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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 293

Transactions Papers

Performance Assessment of DC-Free

Multimode Codes

Kees A. Schouhamer Immink, Fellow, IEEE, and Levente P´

atrovics

Abstract—We report on a class of high-rate dc-free codes, called

multimode codes, where each source word can be represented by

a codeword taken from a selection set of codeword alternatives.

Conventional multimode codes will be analyzed using a simple

mathematical model. The criterion used to select the “best”

codeword from the selection set available has a signiﬁcant bearing

on the performance. Various selection criteria are introduced

and their effect on the performance of multimode codes will be

examined.

I. INTRODUCTION

BINARY sequences with spectral nulls at zero frequency

have found widespread application in optical and mag-

netic recording systems. The dc-balanced or dc-free codes, as

they are often called, have a long history and their application

is certainly not conﬁned to recording practice. Since the early

days of digital communication over cable, dc-balanced codes

have been employed to counter the effects of low-frequency

cut-off due to coupling components, isolating transformers,

etc. In optical recording, dc-balanced codes are employed to

circumvent or reduce interaction between the data written on

the disc and the servo systems that follow the track [1]. In

the literature, code implementations have been concentrated

on byte-oriented dc-free codes of rate 8/10 or 8/9. (see, for

example, [2]–[4]). For certain application it is desirable that

the code rate is much higher than 8/9. The construction of

such high-rate codes is far from obvious, as table look-up

for encoding and decoding is an engineering impracticality.

Two methods for high-rate code design have been described

in the literature [5], [6]. Both methods utilize the idea that the

correspondence between source words and the codewords is

as simple as possible. A serious drawback of both methods

is that the performance, in terms of suppression of low-

frequency components, is far from what could be obtained

according to the tenets of information theory [1], [7], but up

till now attempts to improve the performance failed. Recently,

however, the publications by Fair et al. on “guided scrambling”

Paper approved by E. Ayanoglu, the editor for Communication Theory

and Coding Application of the IEEE Communications Society. Manuscript

received May 7, 1996; revised August 2, 1996. This paper was presented

in part at the IEEE 3rd Symposium on Communications and Vehicular

Technology, Eindenhoven, The Netherlands, October 25–26, 1995.

The authors are with the Philips Research Laboratories, 5656 AA Eind-

hoven, The Netherlands.

Publisher Item Identiﬁer: S 0090-6778(97)01970-3.

stimulated us to investigate the performance of their method

and its varieties. In our context, guided scrambling is a

member of a larger class of related coding schemes called

multimode code. In multimode codes, each source word can

be represented by a member of a selection set consisting of

codewords. The encoder opts for transmitting that codeword

that minimizes, according to a criterion to be deﬁned, the low-

frequency spectral contents of the encoded sequence. There are

two key elements which need to be chosen judiciously: 1) the

mapping between the source words and their corresponding

selection sets and 2) the criterion used to select the “best”

word. The spectral performance of the code greatly depends

on both issues. We start with some preliminaries, followed by

a section providing the state of the art. Thereafter, we will

outline the new multimode schemes and analyze their spectral

performance.

II. PRELIMINARIES

The running digital sum of a sequence (RDS) plays a

signiﬁcant role in the analysis and synthesis of codes whose

spectrum vanishes at the low-frequency end. Let

be a bipolar se-

quence. Note that in the sequel, we will denote the value of

by its logical equivalents ‘0’ or “1.” The (running) digital

sum is deﬁned as

It is an elementary exercise to show that if is bounded, the

spectral density vanishes at zero frequency [1]. The number

of RDS values that the sequence assumes is often called the

digital sum variation and denoted by . The value of

should be as small as possible as it has a direct bearing on

the amount of power at the low-frequency end. Given the

parameter it is possible to compute the maximum value

of the rate, , of any code, irrespective of its complexity,

that translates arbitrary source input into sequences obeying

the given constraint. Results of computation, taken from [1],

are listed in Table I. It can be seen that the sum constraint is not

very expensive in terms of rate loss when is relatively large.

For instance, a sequence that takes at maximum sum

values has a capacity , which implies a rate loss

0090–6778/97$10.00 1997 IEEE

294 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

TABLE I

CAPACITY AND SUM VARIANCE OF MAXENTROPIC SEQUENCES TAKING AT

MOST RDS VALUES VERSUS DIGITAL SUM VARIATION

of less than 6%. The quantity called sum variance plays

an important role in the evaluation of the spectral properties

of a code. Before explaining the relevance of the parameter

, a few words are in order regarding the low-frequency

properties of dc-free codes.

If is ﬁnite, the spectral density is zero at zero frequency,

but it is more relevant to note that there is a region of

frequencies, close to the zero frequency, where the spectral

density is low. The width of this region, termed the notch

width, is of great engineering relevance. The width of the

spectral notch, can be quantiﬁed by a parameter called the

cutoff frequency,. According to the work by Justesen [8]

there is a simple approximate relationship between and the

sum variance, , of the encoded sequence, namely

(1)

Sequences of maximum entropy assuming at most RDS

values obey the following fundamental relationship between

the sum variance and the redundancy [1]

(2)

The above relationship can be employed to derive a simple

yard stick for measuring the performance of implemented

codes. The encoder efﬁciency is deﬁned as

(3)

The encoder efﬁciency , as deﬁned in (3), compares the

“redundancy-sum variance products” of the implemented code

and the maxentropic sequence with the same digital sum

variation as the implemented code. The efﬁciency will be

used in the sequel to measure the performance of dc-free codes.

III. PRIOR ART

Essentially, there are three basic methods for generating dc-

free sequences which are relevant to the ensuing discussion.

These methods are brieﬂy reviewed below.

A. Monomode Codes

In monomode codes, there is a one-to-one relationship

between source words and codewords. By necessity, the code-

words have equal numbers of 1’s and 1’s. There are two

methods available for translating source words into codewords.

The ﬁrst method uses an algebraic technique, called enumera-

tion [1], and in the second method, devised by Knuth [6], -bit

source words are translated into -bit codewords. The

translation is achieved by selecting a bit position within the

-bit word which deﬁnes two segments, each having one half

of the total disparity of the -bit word, where the disparity of

a codeword is deﬁned as the difference between the numbers

of 1’s and 1’s in that codeword. A zero-disparity codeword,

i.e., a codeword with an equal number of 1’s and 1’s, is now

generated by the inversion of all the bits within one segment.

The position information which deﬁnes the two segments is

encoded in the bits.

B. Bimode Codes

Bimode codes ensure balanced transmission by providing

for each source word two alternative channel representations.

From the alternatives available, that codeword is transmitted

that minimizes the absolute value of the RDS after transmis-

sion of the new word. This selection criterion will be termed

MRDS selection criterion. An archetypical example of a bi-

mode code is the polarity switch code [5]. The encoder and

decoder circuits of the polarity switch code are very simple

as no look-up tables are required. Under polarity switch rules,

source symbols are supplemented by one symbol called

the polarity bit. The encoder has the option to transmit the -

bit words without modiﬁcation or to invert all symbols. The

choice of a speciﬁc translation is made in such a way that

the running digital sum after transmission of the new word is

as close to zero as possible. The polarity bit is used at the

decoder site to identify whether the transmitted codeword has

been inverted or not, and can easily be reconstituted. Properties

of the polarity bit code have been described in [1]. The

performance of the polarity switch code can be summarized

as follows. The rate of the polarity bit code is

The sum variance of the code [1] is

so that the efﬁciency is

(4)

From the above, we conclude that polarity switch codes are a

far cry from the optimal situation.

It is not difﬁcult to generalize the above principle of

bimode codes to multimode codes, which, as the name already

suggests, cater for more than two channel representations.

C. Multimode Codes

In multimode codes, each source word can be represented

by a member of a selection set, denoted by , consisting of

codewords. The MRDS selection criterion can be used to select

the “best” codeword. More sophisticated selection criteria will

be described in Section V. It should be appreciated that the

usage of multimode codes is not conﬁned to the generation

of dc-free sequences. Provided that is large enough and

the selection sets contain sufﬁciently different codewords,

IMMINK AND P ´

ATROVICS: DC-FREE MULTIMODE CODES 295

multimode codes can also be used to satisfy almost any

channel constraint with a suitably chosen selection method.

A basic element of multimode codes is the one-to- in-

vertible mapping between the source and its selection set

. Examples of such mappings are the guided scrambling

algorithm presented by Fair et al. [9], the dc-free coset

codes of Deng and Herro [10], and the scrambling using a

Reed–Solomon code by Kunisa et al. [11]. In our context, a

mapping is considered to be “good” if the sets contain suf-

ﬁciently distinct codewords. The guided scrambling algorithm

is brieﬂy described below.

1) Guided Scrambling: The guided scrambling algorithm

uses selection sets of size , where is the number

of redundant bits. Guided scrambling is summarized below.

1) In the ﬁrst step, called augmenting, the source word is

preceded by all the possible binary sequences of length

to produce the set . Hence

2) The selection set is obtained by

scrambling all vectors in . Let the scrambler poly-

nomial be denoted by

where denotes the register length of the scrambler. The

scrambler translates each vector

into using the recursion

(5)

3) The “best” codeword in is selected for transmission.

4) At the receiver’s site, the inverse operation

is

The source word is found by deleting the ﬁrst bits.

In the guided scrambling algorithm described above, trans-

lation of source words into random-like channel represen-

tations is done in a fairly simple way. This basic algorithm

is, however, prone to worst case situations since there is a

probability that consecutive source words have representation

sets whose members all have the same polarity of the disparity.

In this vexatious situation, the RDS cannot be controlled,

and long-term low-frequency components can build up. This

ﬂaw can be solved by a construction where each selection set

consists of pairs of words of opposite disparity. As a result,

there is always a codeword in the selection set that can control

the RDS. A simple method embodying this idea combines the

features of guided scrambling and the polarity bit code. The

improved algorithm using redundant bits is executed in

six steps. In Steps 1), 2), and 5) the original guided scrambling

principle is executed while Steps 3) and 4) embody the polarity

bit code.

1) The source word is preceded by all the possible binary

sequences of length to produce the

elements of the set . Hence:

2) The selection set is obtained by

scrambling all vectors in .

3) By preceding the vectors in with both a ‘one’ and a

“zero,” we get the set , with elements.

4) The selection set is obtained by scrambling (pre-

coding) the vectors in using the scrambler with

polynomial . This embodies the polarity bit prin-

ciple.

5) The “best” codeword in is selected.

6) At the receiver end, the codeword is ﬁrst descrambled

using the polynomial, then after removing the ﬁrst

bit, it is descrambled. The original source word is

eventually reconstituted by removing the ﬁrst

bits.

All simulations and analyses discussed below assume the

above structure where the selection set consists of pairs of

words of opposite disparity.

IV. ANALYSIS OF MULTI-MODE CODES

A precise mathematical analysis of the performance of

multimode codes is, considering the complexity of the code,

out of the question. We can either rely on computer simulation

to facilitate an understanding of the operation of the coding

system or try to deﬁne a simple mathematical model, which

embraces the essential characteristics of the code and is also

analytically tractable. We followed both approaches, and we

commence by describing the mathematical model.

A. The Random Drawing Model

The key characteristic of a multimode code is that each

source word can be represented by a codeword taken from a set

containing “random” alternatives. As the precise structure of

the encoder is extremely difﬁcult to analyze, we assume, in our

mathematical model, that for each source block the channel

set is obtained by randomly drawing -bit words plus

their complementary -bit words. The precise structure

of the scrambler is ignored in our model. The “best” word in

the set, according to the MRDS criterion, is transmitted. The

MRDS criterion ensures that the state space of the encoder,

that is, the number of possible word-end running digital sum

(WRDS) values the encoded sequence may take, is ﬁnite.

However, if the codewords are relatively long, the number

of states and the resulting transition matrix are still too large

for a simple mathematical analysis. We therefore truncated

the state space by omitting those states that do not contribute

signiﬁcantly to the sum variance.

296 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

B. Transition Probabilities of the Finite-State Machine

The implemented encoder schemes can be simply treated in

terms of Markov models. The set of values that WRDS take

prior to transmission of a codeword deﬁnes a set of states of a

ﬁnite-state machine. We will use the shorthand notation

to denote both the WRDS at the start of the th codeword and

to refer to the encoder state itself. We commence our analysis

with a computation of the state transition probabilities.

Assume the th codeword starts with RDS . Then

the multimode code can be cast into a Markov chain model

whose state transition probabilities matrix, , is given by

We make the following remarks concerning the state transition

matrix.

1) For the sake of simplicity, only codes using codewords

of even length are considered.

2) It is assumed that at the start of the transmission WRDS

is set to . As a result, since the codeword length is

even, .

3) For reasons of symmetry, only the probabilities for

need to be calculated.

4) To reduce the computational load, we truncated the state

space. Only those states are considered that can be

reached from the , or the , state with

probability greater than , where is chosen suitably

small, say 10 6. Other values of have been tried

without, however, causing signiﬁcant differences in the

results obtained. The remaining states will be termed

principal states.

We now introduce several notations. If WRDS is positive,

then, according to the simple MRDS criterion, the next code-

word will be of zero or negative disparity. Therefore, assuming

that the encoder occupies state , the set of possible next

states is . Let denote the

probability of a codeword pair having disparity and .

The probability of the next-state candidate in a draw being

is

if

otherwise. (6)

The next-state candidate in the th draw is denoted by

. According to the MRDS criterion, if the next state

is , then for all . The probability that during

a draw the next-state candidate is “worse” than , denoted

by , is given by

Now, the expression for the transition matrix is given by

(7)

The transition probabilities for each pair of WRDS states

can be numerically determined by invoking (7). In order to

make the analysis more tractable, those states are removed

that can be reached from the , or the , state

Fig. 1. Efﬁciency of random drawing algorithm using the MRDS selection

criterion.

with probability less than . The remaining set of states, the

principal states, denoted by

, and the truncated transition probability matrix with

elements can easily be found. Thereafter, vector

of the stationary probabilities, with elements ,is

found by solving . The calculation of the variance

of the digital sum at the start of the codewords is now

straightforward. The computation of the sum variance within

the codewords is more complex, and therefore given in the

Appendix.

C. Computational Results

Using (13), we calculated the efﬁciency of the random

drawing algorithm for selected values of the codeword length

and redundancy. Fig. 1 shows the results. The connected points

have the same redundancy , and the th point on a

curve corresponds to a code having redundant bits, codeword

length , and selection sets of size . For comparison

purposes, we also plotted the efﬁciency of the polarity bit

code [see (4)]. By comparing the efﬁciency values at the

th point on each curve, we can see that these values are

approximately the same. The efﬁciency of the random coding

algorithm is practically independent of the codeword length

and is essentially determined by the number of redundant bits

used. It can be seen that codes with two or three redundant bits

are clearly more efﬁcient than the polarity bit code. With an

increasing number of redundant bits, however, the efﬁciency

decreases. The decrease in performance, as will be explained

in the next section, is due to the shortcomings of the MRDS

criterion.

V. ALTERNATIVE SELECTION CRITERIA

The results, plotted in Fig. 1, reveal that using more than

two redundant bits does not lead to improved performance. The

reason that performance decreases with an increasing number

of redundant bits can easily be understood. A quick calculation

will make it clear that a large selection set contains with great

probability at least one zero-disparity word. On the basis of

IMMINK AND P ´

ATROVICS: DC-FREE MULTIMODE CODES 297

Fig. 2. Simulation results for the random drawing algorithm with ﬁxed

redundancy 1/128 with different selection criteria (a) MRDS, (b) MMRDS,

and (c) MSW.

the simple MRDS criterion one of the zero-disparity words

is randomly chosen and transmitted. As the sum variance of

zero-disparity codewords equals , [1] irrespective

of the rate of the code, we conclude that the efﬁciency will

asymptotically approach zero. More sophisticated selection

criteria, which take account of the running digital sum within

the codeword, and not only at the end of the word, may

result in increased performance. In order to describe these

more sophisticated selection criteria, we introduce the squared

weight,, of a codeword, deﬁned as the sum of the squared

RDS values at each bit position of the codeword. The two

selection criteria examined are as follows:

1) modiﬁed MRDS (MMRDS) criterion: from the code-

words with minimal WRDS , the one with minimum

is selected;

2) minimum squared weight (MSW) criterion: the code-

word of minimal is selected from the selection set,

irrespective of the WRDS of the codeword.

Fig. 2 shows the simulation results obtained for redundancy

1/128. Simulations of codes with other values of the redun-

dancy produced similar results. From the curves, we infer the

following.

• The MRDS method wastes the opportunity offered by the

broader selection sets. By properly selecting the codeword

from the ones with minimal WRDS , the efﬁciency of the

MMRDS scheme tends to unity.

• As indicated by the curve of the MSW criterion, the

best codewords do not necessarily minimize the WRDS .

Selecting the codeword with minimal squared weight

clearly results in more efﬁcient codes.

Based on the above observations, we searched for a criterion

that is simple to implement while its efﬁciency approaches that

of the MSW criterion. The outcome is described in the next

section.

A. The Minimum Threshold Overrun Criterion

Our objective, in this section, is to construct a selection

criterion which takes into account the RDS values within

Fig. 3. Simulation results for the random drawing algorithm having ﬁxed

redundancy 1/128 with (a) the MSW criterion and (b) the MTO criterion. The

dotted line shows the results obtained for the implemented encoding scheme

using a scramblers with polynomial .

the codeword while having a structure that is also easy to

implement. The proposed selection scheme, termed minimum

threshold overrun (MTO) criterion, utilizes the parameter

“RDS threshold,” denoted by . The MTO penalty

is simply the number of times the absolute value of the

running digital sum within a word is larger than . As the

squaring operation needed for the MSW criterion is avoided,

the implementation of the MTO criterion is not more complex

than the MRDS method. The codeword with minimum penalty

is transmitted. If two or more codewords have the same

penalty, one of them is chosen randomly and transmitted.

This procedure does not seriously deteriorate the performance

as it is fairly improbable that two or more codewords in the

selection set have the same penalty value. Fig. 3, curve (b),

shows simulation results obtained with the MTO criterion.

Optimal values of the threshold were found by trial and

error. We can see that the MTO criterion is only slightly less

efﬁcient than the MSW criterion. All results shown so far have

been obtained by a simulation program of the random drawing

algorithm. As a ﬁnal check we also conducted simulations

with a full-ﬂedged implementation using a scramblers with

polynomial . Experiments with other scrambler

polynomials did not reveal signiﬁcant differences. The dotted

curve, Fig. 3, gives results on the basis of the MTO criterion.

The curve shows a nice agreement with results obtained with

the random drawing algorithm. As the proof of the pudding

we have computed the power spectral density (PSD) of two

typical examples. The results are displayed in Fig. 4.

VI. CONCLUSION

Multimode codes have been mathematically analyzed by

introducing a simple random drawing model. We have pre-

sented alternative selection criteria and examined their effect

on the spectral efﬁciency. Multimode codes are excellent

candidate dc-free codes when both low-spectral content at

the low-frequency end and high rate are at a premium. For

given rate and proper selection criteria, the spectral content of

multimode codes is very close to the minimal content promised

298 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

Fig. 4. Spectra of encoded sequences generated by the polarity switch code (upper curve) and multimode code (lower curve). The redundancy is in both

cases 1/128. The multimode code has six redundant bits (codeword length is ) and it uses the MTO selection criterion.

by information theory.

APPENDIX

In this Appendix, we will compute the sum variance of se-

quences encoded with the random drawing model. A codeword

with binary elements is translated into the -tuple

where .

Suppose the th codeword in the sequence,

, starts with initial RDS . The RDS at

the th symbol position of , denoted by , equals

The running sum variance at the th position, given , equals

where the operator averages over all codewords

that start with an initial RDS . As the source population

of codewords is the full set of vectors of nonpositive (nonneg-

ative) disparity, the expectations and

, are independent of the symbol positions

and . For the sake of convenience, we use the shorthand

notation and

. Substitution yields the running sum variance at

the th symbol position

(8)

The sum variance of a codeword starting with initial RDS ,

designated by , is found by averaging the running digital

sum variance over all symbol positions of the codeword or

The probability that a codeword starts with an RDS

equals the stationary probability , so that by taking

the probability into account that a codeword starts with RDS

and averaging over all initial states in , the following

expression is found for the sum variance :

(9)

The variance of the initial sum values, , equals

The quantity can be estimated by noting the periodicity, i.e.,

. Evaluating (8) yields

and after averaging, where the the probability of starting with

an initial RDS is taken into account, we obtain

so that with we ﬁnd

Substitution in (9) yields

(10)

1) Computation of the Correlation: We next calculate the

correlation of the symbols at the th and

the th symbol position within the same codeword. It is

IMMINK AND P ´

ATROVICS: DC-FREE MULTIMODE CODES 299

obvious that .If , some more work

is needed. In that case,

(11)

Assume a codeword to be of disparity . Then the prob-

ability that a symbol at position in the codeword equals

1is

The probability that another symbol at position

within the same codeword equals 1is

Hence

and (11) yields the correlation for codewords of disparity

Using the above, we ﬁnd that

(12)

The sum variance can be determined using (10) and (12)

(13)

ACKNOWLEDGMENT

The authors wish to thank I. Fair for his valuable remarks

that helped improve the contents of this paper.

REFERENCES

[1] K. A. S. Immink, Coding Techniques for Digital Recorders. Engle-

wood Cliffs, NJ: Prentice-Hall International, 1991.

[2] S. Fukuda, Y. Kojima, Y. Shimpuku, and K. Odaka, “8/10 modulation

codes for digital magnetic recording,” IEEE Trans. Magn., vol. MAG-22,

pp. 1194–1196, Sept. 1986.

[3] A. X. Widmer and P. A. Franaszek, “A dc-balanced, partitioned-block,

8b/10b transmission code,” IBM J. Res. Develop., vol. 27, no. 5, pp.

440–451, Sept. 1983.

[4] H. Yoshida, T. Shimada, and Y. Hashimoto, “8-9 block code: A dc-free

channel code for digital magnetic recording,” SMPTE J., vol. 92, pp.

918–922, Sept. 1983.

[5] F. K. Bowers, U.S. Patent 2 957947, 1960.

[6] D. E. Knuth, “Efﬁcient balanced codes,” IEEE Trans. Inform. Theory,

vol. IT-32, pp. 51–53, Jan. 1986. See also P. S. Henry, “Zero disparity

coding system,” U.S. Patent 4 309694, Jan. 1982.

[7] H. Hollmann and K. A. S. Immink, “Performance of efﬁcient balanced

codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 913–918, May 1991.

[8] J. Justesen, “Information rates and power spectra of digital codes,” IEEE

Trans. Inform. Theory, vol. IT-28, pp. 457–472, May 1982.

[9] I. J. Fair, W. D. Gover, W. A. Krzymien, and R. I. MacDonald, “Guided

scrambling: A new line coding technique for high bit rate ﬁber optic

transmission systems,” IEEE Trans. Commun., vol. 39, pp. 289–297,

Feb. 1991.

[10] R. H. Deng and M. A. Herro, “DC-free coset codes,” IEEE Trans.

Inform. Theory, vol. 34, pp. 786–792, July 1988.

[11] A. Kunisa, S. Takahashi, and N. Itoh, “Digital modulation method for

recordable digital video disc,” in Proc. 1996 IEEE Int. Conf. Consumer

Electron., June 1996, pp. 418–419.

Kees A. Schouhamer Immink (M’81–SM’86–

F’90) received the M.S. and Ph.D. degrees from the

Eindhoven University of Technology, Eindenhoven,

The Netherlands.

He joined the Philips Research Laboratories,

Eindhoven, in 1968, where he currently holds the

position of Research Fellow. He has contributed

to the design and development of a wide variety

of digital consumer-type audio and video-recorders

such as the compact disc, compact disc video, R-

DAT, DCC, and DVD. He holds 32 U.S. patents

and has written numerous papers in the ﬁeld of coding techniques for optical

and magnetic recorders.

Dr. Immink is the Chairman of the IEEE Benelux Chapter on Consumer

Electronics, a Governor of the Audio Engineering Society (AES), and a

Governor of the IEEE Information Theory Society. He was named a Fellow

of the AES, SMPTE, and IEE. Furthermore, he was recognized and awarded

with the AES Silver Medal in 1992, the IEE Sir Thomson Medal in 1993,

the SMPTE Poniatoff Gold Medal in 1994, and the IEEE Ibuka Consumer

Electronics Award in 1996. He is a member of the Royal Netherlands

Academy of Arts and Sciences.

Levente P´

atrovics was born in Budapest, Hungary.

He received the M.Sc. degree in electrical engi-

neering and computer science in 1994 from the

Technical University of Budapest, Hungary. From

May to December 1995, he was working towards

the Ph.D. degree at Philips Research Laboratories,

Eindhoven, The Netherlands.

His research interests are in information theory,

particular construction, and analysis of constrained

codes. Currently, he is with Lufthansa Systems,

Hungaria Kft.