrum of the IEEE Communications Society. Manuscript received

July 17, 2002; revised May 30, 2003.

S. ten Brink was with Bell Laboratories, Lucent Technologies, Crawford, NJ.

He is now with Realtek, Irvine, CA 92618 USA (e-mail: stenbrink@realtek-

us.com).

G. Kramer and A. Ashikhmin are with Bell Laboratories, Lucent

Technologies, Murray Hill, NJ 07974 USA (e-mail: gkr@bell-labs.com;

aea@bell-labs.com).

Digital Object Identifier 10.1109/TCOMM.2004.826370

be possible using other chart techniques, see, e.g., [5], [7],

and [8]. We refer to [9] for a comparison of some of these

tools. Another alternative is to use numerical optimization with

density evolution

. Transfer charts and density evolution

complement each other in that the former are easier to visualize

and program, giving insight and good initial code designs,

while the latter can be used to verify the graphical analysis and

to refine the designs.

There are several existing approaches to combining coding

and modulation, for example, trellis-coded modulation (TCM)

[10], multilevel coding [11], bit-interleaved coded modulation

(BICM) [12], and space–time block-coded (STBC) modulation

[13], [14] (see also [15] and references therein). A growing

body of work uses BICM with turbo and LDPC codes, see, e.g.,

[16]–[29]. The EXIT curve-fitting approach described here was

motivated by results for erasure channels [30] and appeared in

[31]. Parallel work using similar ideas was reported in [32] and

[33]. The method was used to design repeat-accumulate (RA)

codes in [34] and [35].

This paper is organized as follows. In Section II, we develop

the curve-fitting procedure for BPSK on the AWGN channel.

In Section III, we extend the technique to other communication

problems, and, in particular, to MIMO fading channels where

the receiver, but not the transmitter, knows the channel. We de-

sign LDPC codes for ergodic fading, and compare their perfor-

mancewithaschemeemployingauniversalmobiletelecommu-

nicationssystem(UMTS)standardturbocode.TheLDPCcodes

are shown to perform substantially better for channels having

more transmit than receive antennas. Such a situation is likely

to occur on the base-to-mobile station link of a wireless com-

munication system. Section IV summarizes our results.

II. CODE DESIGN FOR AWGN CHANNELS

ConsideranLDPCcodeoflength

An iterative decoder for this code can be viewed as a graph that

has

variablenodes,anedgeinterleaver,and

The th variable node represents the th bit of the codeword.

This bit is involved in

parity checks, so that its node has

edges going into the edge interleaver. The edge interleaver

connects the variable nodes to the check nodes, each of which

represents a parity-check equation. The th check node checks

bits so that it hasedges. The sets of variable and check

nodes are referred to as the variable-node decoder (VND) and

check-node decoder (CND), respectively. Iterative decoding is

performed by passing messages between the VND and CND.

The decoder structure is shown in Fig. 1, and its operation is

explained in more detail below. We remark that this structure is

similar to that of an iterative decoder for a serially concatenated

anddesignrate.

checknodes.

0090-6778/04$20.00 © 2004 IEEE

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TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION671

Fig. 1.Iterative decoder for an LDPC code.

code that is based on a mixture of inner repetition codes and

a mixture of outer single parity-check codes. This observation

illustrates the close relation between LDPC decoding and other

iterative schemes, such as turbo decoding [36].

A. EXIT Charts

An a posteriori probability (APP) decoder converts channel

and a priori log-likelihood ratios (LLRs or

posteriori

-values. This is shown in Fig. 1, where a poste-

riori -values come out of the VND and CND. The a posteriori

-values minus the a priori -values are the extrinsic -values,

which are passed on and interpreted as a priori information by a

second decoder. We refer to [37] for further details on extrinsic

information processing.

An EXIT chart characterizes the decoder’s operation. We use

the notation of [6] and write

tion between the bits on the decoder graph edges (which are the

bits about which extrinsic -values are passed) and the a priori

-values. Similarly, we write

mation between the bits on the graph edges and the extrinsic

-values. We refer to [6] for further details on how to interpret

these quantities.

-values) into a

for the average mutual informa-

for the average mutual infor-

B. EXIT Curve of the Inner VND

A variable node of degree

fromtheedgeinterleaverandonefromthechannel.Thevari-

able node decodes by computing, for

has incoming messages,

(1)

where

node,

able node, and

Consider the AWGN channel with BPSK (

andnoisevariance

.Wedefinethenormalizedsignal-to-noise

ratio (SNR) as

is the th a priori

is the th extrinsic -value coming out of the vari-

is the channel-value.

-value going into the variable

) modulation

. The channel -value is

(2)

where

function (pdf) evaluated at the output

and be random variables representing the respective channel

input and output. The variance of

is the channel conditional probability density

given the input . Let

conditioned onis

(3)

Fig. 2. VND EXIT curves for ? ?? ? ? dB and ? ? ???.

To compute an EXIT function, we model

-value of an AWGN channel whose input is the th interleaver

bit transmitted using BPSK. The EXIT function of a degree-

variable node is then

as the output

(4)

where the functions

pendix (see also [6]). Fig. 2 plots several variable node curves

when

and

andare given in the Ap-

dB. The quantity

(5)

is the capacity of the channel at the

which in our case is the capacity of the AWGN channel with

BPSK modulation (see the Appendix).

that is being considered,

C. EXIT Curve of the Outer CND

The decoding of a degree

decoding of a length

code. The output -values are thus (see [37, Sec. II.A])

check node corresponds to the

) single parity-check(or rate

(6)

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672IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

Fig. 3.

swapped, as compared with Fig. 2.

CND EXIT curves. Observe that the a priori and extrinsic axes are

In the

“box-plus” operation

-value notation of [37], this can also be written as a

(7)

We again model

channel whose input is the th interleaver bit transmitted using

BPSK. The check node EXIT curves can be computed in closed

form [38], [46] or by simulation. Alternatively, for the binary

erasure channel, a duality property exists [5], [30] that gives the

EXIT curve

of the length

in terms of the EXIT curve

) repetition code, i.e.,

as the output -value of an AWGN

single parity-check code

of the length (or rate

(8)

This property is not exact for BPSK/AWGN a priori inputs, but

it is very accurate [38], [46]. For convenience, we use (8) and

write

(9)

where the second step follows from (4) with

further useful to express (9) in terms of its inverse function, i.e.,

. It is

(10)

Fig. 3 plots several check node curves. Observe that the curves

are similar to the VND curves of Fig. 2, except that they all start

from the origin.

D. EXIT Curves for Code Mixtures

We will consider only check-regular LDPC codes, i.e., all

check nodes have degree

(in [39], these are called right-reg-

ular codes). After choosing, the remaining LDPC design in-

volves specifying the variable node degrees

,.

Let

note these degrees by

node degree is

be the number of different variable node degrees, and de-

,. The average variable

(11)

where

the

VND and CND are the same, we have

is the fraction of nodes having degree

must satisfy

. Note that

. Since the number of edges at the

or

(12)

Let

be the fraction of edges incident to variable nodes of

. There are, in total,

such nodes, so we have

degree

edges involved with

(13)

Note that the

shown that the EXIT curve of a mixture of codes is an average

of the component EXIT curves. We must here average using the

(and not the) because it is the edges that carry the extrinsic

messages. The effective VND transfer curve is thus

must satisfy. In [30] and [40], it is

(14)

i.e.,(14)isaweightedsumofthecurvesofFig.2.Notethatonly

edge fractions can be adjusted because we must enforce

(12) and

. Thus, in order to have any flexibility, we

must choose

. We shall see that

surprisingly good results [4, p. 634].

already gives

E. Design Example

Thepaper[30](seealso[40])showsthattoapproachcapacity

onerasurechannels,onemustmatchtheVNDandCNDtransfer

curves. Empirically, the same is true for other channels. We il-

lustrate this for BPSK modulation on an AWGN channel, and

in the next section for MIMO channels.

Recallthat

the Appendix). We design a code with

dB, so that

is slightly larger than. We further choose

the CND transfer curve has a reasonable distance from the

axis at. This approach simplifies finding a VND

curvethatliesabovetheCNDcurve.Furthermore,thisgivesthe

decoder a good “head start” at the first iteration.

For simplicity, we restrict the VND to have only three dif-

ferent variable node degrees (

can be chosen freely, and Fig. 4 shows a manual curve fit

whose variable node parameters are as follows:

isthechannelcapacity(see

, and choose

dB

so that

). This means that only one

For this simple example, we obtain a convergence threshold of

about 0.5 dB, while the capacity is at 0.19 dB. A simulation

with

, a random edge interleaver, and 100 iterations

shows a turbo cliff at about 0.55 dB (we measure the turbo cliff

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TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION 673

Fig. 4.Curve fit for an LDPC code with ? ? ???.

at a bit-error rate (BER) of

iterations,theturbocliffisat0.5dB,whichverifiestheaccuracy

of the EXIT chart and our approximations.

Weremarkthatbychoosinglarger

curve more closely to the CND curve. One can also model the

a priori

-values more carefully than as outputs of an AWGN

channel. Finally, code design is just as easy for other rates. For

instance, suppose that

start at about

that

can yield a good curve fit. We would now again

choose the

to ensure that the

curve.

). For longer codes and more

,onecanmatchtheVND

so that the VND curve should

. Fig. 3 suggests

curve lies above the

III. CODE DESIGN FOR MIMO CHANNELS

We next turn to multiantenna modulation and detection. The

techniques described here can be applied to many other modu-

lators, channels, and detectors.

A. MIMO Channel Model

For MIMO fading, one often distinguishes between three

practical cases based on channel knowledge. The first case is

that both the transmitter and receiver know the channel, the

second that only the receiver knows the channel, and the third

that neither terminal knows the channel [41, pp. 2627–2629].

The first case can be dealt with in a manner similar to the

AWGN channel, with the addition of “water pouring” over the

antennas and time. The third case seems to be the most difficult

and is not dealt with here. We will consider only the second

case.

Consider the setup of Fig. 5, where there are

receiveantennas.Eachtransmittersymbolisan

whose entries take on complex values in a

constellation set. We consider constellations of size

each vector symbol carries

for QPSK, we have

. The average energy per transmit

transmit and

1vector

so that

coded bits. For example,

Fig. 5.MIMO model.

symbol is limited to

for

The receiver sees

is the

Suppose the entries of

Gaussianrandomvariableswithindependentrealandimaginary

parts each having variance

ized SNR

as

, and we will use

.

1 vectors, where

channel matrix and

are independent, complex, zero-mean,

is a 1 noise vector.

. We define the normal-

(15)

The value

to keep the

ferent

consider a Rayleigh fading channel, so that the entries of

independent, complex, zero-mean, Gaussian random variables

with independent real and imaginary parts each having variance

1/2 [42, Sec. 4]. For a quasi-static channel, the matrix

mains unchanged over long time intervals, while for an ergodic

channel,

changes for every symbol . We consider only the

ergodic model whose capacity is (see [42] and [43])

appears in (15) by convention; this is done simply

at capacity close to each other for dif-

. We assume thatis known to the receiver only, and

are

re-

(16)

where

gate transpose of

tributed . We will consider only QPSK for simplicity, but the

design procedure described below is the same for other modu-

lation sets and mappings (see [34]).

is the identity matrix and

. One achieves capacity with Gaussian dis-

is the complex-conju-

B. EXIT Curve of the MIMO Detector

We proceed by showing how to compute EXIT curves for

the MIMO detector. We then describe a combined demodula-

tion/decoding structure that is flexible enough to closely ap-

proach MIMO capacity. This structure automatically specifies

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674IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

Fig. 6.MIMO detector EXIT curves at ? ?? ? ? dB and ? ? ???.

how the encoding/modulation is done, i.e., one maps the coded

bits directly onto the modulation signal set. Finally, we show

how to match EXIT curves.

The MIMO detector performs APP detection by considering

all

possible hypotheses on

detector computes the

-values

(see [16]–[20]). That is, the

(17)

for

onto the vector symbol

tector EXIT curve cannot be described in closed form, so we

measure it by Monte Carlo simulation [20]. We denote the de-

tector by DET and its EXIT curve by

, where

whose channel output is

is the th coded bit mapped

. The de-

(18)

Fig. 6 shows simulated EXIT curves of MIMO detectors for

Gray-mapped QPSK, and different numbers of transmit and

receive antennas. Observe that most of the curves resemble

straight lines. The 1

1 curve is, in fact, the horizontal line

(19)

where

random variable, and where we have used (5). We remark that

any 1

curve is a horizontal line. Furthermore, the

curves meet at

, and the curves decrease with

when. Similarly, one can show analytically that the

curves meet the 1

they decrease with

for fixed

is a zero-mean, unit-variance, complex Gaussian

1

curve at, and that

and.

C. EXIT Curve of the Combined MIMO Detector and VND

We combinetheMIMO detectorand theLDPC variablenode

decoder as shown in Fig. 5. The detector consists of

individual detectors (or detector nodes) that are each connected

to

variable nodes. We further choose allvariable

nodes connected to a common detector node to have the same

degree

. This restriction is not necessary, but it simplifies the

design by focusing on a smaller set of EXIT curves.

The decoder structure specifies that the LDPC coded bits be

mapped directly onto the modulation set. A similar technique

has been considered by others, e.g., [26] and [27], but without

the structured combination of variable and detector nodes, and

without the EXIT curve matching approach to code design. We

show that curve matching gives coded modulations that closely

approach capacity.

Consider the three boxes in Fig. 5 corresponding to one of

the

individual detector/VNDs. We denote the EXIT

curve of this structure by

priori

-values as the output

whose inputs are the interleaver bits transmitted using BPSK.

Using (4), the lower VND in Fig. 5 maps

. We again model the a

-values of an AWGN channel

into

(20)

We next approximate the detector curve (18) by a third-order

polynomial. For example, the EXIT curve of the 4

detector with QPSK at

approximated as

1 MIMO

is well dB and

dB

(21)

The polynomial (21) lets us express the combined de-

tector/VND curve in closed form, which is convenient for curve

fitting.

The third step is to consider the upper VND in Fig. 5. We

use (4) with

detector/VND EXIT curve as

and write the combined

(22)

Finally, inserting (18), (20), and the detector polynomial (21)

into (22), we obtain the desired transfer curve in the form

(23)

Fig. 7 shows some combined detector and VND transfer curves.

Observe that by setting

detector transfer curves of Fig. 6.

in (23) we recover the “pure”

D. Design Examples

We design check-regular LDPC codes by matching the curve

(23)toaCNDcurve(9).Weagainrestrictourselvestojustthree

different variable node degrees (

fit manually. Table I shows the parameters of our curve fitting

for several MIMO channels, and Fig. 8 plots the resulting EXIT

chart for a 4

1 MIMO channel.

Fig.9andTableIgivesimulationresultsforLDPCcodeswith

Gray-mapped QPSK,

,

leaver, and 100 decoder iterations. All schemes operate within

1.25dBoftheirrespectivecapacitylimits(duetothecomplexity

of the APP processing, the number of simulated blocks for the

) and perform the curve

, a random edge inter-

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TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION675

Fig. 7.

and ? ? ???.

Combined 4?1-detector and VND EXIT curves at ? ??

? ? dB

TABLE I

LDPC CODE PARAMETERS FOR ? ? ? MIMO CHANNELS, QPSK, AND

? ? ???. OVERALL RATE: ? BITS/CHANNEL USE

4

pacity can be narrowed further by using

fits, and by using longer codes and more iterations.

Fig.9alsoplotstheBERcurvesofBICMemployingaUMTS

standard turbo code, denoted PCC for “parallel concatenated

code.”ThePCChasmemorythreeconstituentcodes,withfeed-

cases was limited to 40). We expect that the gaps to ca-

and better curve

Fig. 8. Curve fit for the 4?1 scheme at ? ?? ? ? dB and ? ? ???.

Fig. 9.

concatenated codes or PCC). All schemes have an overall rate of ? bits per

channel use.

BERs for LDPC codes (LDPCC) and turbo codes (parallel

back and feedforward generator polynomials

,respectively.ThemodulationisagainGray-mapped

QPSK and the code has

terleaver. The iterative processing is done by performing 20 in-

ternaldecoder iterationsper detector/decoderiteration, and four

detector/decoder iterations. More internal or detector/decoder

iterations hardly improve the BER.

Observe that the LDPC scheme outperforms the turbo-coded

scheme by wide margins for

for 4

1 MIMO). This happens because the turbo code EXIT

curve is almost a horizontal line, as will be the case for any

strong code, for the AWGN channel. The code EXIT curve is,

therefore, poorly matched to a steep detector EXIT curve [20].

and

,, and a random in-

(the gain is over 5 dB

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676IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

E. Fading Model and Finite Modulation Sets

We discuss two practical issues. First, the ergodic model is

suitable if there appear many independent realizations of

and the receiver can accurately estimate these. This occurs in

practice when there is block fading (such as with time-divi-

sionmultipleaccess,orwithmulticarriermodulationemploying

time/frequency interleaving), and one can afford to code over

many blocks. However, in some cases, the delay constraints are

too severe to code over many blocks, and the quasi-static model

is appropriate [42, Sec. 5]. For such cases, one can still apply

curve fitting, but one now shapes the EXIT curves to minimize

outage probability. The question of how to do this precisely for

given fading statistics is an interesting open problem.

Second, the capacities of both the ergodic and quasi-static

models are achieved with Gaussian input distributions. How-

ever, in practice, one does not use such distributions. Some rea-

sons for this are the transmitter and receiver amplifiers have

peak energy constraints, the amplifiers are linear over limited

ranges of inputs, the receiver must acquire and maintain phase

synchronization, and the APP detector complexity grows expo-

nentially with the number of bits per input symbol. These rea-

sons make small modulation sets such as QPSK attractive for

wirelesscommunication. Ofcourse,thecapacities of suchmod-

ulation sets are smaller than for Gaussian inputs. We have com-

pared the performance of our codes with the finite modulation

set capacities, which is the fairest benchmark.

,

F. Comparison With STBCs

STBCs are attractive because they improve diversity and

are easy to encode and decode (see [13]–[15] and references

therein). However, the gains come at the expense of spectral

efficiency. For example,consider the 2

QPSK and a rate-1/2 LDPC code. The spectral efficiency is

two bits per channel use. Suppose next that we use Alamouti’s

space–time code [13] as an inner code. This scheme obtains a

diversity gain with what is, in effect, a rate-1/2 repetition code.

Thus, to achieve two bits per channel use with a rate-1/2 outer

code, one must compensate for the inner-code rate loss by

changing the constellation from QPSK to 16-quadrature ampli-

tude modulation (QAM). But recall that 16-QAM complicates

amplification, synchronization, and detection.

These issues will be even more pronounced for 4

channels. For example, the scheme proposed here achieves a

spectral efficiency of four bits per channel use with QPSK and

arate-1/2LDPCcode.AcorrespondingSTBC[14],whichis,in

effect, a rate-1/4 repetition code, would have to use 256-QAM

to achieve four bits per channel use with a rate-1/2 outer code.

Furthermore, no rate-1/4 orthogonal STBC exists for the 4

MIMO channel [14]. These considerations suggest that diver-

sity is best achieved by designing the coding and modulation

together rather than separately.

1 MIMO channel with

1 MIMO

1

IV. SUMMARY

We described a method for combining an irregular LDPC

code with a modulator and detector. We further introduced a

pragmatic technique for designing LDPC degree distributions

that are well matched to detectors. The design was based on

curve fitting on EXIT charts. For MIMO communication, sim-

ulations verified that all our codes operate within 1.25 dB of

their respective capacity limits at a BER of

that EXIT curve fitting can be used to construct equally good

coded modulations with RA codes [34], [35].

Finally, the goal of this paper was to present a coding and

modulation technique, as well as a design methodology, that

gives good performance for many communication systems. We

were less ambitious with our code designs. The LDPC degree

distributions listed in Table I can almost certainly be improved

upon, especially for the 4

1 MIMO channel. Other open prob-

lems are EXIT curve fitting for quasi-static channels, EXIT

curve fitting for suboptimal detectors, e.g., the detectors of sev-

eral papers in [44], and improving the EXIT approximations.

. We remark

APPENDIX

Consider

is zero-mean, Gaussian noise with variance

-value of (2) is a function of , and we write this as

Note that

conditioned on

mean

and variance

, where and

. The channel

.

is Gaussian with

. We thus have

Letbe the mutual information. We have

(24)

where

tropy of

the same as

is, therefore,

For computer implementation, we split

corresponding to the intervals

. We used a polynomial fit for the left interval and

an exponential fit for the right interval. We applied the Mar-

quardt–Levenberg algorithm (see [45]) to obtain

is the entropy of

conditioned on

. The capacity of our

and

. Note that

is the en-

is

channel

.

into two parts,

andwhere

where

For the inverse

tervals at

-function we split the curve into two in-

. Our approximation is

where

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TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION677

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for publication.

Stephan ten Brink received the Dipl.-Ing. and the

Dr.-Ing. degrees in electrical engineering from the

University of Stuttgart, Stuttgart, Germany, in 1997

and 2000, respectively.

From 2000 to June 2003, he was with the Wireless

Research Laboratory, Bell Laboratories, Lucent

Technologies, Holmdel, NJ, conducting research

on channel coding for multiple-antenna systems.

Since July 2003, he has been with Realtek, Irvine,

CA, where he is involved in the development and

standardization of high-throughput WLAN systems.

His research interests include error-correcting coding, iterative decoding,

multiple-antenna communications, and watermarking.

Page 9

678IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

Gerhard Kramer (S’91–M’94) received the B.Sc.

and M.Sc. degrees in electrical engineering from

the University of Manitoba, Winnipeg, MB, Canada,

in 1991 and 1992, respectively, and the Dr.Sc.

Techn. degree from the Swiss Federal Institute of

Technology (ETH), Zürich, Switzerland, in 1998.

FromAugust1998toMarch2000,hewaswithEn-

dora Tech AG, Basel, Switzerland, as a communica-

tions engineeringconsultant. SinceMay2000, hehas

been with the Mathematics of Communications Re-

search Department, Bell Laboratories, Lucent Tech-

nologies, Murray Hill, NJ.

Alexei Ashikhmin(M’00) received the Ph.D. degree

in electrical engineering from the Institute for Infor-

mationTransmission Problems, Russian Academyof

Science, Moscow, Russia, in 1994.

From September 1995 to September 1996, he was

with the Mathematics Department, Delft University

of Technology, Delft, The Netherlands. From Jan-

uary 1997 to July 1999, he was a Postdoctoral Fellow

at the Computer, Information, and Communication

Division of Los Alamos National Laboratory, Los

Alamos, NM. Since 1999, he has been with the

Mathematics of Communications Research Department, Bell Laboratories, Lu-

cent Technologies, Murray Hill, NJ. His research interests include information

and communication theory, with emphasis on error-correcting codes.

Dr. Ashikhmin currently serves as an Associate Editor for Coding Theory for

the IEEE TRANSACTIONS ON INFORMATION THEORY.