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An enumerative coding technique for DC-free runlength-limited seqences

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We present an enumerative technique for encoding and decoding DC-free runlength-limited sequences. This technique enables the encoding and decoding of sequences approaching the maxentropic performance bounds very closely in terms of the code rate and low-frequency suppression capability. Use of finite-precision floating-point notation to express the weight coefficients results in channel encoders and decoders of moderate complexity. For channel constraints of practical interest, the hardware required for implementing such a quasi-maxentropic coding scheme consists mainly of a ROM of at most 5 kB
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2024 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 12, DECEMBER 2000
An Enumerative Coding Technique for DC-Free
Runlength-Limited Sequences
Volker Braun, Member, IEEE, and Kees A. Schouhamer Immink, Fellow, IEEE
Abstract—We present an enumerative technique for encoding
and decoding dc-free runlength-limited sequences. This technique
enables the encoding and decoding of sequences approaching the
maxentropic performance bounds very closely in terms of code
rate and low-frequency suppression capability. Use of finite-pre-
cision floating-point notation to express the weight coefficients
results in channel encoders and decoders of moderate complexity.
For channel constraints of practical interest, the hardware re-
quired for implementing such a quasi-maxentropic coding scheme
consists mainly of a ROM of at most 5 kB.
Index Terms—Enumeration, modulation coding, recording.
I. INTRODUCTION
DC-FREE runlength-limited (DCRLL) modulation codes
have found application in magnetic and optical recording
systems, for example, in the compact disk (CD) player [1].
DCRLL codes are used to transform the digital user bit stream
into a sequence of binary channel symbols that is suitable for
the specific recording requirements. The runlength is known
as the number of consecutive identical symbols occurring in a
sequence. Runlength-limited (RLL) sequences are character-
ized by two parameters, and ,
denoting the minimum and maximum runlengths occurring
in the sequence. DC-free sequences have the properties that
their power spectral density (PSD) vanishes at the zero fre-
quency and that there is a region of frequencies close to the
zero frequency in which the PSD is low. The restrictions
imposed on the DCRLL sequences are usually called the
“channel constraints.” In optical disk systems, suppression of
the low-frequency content of the RLL modulation sequence
is employed primarily to circumvent or reduce interaction
between the data written on the disc and the servo systems that
follow the track [1]. Efficient suppression of the low-frequency
components is essential, as error correction is totally useless
if track or clock loss occurs [2]. Low-frequency suppression
should be achieved at minimum cost in code rate, i.e., the ratio
between user bit rate and channel bit rate. The maximum code
rate, called the capacity, is a function of the channel constraints
Paper approved by E. Ayanoglu, the Editor for Communication Theory and
Coding Application of the IEEE Communications Society. Manuscript received
May 1, 1998; revised October 18, 1999. This paper was presented in part at the
1997 Symposium on Information Theory and Its Applications (SITA’97), Mat-
suyama, Japan, December 2–5, 1997, and in part at the 1998 IEEE International
Symposium on Information Theory, Cambridge, MA, August 16–21, 1998.
V. Braun was with Hitachi Central Research Laboratory, Tokyo 185-8601,
Japan. He is now with Alcatel Corporate Research Center, 70499 Stuttgart, Ger-
many.
K. A. S. Immink is with the Digital Communications Group, Institute for
Experimental Mathematics, University of Essen, 45326 Essen, Germany.
Publisher Item Identifier S 0090-6778(00)10914-6.
in force. The quotient of rate and capacity is usually called the
rate efficiency. Implemented DCRLL codes, for example, the
EFM code [1] applied in the CD system or the codes designed
for magnetic disk or tape drive applications described in [3]
and [4], often achieve rate efficiencies in the order of only
90%. The EFMPlus code [2] applied in the digital versatile
disk (DVD) system has a rate efficiency of about 92.5%. In
other words, many of the conventionalDCRLL codes currently
implemented can still be improved significantly in terms of
code rate.
Enumerative coding techniques [5] make it possible to trans-
late source words into codewords and vice versa by invoking
an algorithmic procedure rather than performing the translation
with a look-up table [6]. Code rates very close to capacity can
be achieved by using enumerative coding and long codewords
[6]. Severe error propagation resulting from the use of long
codewords can be avoided by reversing the conventional hier-
archy of outer error correcting code and inner modulation code
[6]. Enumerative decoding is done by forming the weighted
sum of the symbols of the codeword received [7]. The integer-
valued weights used in forming this sum are a function of the
channel constraints in force. Encoding is done with the aid of a
method that is similar to decimal-to-binary conversion in which
the weights are used instead of the usual powers of 2. The hard-
ware which implements an enumerative coding scheme mainly
consists of a ROM to store the weight coefficients, a binary
adder, and subtracter. In order to obtain a feasible ROM size,
the weight coefficients can, without relevant losses in code rate,
be expressed in finite-precision floating-point notation [6].
The outline of the next sections is as follows. In Section II,
we will introduce an efficient runlength graph representation
of the DCRLL constraints. To enable the enumeration of the
DCRLL sequences, knowledge of the number of distinct valid
sequences is required. In Section III, we will derive these num-
bers from the underlying runlength graph. An enumerative tech-
nique for encoding and decoding DCRLL sequences will be in-
troduced in Section IV. In Section V, we will discuss the effects
of using finite-precision floating-point arithmetic on achievable
code rates and on the size of the required weight set. In addition,
the low-frequency suppression capabilities of selected enumer-
ative DCRLL codes will be evaluated in computer simulations.
II. PRELIMINARIES
We will assume that binary user information with a bit rate of
is translated into a coded channel sequence having
the channel bit rate , where denotes the rate
of the code. Let ,
0090–6778/00$10.00 © 2000 IEEE
BRAUN AND IMMINK: AN ENUMERATIVE CODING TECHNIQUE FOR DCRLL SEQUENCES 2025
denote the coded channel sequence, representing the positive or
negativemagnetizationoftherecordingmedium,or pits or lands
in the case of optical recording. The running digital sum (RDS)
at time of the sequence is defined as
(1)
A sequence is dc-free if, and only if, the RDS assumes a finite
number of values [8]. This number is called the digital sum vari-
ation(DSV), denoted by . Wewillstartbyconfiningourselves
to odd values of DSV, so . DCRLL constraints
will be characterized by the three integer parameters .
As a finite RDS implies a constraint on the maximum runlength,
these parameters satisfy .
RLL sequences are often encoded in two consecutive steps.
In the first step, the binary user information bits are translated
into a sequence , having at least and at
most “zeros” (i.e., ) between consecutive “ones” (i.e.,
). The sequence is often called a sequence
or a runlength-limited sequence in nonreturn-to-zero-inverse
(NRZI) format. Prior to the recording, the sequence
is converted into the bipolar RLL channel sequence such
that the logical “ones” in the sequence indicate the posi-
tions of a 1 1or 1 1 transition of the corresponding
RLL channel sequence. This conversion step is commonly
called precoding. The sequence
(2)
would be converted, e.g., into the RLL channel sequence
(3)
Another representation of the sequence can be given as
a sequence of runlengths , where
. We define a “run” in the sequence
as a logical “one” followed by a sequence of “zeros.” The
runlength sequence corresponding to the sequence in (2)
would be . Kerpez et al. [9] introduced
another sequence , defined by , where
. The sequence corresponding to the se-
quence (2) would be . Let denote the
RDS of the channel sequence after the th run in the corre-
sponding sequence . The sequence corresponding to
theRLL channel sequence (3) would be .
Between and , we find the relation , and
hence .As , we find
for all
(4)
i.e., assumes a finite number of values
(5)
A compact description of the constraints by means
of a “runlength graph” has been presented by Kerpez et al. [9].
The states of this graph are associated with values and the
edges with runlengths. As an example, Fig. 1 depicts the run-
length graph representing the constraint. Although not
formally defined, we will denote the runlength graph underlying
the DCRLL constraints by . The states of will be denoted
Fig. 1. Runlength graph representing the constraint. The states are
denoted by their values, and the edges are labeled by their lengths [9].
by their values, where in the sequel
. Note that the states of are related with RDS
values. The sequences (2) and (3), for example, may emanate
from state 0 and terminate in state 0 of the graph in Fig. 1.
Associated with the runlength graph is an ad-
jacency matrix denoted by ( -transform notation). The
adjacency matrix associated with the runlength graph turns
out to have a Hankel structure, i.e., it is constant on the antidi-
agonals [9]. As an example, we present the adjacency matrix
associated with the runlength graph in Fig. 1, given by
Note that is symmetric, and that the largest exponent,
denoted by , of the nonzero entries of , occurring
in the th row (or column) which corresponds to the state of
, is given by
III. NUMBER OF DCRLL SEQUENCES
The number of distinct constrained sequences can
be described using recursive relations obtained from the run-
length graph . Let denote the number of dis-
tinct sequences of length emanating from state and ter-
minating in state of the graph . The number of sequences
can be determined using the following recursive
relations. Let
For and for all let
(6)
where denotes the set of states of . Note that
as is symmetric (we call
symmetric if the adjacency matrix is symmetric), and
2026 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 12, DECEMBER 2000
if is odd. This property states
that if and are both even or both odd, then all sequences
emanating from state and terminating in state of have
an even length .If is even and odd, or vice versa, then
all sequences emanating from state and terminating in state
of have an odd length . Knowledge of the numbers
enables the application of enumerative coding
techniques.
IV. ENUMERATIVE ENCODING AND DECODING
A. Decoding
A general enumerative technique for encoding and decoding
binary constrained sequences has been presented by Cover [5].
Let denote the set of binary sequences of length and
let be any (constrained) subset of . We establish a 1-1
mapping from set onto the set of integers ,
where is the cardinality of (i.e., the number of distinct
sequences in ). Set can be ordered lexicographically as fol-
lows: if and ,
then is called less than , in short, , if there exists an
, such that and .
For example, . The position of in the lexico-
graphical ordering of is defined to be the rank of , denoted
by , i.e., is the number of all in with . Let
be the number of elements in for which
the first coordinates are . The rank of
can be obtained by using
(7)
An alternative of Cover’s enumeration scheme can be given by
counting the number of elements in that have a lexicographic
index higher than , the inverse rank of [7]. The inverse rank
of can be obtained by using
(8)
where , the complement of . The algorithms
(7) and (8) implement the decoding operation, i.e., given the
constrained sequence , find the corresponding lexicographic
indexinset .Theinverserankhasthevirtuethatthe same set of
weight coefficients can be used for encoding and decoding [7].
We will now consider the inverse rank for enumerative decoding
of DCRLL sequences.
Let denote a sequence of
length in NRZI notation emanating from state and ter-
minating in state of the graph . Note that . Let
denote the number of trailing “zeros” of the subvector
, i.e.,
Let be the length of the trailing run of the
subvector . By observing , we can uniquely
determine the sequence of states of graph corresponding to
. To this end, we define the sequence of states through
graph associated with the sequence as
if
if and
if and
Proposition: The inverse rank of a constrained se-
quence of length emanating from state and terminating
in state of the runlength graph is
(9)
where
if and and odd
otherwise
Proof: Let .We
can observe that for a given can take
one of two different values depending on .If then
,because in this case the numberof“zeros”between
thelast two “ones” of is less than , violating the constraint.
Therefore, no constrained sequence can begin with
,so .If , then does not violate
the constraint. In this case, the sequence leads graph to
the state , so that equals the number of
sequences of length (including the symbol )
emanating from the state of and terminating
in , i.e., .
Using the symmetry of , we obtain
.If
is even, then . In this
case, we do not perform an addition in (9).
B. Encoding
The encoding operation, i.e., given the inverse lexicographic
index , , find the corresponding , is described by
the following algorithm:
;
for to do
if odd and
then ;
else ;
end if
ifthen ;
else ;
end if ;
end for
C. Multiple State Coding
The ranking procedure can be generalized for a set of
sequences of length that emanate from a common
state, denoted by , and terminate in a state which is a
member of a predefined set of states. Consider terminating
BRAUN AND IMMINK: AN ENUMERATIVE CODING TECHNIQUE FOR DCRLL SEQUENCES 2027
states given by . Ranking of this set
of sequences can be accomplished by
(10)
Equation (10) holds because the sets of sequences emanating
from a common state and terminating in ,
are disjoint.
The enumerative technique described above enables the
design of channel encoders and decoders of moderate com-
plexity. Storage capacity is required for approximately
nonzero weight coefficients. The storage capacity required for
implementing the presented enumerative coding scheme for
combined dc-free runlength-limited sequences is about the
same as that required for implementing the enumerative coding
scheme for pure dc-free sequences [1].
D. Even Values of DSV
So far we have confined ourselves to DCRLL constraints
having odd values of DSV. For even, a runlength graph can be
derived from a Hankel-type adjacency matrix having
states, and enumerative encoding and decoding algorithms are
similar as described above. However, there may exist paths of
even length and paths of odd length emanating from a state
and terminating in a state of the corresponding runlength
graph. Thus, storage capacity is required for approximately
nonzero weight coefficients.
V. IMPLEMENTATION ASPECTS
In the following, we will consider coding schemes which
translate user information bits into codewords of an even
length . Using the technique described in [10], the code
construction can be extended to odd codeword lengths. The
ratio is called the code rate, and denotes
the capacity. We will distinguish two separate coding schemes.
In the first coding scheme, denoted by , we assume that
all codewords emanate from a state and terminate in the
same state of graph . We will call the state the
principal state of the coding scheme . In the second coding
scheme, denoted by , we assume that either of the codewords
emanates from a state which is a member of the set of states
and terminates in a state which is a
member of the same set . We will call the set of principal
states of the coding scheme . In both cases, transmission
errors that may occur in the DCRLL channel sequence must be
corrected before the decoding in order to prevent severe error
propagation. In the case of coding scheme , transmission
errors can cause catastrophic error propagation, which is why
the encoder must regularly be forced into a predefined principal
state.
A. Principal States and Feasible Code Rates
We will start by determining the principal state of coding
scheme and the set of principal states of coding scheme
that result in the maximum number of codewords. We
will also investigate at what codeword lengths the code rate
approaches the capacity of the DCRLL constraints to within a
small value, say 1%, and we will determine feasible code rates
for constraints of practical interest.
Let denote the number of distinct sequences of length
emanating from the state of . In a coding scheme that
enables bit encoding, must be satisfied for all
principal states of this scheme. For large can be
approximated by
where is the largest real root of the characteristic polyno-
mial , denotes the identity matrix,
and is a positive eigenvector of the matrix
associated with eigenvalue 1, i.e., [1].
The number of distinct DCRLL sequences of length ema-
nating from a state and terminating in a state of can
hence be approximated by
(11)
(if even, else 0). The accuracy of this approxima-
tion has been computed for constraints of practical in-
terest and codeword lengths , where the magni-
tude of vector has been evaluated numerically from the actual
number of sequences of a certain length . In all the
computed examples, approximation (11) was accurate to within
0.15%. Assuming equality in (11), it follows that the principal
state of coding scheme that results in maximum cardi-
nality is associated with the maximum entry of vector
. We define and
. In coding scheme , we intend to determine
the set of principal states so that the minimum number
of valid codewords emanating from a state is max-
imum among all possible sets . Assuming equality in (11), it
follows that this set is associated with:
The optimum set can be determined by starting with
odd , and by iteratively removing the state asso-
ciated with from this set. The procedure is then repeated
starting with even . Finally, the optimum
set is selected from the best subsets of and . The
minimum number of valid codewords of length ema-
nating from a state that is a member of the set of principal states
can thus be approximated by . The rate
of an implemented code is , and the ca-
pacity is given by . The difference between code rate
and capacity is
Examples of optimum principal state configurations are shown
in Table I for coding scheme and in Table II for coding
scheme . Also presented are the corresponding values
and the codeword lengths .
The codeword length ensures that . Table I
also shows the capacities of the corresponding con-
straints. We see that for many DCRLL constraints of practical
2028 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 12, DECEMBER 2000
TABLE I
CODING SCHEME:CAPACITY ,OPTIMUM PRINCIPAL STATE ,
CODEWORD LENGTH ,FEASIBLE CODE RATES FOR ,
RATE EFFICIENCIES ,NUMBER OF BITS ,AND APPROXIMATE
STORAGE CAPACITY IN kB FOR SEVERAL CONSTRAINTS
OF PRACTICAL INTEREST
TABLE II
CODING SCHEME:OPTIMUM PRINCIPAL STATE SET ,
CODEWORD LENGTH ,FEASIBLE CODE RATES FOR ,
RATE EFFICIENCIES ,NUMBER OF BITS ,AND APPROXIMATE
STORAGE CAPACITY IN kB FOR SEVERAL CONSTRAINTS
OF PRACTICAL INTEREST. : DENOTES THE PRINCIPAL STATE
SET
interest a codeword length in the order of 500 bits is sufficient
for achieving rate losses versus capacity of less than 1%.
Also shown in Tables I and II are feasible code rates and
rate efficiencies, defined by , where we have as-
sumed the optimum principal state configurations and the en-
coding of 256 bits. We see that 256 bit encoding using coding
scheme results in rate efficiencies in the order of 98%. The
rate efficiency can be further increased by using larger code-
word lengths or, for fixed , by applying coding scheme .
We add that the same code rates as in Tables I and II can often
be achieved by using principal states other than the optimum
configurations, for example, by using in coding scheme
.
B. Weight Truncation
In order to enable the implementation of the presented
DCRLL coding scheme, we will in the following express the
weights in truncated radix-2 representation. Let denote one
of the integer weights . An integer can be
uniquely represented by a binary -tuple ,
where . Let be the position
of the leading “one” element or “most significant bit” of .
We define the -bit truncated representation of , denoted by
,by
(12)
In words, is obtained from by leaving the most sig-
nificant bits unchanged and setting all the other trailing bits to 0.
The -bit truncated weight can be expressed in two-part
radix-2 representation , where the in-
tegers and are called the mantissa and exponent of ,
respectively. Apparently, bits are required to express the man-
tissa . The number of bits required to express the exponent is
in the order of . However, the exponents and of two
weight coefficients of similar magnitude
can usually be represented much more efficiently, for example,
by using the difference . The capacity of the memory
unit for the storage of the exponents of the weight coefficients
is hence negligible compared with the storage capacity required
for the mantissa. Therefore, the exponents of the weight coeffi-
cients will not be considered any further here.
When use is made of truncated weights, denoted by , recur-
sion (6) results in
In the enumerative encoding and decoding algorithms, the trun-
cated weights are used instead of the full-preci-
sion weights . The effect on the set of codewords
will be that the highest ranking
words of length will be recursively discarded from the set of
all the lexicographically ordered DCRLL sequences.
Speaking in terms of runlengths, the effect of using trun-
cated weights will be that short runs will occur more frequently
than in the case of untruncated weights. As an illustrative ex-
ample, Fig. 2 displays the accumulated runlength distributions
of two constrained enumerative codes having a rate
of and a rate efficiency of 98.1%, using either or
bits to express the mantissa of the weights. Also shown
in Fig. 2 is the accumulated runlength distribution of an ideal,
“maxentropic” constrainedsequence,i.e.,in this case
the code rate equals the capacity.
We will briefly consider the effect of using a truncated weight
representation on the achievable code rates. We define as
the minimum number of bits required to express the mantissa of
the truncated weights, so that there is no rate loss as in the full-
precision scheme. Values of are collected in Tables I and
II for optimum principal state configurations. We see that for
constraints and codeword lengths of practical interest,
values of in the order of 8–10 are usually sufficient to avoid
rate losses as in the full-precision scheme.
As shown in [6], the mantissa of the truncated weights
, will for increasing become (and
remain) periodic. That is, for , there are integers and
such that
We will call the period of the mantissa of and
the preamble. The period of the weight coefficients is a
BRAUN AND IMMINK: AN ENUMERATIVE CODING TECHNIQUE FOR DCRLL SEQUENCES 2029
Fig. 2. Accumulated runlength distributions of two constrained
enumerative codes having a rate of (obtained in computer simulations;
solid, dashed) and the accumulated runlength distribution of the corresponding
maxentropic sequence (obtained with the method developed by Kerpez et al.
[9], denoted by the dotted line). The step functions indicate the corresponding
average runlengths.
function of the constraints and the number of bits
used to express the mantissa of the weights. The preamble
also depends on the set of principal states used for coding. By
exploiting the periodicity of the weight coefficients, a signifi-
cant saving in storage hardware can often be realized [6]. The
ROM size required for the storage of the mantissa of the weight
coefficients is approximately given by
if odd
if even (13)
where . The factor in (13) is an
immediate consequence of the structure of the runlength graph
. We can see in Fig. 1 that ,
i.e., there is no need to store .InTa-
bles I and II, useful upper bounds of the ROM size are pre-
sented in kilobyte (kB) units, where we assumed
and . We can conclude that for many con-
straints of practical interest the presented enumerative coding
technique can be implemented by using a ROM of at most 5 kB
for storing the mantissa of the weight coefficients. In many
cases, the required ROM size is significantly lower than 5 kB.
C. Low-Frequency Suppression
Suppression of the low-frequency components is an essen-
tial performance criterion of DCRLL codes [1]. The PSD func-
tions at the low-frequency end are depicted in Fig. 3 for several
constrained coded sequences. Both axes of the power
spectra in Fig. 3 have been normalized for a fixed-user bit rate,
i.e., we consider . The dotted curve
in Fig. 3 represents the PSD of a maxentropic con-
strained sequence. Maxentropic DCRLL sequences provide an
upper bound in low-frequency suppression capability, given a
certain constraintand code rate [11]. The solidanddashed
curves in Fig. 3 represent the PSD functions of con-
strained sequences generated with the aid of the presented enu-
merative coding technique. These codes are based on construc-
tion using the principal state and have a rate of
and a rate efficiency of 98.1%. The upper curve in
Fig. 3 shows the PSD function of the coded sequence when the
weights are expressed in radix-2 representation using
bits to express the mantissa. Apparently the use of re-
sults in a loss of about 5 dB in low-frequency suppression rel-
ative to the maxentropic bound. When values of in the order
Fig. 3. Power spectra of several constrained enumerative codes
having a rate of (obtained in computer simulations; solid, dashed) and
power spectrum of the corresponding maxentropic sequence (evaluated using
the method developed by Kerpez et al. [9], denoted by the dotted line).
of 10–12 or untruncated weights are used, the PSD function of
the coded sequence approximates the maxentropic performance
bound very closely, as can be seen in the lower solid and dashed
curves in Fig. 3. Similar behavior of the PSD function at the
low-frequency end has been observed for constraints
other than and for sets of principal states other than
. We conclude that values of bits to ex-
press the mantissa of the truncated weights are suitable for ob-
taining a low-frequency suppression performance very close to
the maxentropic bound.
We suppose that the loss in low-frequency suppression versus
the maxentropic bound represented by the upper curve in Fig. 3
is a result of the runlength distribution, which, as shown in
Fig. 2, differs remarkably from the maxentropic runlength
distribution. Given a certain constraint and value of ,
the loss in low-frequency suppression versus the maxentropic
bound tends to become more pronounced with a decreasing
DSV or with an increasing codeword length, as we observed in
computer simulations. For , we never observed relevant
losses in low-frequency suppression versus the maxentropic
bound. The variance of the RDS, in short sum variance,is
often used to characterize the low-frequency characteristic of
dc-free sequences [1]. We would like to add that the
constrained sequence associated with the upper curve in Fig. 3
exhibits a sum variance that is only about 2.5% larger than for
a maxentropic constrained sequence, whereas from
the low-frequency characteristic and the theory of maxentropic
DCRLL sequences [11], we would roughly expect additional
60%.
D. Performances of Implemented Codes
In the following, we will briefly assess the performances of
several selected DCRLL codes with respect to their low-fre-
quency suppression capabilities. In order to obtain a fair
comparison of different DCRLL codes, both axes of the power
spectra of these codes have been normalized for a fixed-user
bit rate. As a performance criterion, we determine the PSD
2030 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 12, DECEMBER 2000
Fig. 4. Comparison of the performance of several constrained
modulation codes.
of the coded sequence at a small fraction of the user bit rate,
for example, at . We will compare the low-frequency
suppression capability of the coded sequence with the corre-
sponding maxentropic performance bound.
Fig. 4 displays the PSD versus a parameter called
the extra redundancy for several dc-free constrained
coded sequences. The extra redundancy has been defined in [11]
as the difference between the capacity of the constraint,
denoted by , and the rate of an implemented dc-free
code satisfying this runlength constraint. The solid curve in
Fig. 4 represents the maxentropic performance bound and the
crosses indicate the low-frequency suppression performances
of implemented codes. Four conventional codes are considered:
the EFM code [1] applied in the CD player; EFMPlus [2]
applied in the DVD system; and two other EFM alternatives
described in [2]. These four codes all have finite values of DSV
and satisfy the runlength constraint. The power spectra
of these codes were evaluated in computer simulations [2]. A
strategy for improving the EFMPlus low-frequency suppression
performance by about 3 dB is presented in [2].
Fig. 4 also shows the low-frequency suppression per-
formances for five DCRLL codes based on the presented
enumerative coding technique. All of these five enumerative
codes, denoted by EMC1-EMC5, have and they are
based on construction . The design parameters of these
codes are listed in Table III. EMC1 and EMC2 have code rates
similar to those of EFMPlus. In low-frequency suppression
performance, they outperform EFMPlus by about 15 dB. EMC3
and EMC4 have a low-frequency suppression performance
similar to that of EFM. They achieve an 11%–12% increase in
recording density relative to EFM, but require a rather large
ROM for storing the mantissa of the weights. As an alterna-
tive, EMC5 achieves a gain of 5.35% in code rate relative to
EFMPlus and it has a comparable low-frequency suppression
capability.
By relaxing the constraint from to ,we
can achieve a 6.67% gain in code rate relative to EFMPlus by
using EMC6. EMC6 is based on construction and its design
parameters are given in Table III. EMC6 achieves
TABLE III
DESIGN PARAMETERS OF SEVERAL ENUMERATIVE CONSTRAINED
MODULATION CODES
dB. The size of the required ROM can be decreased from
about 5 kB to about 2 kB by using instead of .
The resulting code, denoted by EMC7, achieves
dB.
VI. CONCLUSIONS
We have presented an enumerative technique for encoding
and decoding DCRLL sequences. The size of the full-precision
weight set required for implementing the presented coding
scheme is about the same as that required for implementing the
enumerative coding scheme for pure dc-free sequences [1]. As
the greater part of the electronics that implements the enumer-
ative coding technique is taken up by the storage of the weight
coefficients, this technique enables the design of channel
encoders and decoders of moderate complexity. To enable
the implementation of the proposed coding technique, we ex-
pressed the weight coefficients in finite-precision floating-point
notation. We have shown that the presented enumerative coding
technique can be used to encode and decode DCRLL sequences
approaching the maxentropic performance bounds very closely
in terms of code rate and low-frequency suppression capability.
For channel constraints of practical interest, the hardware
required for implementing such a quasi-maxentropic coding
scheme consists mainly of a ROM of at most 5 kB. In many
cases of practical relevance, the size of the required ROM is
significantly lower.
ACKNOWLEDGMENT
The authors would like to thank A. J. H. Vinck, H. Morita,
and M. Umemoto for useful discussions and comments. The
authors would also like to thank the two anonymous reviewers
for providing valuable comments on an earlier version of the
manuscript.
REFERENCES
[1] K. A. S. Immink, Codes for Mass Data Storage Systems, Eindhoven,
The Netherlands: Shannon Foundation, 1999.
[2] , “EFMPlus: The coding format of the multimedia compact disc,”
IEEE Trans. Consumer Electron., vol. 41, pp. 491–497, Aug. 1995.
[3] T. Uehara and Y. Oba, “A new 8–14 channel coding for D-3 format
VTR,” IEEE Trans. Broadcast., vol. 39, pp. 265–271, June 1993.
[4] J. Li and J. Moon, “DC-free run-length-limited codes for magnetic
recording,” IEEE Trans. Magn., vol. 33, pp. 868–874, Jan. 1997.
[5] T. M. Cover, “Enumerative source coding,” IEEE Trans. Inform. Theory,
vol. IT-19, pp. 73–77, Jan. 1973.
[6] K. A. S. Immink, “A practical method for approaching the channel ca-
pacity of constrained channels,” IEEE Trans. Inform. Theory, vol. 43,
pp. 1389–1399, Sept. 1997.
BRAUN AND IMMINK: AN ENUMERATIVE CODING TECHNIQUE FOR DCRLL SEQUENCES 2031
[7] L. Pátrovics and K. A. S. Immink, “Encoding of -sequences using
one weight set,” IEEE Trans. Inform. Theory, vol. 42, pp. 1553–1554,
Sept. 1996.
[8] G. L. Pierobon, “Codes for zero spectral density at zero frequency,”
IEEE Trans. Inform. Theory, vol. IT-30, pp. 435–439, Mar. 1984.
[9] K. J. Kerpez, A. Gallopoulos, and C. Heegard, “Maximum entropy
charge-constrained run-length codes,” IEEE J. Select. Areas Commun.,
vol. 10, pp. 242–252, Jan. 1992.
[10] K. A. S. Immink, “DC-free codes of rate odd,” in Proc.
1998 IEEE Int. Symp. Information Theory (ISIT’98), Cambridge, MA,
Aug. 16–21, 1998, p. 111.
[11] V. Braun and A. J. E. M. Janssen, “On the low-frequency suppression
performance of dc-free runlength-limited modulation codes,” IEEE
Trans. Consumer Electron., vol. 42, pp. 939–945, Nov. 1996.
Volker Braun (S’93–M’97) was born in Saarland,
Germany, in 1967. He received the Dipl.-Ing. degree
from the University of Kaiserslautern, Kaiser-
slautern, Germany, in 1992, and the Dr.-Ing. degree
from the University of Essen, Essen, Germany, in
1997.
From 1992 to 1997, he was a member of the
Digital Communications Group at the Institute for
Experimental Mathematics, University of Essen.
During that period, he was a regular Visiting Sci-
entist at Philips Research Laboratories, Eindhoven,
The Netherlands, and he was contributing to the ACTS Project SMASH of
the European Community. From March 1997 to March 1998, he was with the
Hitachi Central Research Laboratories, Tokyo, Japan, as a Visiting Scientist.
His research activities from 1992 to 1998 focused on modulation, coding, and
signal processing related to magnetic and optical recording systems. In 1998,
he joined Alcatel SEL AG, Stuttgart, Germany, where he started working in
the Switching Division Hardware Architecture Design Group on optimizing
implementation aspects of Alcatel’s MPSR broad-band switch. Since June
1999, he has been with the Radio Communications Department at Alcatel Cor-
porate Research Center, Stuttgart, Germany, where he is involved in research
and development activities regarding channel coding and multiple-antenna
techniques for applications in second- and third-generation mobile radio
communications systems.
Kees A. Schouhamer Immink (SM’86–F’90)
received the M.S. and Ph.D. degrees from the
Eindhoven University of Technology, Eindhoven,
The Netherlands.
He is Director of Turing Machines, Inc. Since
1995, he has been an Adjunct Professor at the
Institute for Experimental Mathematics, Essen Uni-
versity, Essen, Germany. In addition, he is affiliated
with the National University of Singapore. He has
contributed to the design and development of a wide
variety of consumer-type audio and video recorders
such as the laser video disc, compact disc, compact disc video, DAT, DV, DCC,
and DVD. He holds 42 issued and pending U.S. patents in variousfields.
Dr. Immink is an elected member of the Royal Netherlands Academy of Arts
and Sciences (KNAW) and holds fellowships of the IEEE, AES, SMPTE, and
IEE. He is Vice President of the Audio Engineering Society (AES) and a Gov-
ernor of the IEEE Consumer Electronics Society. He is a member of the IEEE
Fellows Committee. For his contributions to the digital audio and video revolu-
tion, he received wide recognition, among others the 1999 IEEE Edison Medal,
AES Gold Medal, IEEE Masaru Ibuka Consumer Electronics Award, and the
Golden Jubilee Award for Technological Innovation awarded by the IEEE In-
formation Theory Society in 1998.
... In 1973, Cover introduced an important result for indexing a sequence within a set of lexicographically-ordered sequences [34]. Later, this result inspired Immink and others to design enumerative constrained codes [35], [36]. Works focusing on the enumeration of constrained sequences include [37] and [38], while other works introducing constrained code constructions based on enumerative approaches include [39], [40], and [41]. ...
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