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Most recording systems encode their data using binary run-length-limited (RLL) codes. Statistics such as the density of 1s, the probabilities of specific code strings or run lengths, and the power spectrum are useful in analyzing the performance of RLL codes in these applications. These statistics are easy to compute for ideal run-length-limited codes, those whose only constraints are the run-length limits, but ideal RLL codes are not usable in practice because their code rates are irrational. Implemented RLL codes achieve rational rates by not using all code sequences which satisfy the run-length constraints, and their statistics are different from those of the ideal RLL codes. Little attention has been paid to the computation of statistics for these practical codes. In this paper a method is presented for computing statistics of implemented codes. The key step is to develop an exact description of the code sequences which are used. A consequence of the code having rational rate is that all the code-string and run-length probabilities are rational. The method is illustrated by applying it to three codes of practical importance: MFM, (2, 7), and (1, 7).

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... Since the capacity of (d, k)-constrained sequences is irrational (see Ashley and Siegel [2]), it is clear that code implementations which, by necessity, operate at rates of the form m / n, where mand nare integers, can never attain 100%' of the capacity. It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences bY,the addition of certain constraints, which he called incidental contraints. He found that certain bit patterns, which could readily be used as a basis for constructing sync patterns, are absent in sequences generated by the popular (l, 7) and (2, 7) codes. ...

... It is, therefore, clear that code implementations can never attain 100% of the capacity . It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences by the addition of a few constraints, which he called incidental constraints, Howell [3] described in detail the incidental constraints ofthree popular (d, k) codes, namely the rate 2/3, (1, 7) code [15], the rate 1/2, (2, 7) code [16], and the rate 1/2, (I, 3) code. He found that eertain bit pattems do not occur in sequenees generated by these codes. ...

... It is, therefore, clear that code implementations can never attain 100% of the capacity . It was noted by Howell [3] that implemented codes differ from maxentropic, "ideal, " sequences by the addition of a few constraints, which he called incidental constraints, Howell [3] described in detail the incidental constraints ofthree popular (d, k) codes, namely the rate 2/3, (1, 7) code [15], the rate 1/2, (2, 7) code [16], and the rate 1/2, (I, 3) code. He found that eertain bit pattems do not occur in sequenees generated by these codes. ...

... The characteristic equation pertaining to the recursion (17) equals (18) The difference between the largest root of (18) and (10) in the working range which means that the loss in capacity resulting from the truncation of the weights can be simply approximated by (19) A numerical comparison of the above result with the outcome of the method of computing the cycle time revealed that (19) can serve as a reliable rule of thumb. ...

... The characteristic equation pertaining to the recursion (17) equals (18) The difference between the largest root of (18) and (10) in the working range which means that the loss in capacity resulting from the truncation of the weights can be simply approximated by (19) A numerical comparison of the above result with the outcome of the method of computing the cycle time revealed that (19) can serve as a reliable rule of thumb. ...

... In this subsection, we will compare codes contructed by the new coding technique with traditional sliding-block codes: the rate , code and the rate , code [19], which have been used in magnetic disk drives. The maximum runlength constraint is imposed to restrict the maximum time between two consecutive transitions in the recorded signal. ...

A new coding technique is proposed that translates user
information into a constrained sequence using very long codewords. Huge
error propagation resulting from the use of long codewords is avoided by
reversing the conventional hierarchy of the error control code and the
constrained code. The new technique is exemplified by focusing on (d,
k)-constrained codes. A storage-effective enumerative encoding scheme is
proposed for translating user data into long dk sequences and vice
versa. For dk runlength-limited codes, estimates are given of the
relationship between coding efficiency versus encoder and decoder
complexity. We show that for most common d, k values, a code rate of
less than 0.5% below channel capacity can be obtained by using hardware
mainly consisting of a ROM lookup table of size 1 kbyte. For selected
values of d and k, the size of the lookup table is much smaller. The
paper is concluded by an illustrative numerical example of a rate
256/466, (d=2, k=15) code, which provides a serviceable 10% increase in
rate with respect to its traditional rate 1/2, (2, 7) counterpart

... That means the process of imposing the incidental constraints will involve discarding states rather than discarding branches between states. There are other ways to arrive at the desired results, but this method was preferred because it facilitated the use of the computational tools that were available [2], [4]. ...

... A one followed by a run of 6 zeros must start with the one in position a or c. There are other choices of incidental constraints that would result in the same restrictions, but with different labels attached to the individual bits of the output symbols [2], [5], [61. For these other choices, the individual bits of the output symbols would be labeled (bca) and (cab). ...

... The statistical properties of a code are embodied in the runlength constraint graph. Because this code is based on the same incidental constraints as the standard IBM (1,7) code, the set of sequences which can be generated is the same [2]. Although the mapping of user bits to channel bits is different, the statistical properties are exactly the same. ...

... spectrum resulting from the graph is shown in Fig. 11 along with the power spectrum for the maxentropic case. It turns out that I. INTRODUCTION the constraint graph in Fig. 5 corresponds exactly to one of the unused components of the cubed (1,7) constraint graph in [2]. ...

... Quite recently [2], a new algorithm for generating zero-disparity codewords was presented by Knuth. The method is based on a simple correspondence between the set of all m-bit source words and the set of all (m + p)-bit balanced codewords. ...

... Most schemes for generating dc-balanced sequences use lookup tables, and are therefore restricted to codewords of medium size. An alternative and easily implementable encoding technique for zero-disparity codewords that is capable of handling (very) large blocks was described by Knuth [2]. The method is based on the idea that there is a simple correspondence between the set of all m-bit binary source words and the set of all (m + p&bit codewords. ...

The problem of appraising the spectral performance of codes based on a new algorithm for generating zero-disparity codewords presented by D.E. Knuth (1986) is addressed. In order to get some insight into the efficiency of Knuth's construction technique, the authors evaluate the spectral properties of its code streams. The structure of Knuth codes allows the derivation a simple expression for (an approximation to) the sum of variance of these codes. This quantity plays a key role in the spectral performance characterization of DC-balanced codes. The authors evaluate this expression and compare the sum variance of Knuth codes with the sum variance of the polarity bit codes for fixed redundancy. Under the premise that the sum variance can serve as a quantity to judge the width of the spectral notch, the authors conclude that codes based on Knuth's algorithm offer less spectral suppression than polarity bit codes with the same redundancy.

... T HE THEORY and design of codes that limit the minimum and maximum runlengths in a modulation code are well developed (e.g., [2]-[4], [12]- [16]). Until recently, most information theoretic models for these codes were noiseless. ...

... (Note that the SNR = 22 db corresponds to the "peak-shift" probability 2Q(A /2a) = 3.2 x lo-".) Popular rate 2/3 (1,7) codes are the Jacoby code [141 and the AHM (IBM) code [15], [16]. For high signal-to-noise ratios, their rate is much lower than our lower bound on C,(2,8, a21 (Fig. 5). ...

... For high signal-to-noise ratios, their rate is much lower than our lower bound on C,(2,8, a21 (Fig. 5). Similar comparisons can be made for rate l/2 IBM [II, [21, [16] and Zerox [13] (2,7) codes (Fig. 6). ...

The authors consider the model for runlength coded systems of P.H. Siegel (1982) that is based on peak or edge detection. The model does not attempt to account for failures of the qualifying circuit; it is assumed that every transition is successfully declared (i.e., no missing or false qualifiers). In this model it is the error in the estimate of the location of the transition that introduces uncertainty at the channel output. The model is motivated by the problem of pulse location in white Gaussian noise. Through a series of lemmas and theorems, bounds on the capacity of this model are obtained. These bounds are evaluated and used to suggest that improvements in storage capacity are possible through the use of codes with designed noise tolerance.

... The sequences described by this FSTD can be interpreted as a "phase-shifted'' version of the sequences produced by the FSTD in Fig. 28, as discussed in Howell [50]. More precisely, if the sequences from The main purpose of this subsection is to show that both of these codes can be constructed "from scratch," so to The first (A(G3), 4)-approximate eigenvector found by the AE algorithm, with L = 3, is speak,' using the techniques discussed in Sections 111-V. ...

The authors provide a self-contained exposition of modulation code
design methods based upon the state splitting algorithm. They review the
necessary background on finite state transition diagrams, constrained
systems, and Shannon (1948) capacity. The state splitting algorithm for
constructing finite state encoders is presented and summarized in a
step-by-step fashion. These encoders automatically have state-dependent
decoders. It is shown that for the class of finite-type constrained
systems, the encoders constructed can be made to have sliding-block
decoders. The authors consider practical techniques for reducing the
number of encoder states as well as the size of the sliding-block
decoder window. They discuss the class of almost-finite-type systems and
state the general results which yield noncatastrophic encoders. The
techniques are applied to the design of several codes of interest in
digital data recording

... As is well known for the case of q = 2 (see, e.g., [14], [36]), its characteristic polynomial is ...

DNA as a data storage medium has several advantages, including far greater data density compared to electronic media. We propose that schemes for data storage in the DNA of living organisms may benefit from studying the reconstruction problem, which is applicable whenever multiple reads of noisy data are available. This strategy is uniquely suited to the medium, which inherently replicates stored data in multiple distinct ways, caused by mutations. We consider noise introduced solely by uniform tandem-duplication, and utilize the relation to constant-weight integer codes in the Manhattan metric. By bounding the intersection of the cross-polytope with hyperplanes, we prove the existence of reconstruction codes with greater capacity than known error-correcting codes, which we can determine analytically for any set of parameters.

... As an example, in figure 2.12, the state transition diagram MFM is shown. See appendix A (and Howel [13]) for relevant data on the practical codes mentioned before. ...

The authors obtain general lower bounds on the number of states in any encoder for a given constrained system and rate. Lower bounds on the number of states are exhibited in a fixed-rate finite-state encoder that maps unconstrained n-ary sequences into a given set of constrained sequences, defined by a finite labeled graph G. In particular, one simple lower bound is given by min<sub>x</sub>max<sub>v</sub>x<sub>v</sub> where x=(x<sub>v</sub>) ranges over certain (nonnegative integer) approximate eigenvectors of the adjacency matrix for G. In some sense, the bounds are close to what can be realized by the state splitting algorithm and in some cases, they are shown to be tight. In particular, these bounds are used to show that the smallest (in number of states) known encoders for the

Coding techniques are used in communication systems to increase the efficiency of the channel. Not only is coding equipment being used in point-to-point communication channels, but coding methods are also used in digital recording devices such as sophisticated computer disk files and numerous domestic electronics such as stationary- and rotary-head digital audio tape recorders, the Compact Disc, and floppy disk drives. Since the early 1970s, coding methods based on runlength-limited sequences have played a key role for increasing the storage capacity of magnetic and optical disks or tapes. A detailed description is furnished of the limiting properties of runlength-limited sequences, and a comprehensive review is given of the practical aspects involved in the translation of arbitrary data into runlength-limited sequences.

In this paper, a novel method is developed for computing formulae for the power spectra associated with runlength-limited (RLL) codes. Explicit use is made of a compact description of the runlength process associated with the RLL code. This association simplifies the general derivation of the power spectrum. The calculation of the spectra of several RLL codes popular in data storage applications is presented. Some of the closed-form expressions for the spectra of these widely used codes are new.

Describes a new technique for constructing fixed-length (d,k)
runlength-limited block codes. The new codes are very close to
block-decodable codes, as decoding of the retrieved sequence can be
accomplished by observing (part of) the received codeword plus a very
small part (usually only a single bit) of the previous codeword. The
basic idea of the new construction is to uniquely represent each source
word by a (d,k) sequence with specific predefined properties, and to
construct a bridge of β, 1⩽β⩽d, merging bits between
every pair of adjacent words. An essential element of the new coding
principle is look ahead. The merging bits are governed by the state of
the encoder (the history), the present source word to be translated, and
by the upcoming source word. The new constructions have the virtue that
only one look-up table is required for encoding and decoding

A new rate 4/6 (d=1, k'=11) runlength-limited code which is well
adapted to byte-oriented storage systems is presented. The new code has
the virtue that it can be decoded on a block basis, i.e., without
knowledge of previous or next codewords, and, therefore, it does not
suffer from error propagation. This code is particularly attractive as
many commercially available Reed-Solomon codes operate in GF(2<sup>8
</sup>)

An input-constrained channel S is defined as the set of words
generated by a finite labeled directed graph. It is shown that every
finite-state encoder with finite anticipation (i.e., with finite
decoding delay) for S can be obtained through state-splitting rounds
applied to some deterministic graph presentation of S, followed by a
reduction of equivalent states. Furthermore, each splitting round can be
restricted to follow a certain prescribed structure. This result, in
turn, provides a necessary and sufficient condition on the existence of
finite-state encoders for S with a given rate p:q and a given
anticipation a. A second condition is derived on the existence of such
encoders; this condition is only necessary, but it applies to every
deterministic graph presentation of S. Based on these two conditions,
lower bounds are derived on the anticipation of finite-state encoders.
Those lower bounds improve on previously known bounds and, in
particular, they are shown to be tight for the common rates used for the
(1,7)-runlength-limited (RLL) and (2,7)-RLL constraints

The benefits of using modulation codes in two dimensions in
multichannel recording include improved clocking and improved ratios of
user bits to recorded bits. Recent work has described a class of d
<sub>x</sub>, k <sub>y</sub> codes in two dimensions which
satisfy the d <sub>x</sub> constraint in one dimension and the
k <sub>y</sub> constraint in the other. In tape systems this
method is extremely vulnerable to dropouts. A new class of
two-dimensional run-length codes that operate using the usual d
<sub>x</sub>, k <sub>y</sub> constraint along the track is
proposed, with an additional k <sub>y</sub> constraint across
the tracks. In this approach the horizontal d <sub>x</sub>,
k <sub>x</sub> code is allowed a much larger k <sub>x
</sub> constraint since it is no longer the sole carrier of clocking
information. Capacities are calculated for a range of codes and number
of channels, and an example of the construction of one such code is
given. An extension of the codes that preserves clocking during channel
loss is described

We compare performance and error propagation of DFE with and
without a d=1 RLL code, at 2.67 user density and with a single
coefficient FIR phase equalizer. Performance without error propagation
is slightly better with d=1 in spite of the rate loss, because precursor
ISI can be completely eliminated. We develop a model to estimate the
effects of error propagation for both d=0 and d=1. The model is in good
agreement with a 20 db SNR simulation. For an overall error rate of 10
<sup>-6</sup>, the probability of a burst of length 50 in the decoded
data is 10<sup>-13</sup> for d=1 and 10<sup>-8</sup> for d=0. This large
difference is due both to the higher code rate and to the larger
postcursor cancellation for d=0. In the model, we rigorously compute
burst error probabilities using a Markov chain derived from our channel
assumptions. We also use the model to compute the decay rate of the
burst error probability and to identify the set of infinitely
propagating sequences. In the simulations, we use random data and a
commonly used (1,7) code for DFE17, to which we added AWGN noise at SNR
20 db. Finally, we compare the results of the model with the simulations

Multilevel decision feedback equalization (MDFE) is a detection
scheme that was developed for (1,k) coded recording channels. The user
signal-to-noise ratio required to achieve a chosen bit error rate (BER)
of 1e-6 has been shown to be about 1.9 dB more than that of the maximum
likelihood lower bound for a Lorentzian channel at user density 2.5.
Recently, an advanced version of MDFE, called M2DFE, was proposed. By
using computer simulations, the BER of M2DFE has been shown to improve
by about 1 dB compared to MDFE. In this paper, we first discuss the
various aspects of M2DFE design and then present its theoretical
analysis. Using the analysis, we show how the two critical parameters in
the design are to be chosen for optimum performance. We also propose a
modified M2DFE detector, which exploits the noise correlation at the
slicer input, to improve the BER performance as well as reduce error
propagation considerably. These MDFE detectors are then compared for
their BERs and error propagation performances

A two step coding scheme for peak-shift correction in ( d ,
k )-constrained sequences is described. The first step is based
on q -ary ( q = k - d +1 is a prime) block
codes that allow correction of specific types of double errors caused by
single peak-shifts. The second step is a simple conversion of
q -ary symbols to binary strings of the type 00. . .01. The
concatenation of these strings satisfies the
( d , k )-constraint within the codeword and in
concatenation with neighboring words. The length of the codewords is
controlled and, if necessary, can be fixed. The rate R <sub>1
</sub> of the overall encoding approaches (2 log<sub>2</sub>
( k - d +1)/( k + d +2) for large codeword
lengths. Codes for correction of peak-shift, deletions, and insertions
of zeros are presented as well. Encoding and decoding are done by simple
algorithms without using look-up tables, enumeration or denumeration
procedures and, therefore, the codelength may be large

DNA as a data storage medium has several advantages, including far greater data density compared to electronic media. We propose that schemes for data storage in the DNA of living organisms may benefit from studying the reconstruction problem, which is applicable whenever multiple reads of noisy data are available. This strategy is uniquely suited to the medium, which inherently replicates stored data in multiple distinct ways, caused by mutations. We consider noise introduced solely by uniform tandem-duplication, and utilize the relation to constantweight integer codes in the Manhattan metric. By bounding the intersection of the cross-polytope with hyperplanes, we prove the existence of reconstruction codes with full rate, as well as suggest a construction for a family of reconstruction codes.

This paper describes the development and application of software
models for the design and optimization of the record, readout and data
recovery processes in all commonly used optical disk storage formats. A
detailed design example for CAD MSR/MAMMOS system is described. Written
bits, readout waveforms, and the performance of PRML for this new format
are presented

In digital recorders, the coded information is commonly grouped in
large blocks, called frames. The authors concentrate on the frame
synchronization problem of run-length-limited sequences, or ( d ,
k ) sequences. They commence with a brief description of ( d
, k )-constrained sequences, and proceed with the
examination of the channel capacity. It is shown that for certain sync
patterns, called repetitive-free sync patterns, the capacity can be
formulated in a simple manner as it is solely a function of the ( d
, k ) parameters and the length of the sync pattern. For
each forbidden pattern and ( d , k ) constraints, methods
for enumerating constrained sequences are given. Design considerations
of schemes for encoding and decoding are addressed. Examples of
prefix-synchronized ( d , k ) codes, based for the
purpose of illustration on the sliding-block coding algorithm, are
presented

The authors present a study of run-length-limiting codes that have
a null at zero frequency or DC. The class of codes or sequences
considered is specified by three parameters: ( d , k ,
c ). The first two constraints, d and k , put
lower and upper bounds on the run-lengths, while the charge constraint,
c , is responsible for the spectral null. A description of the
combined ( d , k , c ) constraints, in terms of a
variable length graph, and its adjacency matrix, A ( D ),
are presented. The maximum entropy description of the constraint
described by a run-length graph is presented as well as the power
spectral density. The results are used to study several examples of
( d , k , c ) constraints. The eigenvalues and
eigenvectors of the classes of ( d , k =2 c -1,
c ) and ( d , k = d +1, c )
constraints for ( c =1,2,. . .), are shown to satisfy certain
second-order recursive equations. These equations are solved using the
theory of Chebyshev polynomials

Given a (1,7) code, a loss of synchronization may occur by
insertion or deletion of a 0 or a 1. That event is catastrophic, i.e.,
it will cause an unlimited number of errors in the data since the moment
of the loss of synchronization. We introduce two methods for recovering
against insertions or deletions of symbols. The first one allows for
identification of up to 3 insertions and/or deletions in a given block,
permitting quick synch recovery. This method is based on variable length
codes. The second method is of block type, and allows for detection of
large numbers of insertions and deletions. Both methods can be extended
to general (d,k) constrained codes

MDFE (multi-level decision feedback equalization) is an excellent
detector for magnetic recording channels using (1,k) codes. However,
being a decision feedback based scheme, MDFE also suffers from the
phenomenon of error propagation. In this paper, a novel threshold
approach is proposed for detection and partial suppression of error
propagation in MDFE. A theoretical analysis of this approach is
presented to quantify the probability of false detection of error
propagation. Computer simulations performed over different user
densities show that the proposed technique reduces error propagation
significantly. In particular, the decay rate of cumulative error
burst-length distribution has been almost doubled by this approach

The effect of data constraints on synchronization is quantified by the use of three simple timing metrics that respectively measure the ensemble average, the worst, and the best timing qualities attainable with a given binary pulse amplitude modulation (PAM) waveform. These timing metrics are computed with the help of a graph which represents the constrained PAM system. The timing metrics of the (0,k) constraint are studied in detail for selected PAM pulses

We introduce the continuous time asynchronous channel as a model for time jitter in a communication system with no common clock between the transmitter and the receiver. We have obtained a simple characterization for an optimal zero-error self-synchronizable code for the asynchronous channel. The capacity of this channel is determined by both a combinatorial approach and a probabilistic approach. Our results unveil the somewhat surprising fact that it is not necessary for the receiver clock to resynchronize with the transmitter clock within a fixed maximum time in order to achieve reliable communication. This means that no upper limit should be imposed on the run lengths of the self-synchronization code as in the case of run-length limited (RLL) codes which are commonly used in magnetic recording.

We discuss the relation between some early techniques for constrained channel coding and more recent ones adapted from the mathematical area of symbolic dynamics. A primary difference between the two is that the latter focus on issues of code existence, whereas the former were primarily concerned with code construction and optimality.

The read channel of a typical Winchester disk is considered. The
primary concern is with the performance of the decoder for the
Reed-Solomon error-correcting code. Decreasing the signal-to-noise ratio
of a disk increases its error rate. Use of an error-correcting code
reduces this error rate. For a fixed error rate, then, the use of a
particular error-correcting code can be balanced by a certain reduction
in signal-to-noise ratio. This reduction is usually referred to as the
coding gain of the code. The author calculates the coding gain of
various Reed-Solomon error-correcting codes. The intent is to determine
how low a signal-to-noise ratio can be supported by a particular
error-correcting code. The results show that coding gains of 3-4 dB are
achievable with codes of modest implementation cost

An analysis is given to determine the maximum areal density
achievable in digital magnetic recording and the associated optimal bit
aspect ratio. Typical transducer-media parameters are selected, a
conventional two-path peak detection data channel is modeled, and error
probability is determined as a function of track density and bit density
so that maximal areal density and optimal user bit aspect ratio can be
determined

The effects of introducing shading bands into the collector path of
optical storage devices to equalize optically the signal to a partial
response target have been investigated. A software simulation of the
optical readout process and the partial-response maximum-likelihood
(PRML) read channel have been developed and used to evaluate different
channel configurations. It is shown that optical filtering can be used
to eliminate the requirement for digital filtering in the PRML channel.

Linear superposition techniques are often used to simulate waveforms in a variety of applications. In particular, superposition techniques have been used extensively to quickly and efficiently analyze readout waveforms from optical storage media. However, as storage densities increase the optical readout process exhibits nonlinear characteristics that are not reproduced using linear superposition techniques. Alternative approaches, based on curve fitting and optimization routines, are often used to predict more accurately the readout waveform. This paper describes a novel technique for accurately simulating readout waveforms from high-density optical storage media, based on the superposition of isolated pulse responses and nonlinear signal components that arise due to inter-symbol-interference in the readout waveform.

This thesis is devoted to the development of timing recovery techniques for digital recording systems. The thesis begins with a detailed review and discussion of timing recovery structures, requirements, performance measures, timing error detector (TED) algorithms, and timing acquisition issues. The main contributions include five parts. The first part examines the timing sensitivity of read channel detectors, and develops a new analytical approach for evaluating the performance under static and random timing errors. The second part examines the TED efficiencies, develops an improved TED for jitter minimization, and studies optimality issues for timing acquisition. The third part investigates false lock and hang up problems, and develops two novel acquisition techniques. The fourth part presents the timing recovery loop design and implementation for an experimental read channel detector. The fifth part develops a new asynchronous equalizer adaptation structure with fully digital interpolative timing recovery (ITR) for digital optical recording systems.

Partial response maximum likelihood (PRML) detection using the Viterbi algorithm involves the calculation of likelihood metrics that determine the most likely sequence of decoded data. In general, it is assumed that branches at each node in the trellis diagram have same probabilities. If modulation code with minimum and maximum run-length constraints is used, the occurrence ratio (Ro) of each particular pattern is different, and therefore the assumption is not true. We present a calculation scheme of the likelihood metrics for the PRML detection using the occurrence ratio. In simulation, we have tested the two (1,7) run-length-limited codes and calculated the occurrence ratios as the orders of PR targets are changed. We can identify that the PRML detections using the occurrence ratio provide more than about 0.5dB gain compared to conventional PRML detections at 10/sup -5/ BER in high-density magnetic recording and optical recording channels.

To compare encoding and decoding schemes requires one to first look into information and coding theory. This article discusses problems and possible solutions in encoding information

Maximum-likelihood sequence estimation of binary coded and uncoded information, stored on an optical disc, corrupted with additive Gaussian noise is considered. We assume the presence of inter-symbol interference and channel/receiver mismatch. The performance of the maximum-likelihood detection of runlength-limited sequences is compared against both
uncoded information and information encoded by Hamming-distance-increasing
convolutional codes.

In this paper, a novel method is developed for computing formulae for the power spectra associated with runlength-limited (RLL) codes. Explicit use is made of a compact description of the runlength process associated with the RLL code. This association simplifies the general derivation of the power spectrum. The calculation of the spectra of several RLL codes popular in data storage applications is presented. Some of the closed-form expressions for the spectra of these widely used codes are new.

Several results on binary ( d , k ) codes are
given. First, a novel derivation for the capacity of these codes based
on information-theoretic principles is given. Based on this result the
spectrum of a ( d , k ) code is computed. Finally, the
problem of computing the capacity of the binary symmetric channel under
the condition that the input sequences satisfy the ( d , k
) constraint is considered. Lower bounds on the capacity of such a
channel are derived

Bell System Technical Journal, also pp. 623-656 (October)

This paper investigates some of the properties of a class of two-level codes with constrained run length, whose use has been proposed for purposes of bandwidth compression. It is shown that such codes can indeed reduce the bandwidth containing a given percentage of the transmitted power. To communicate information, however, different transmitted codewords must be distinguishable at the receiver, and this requires that the channel bandwidth be sufficiently wide to allow the difference waveform to propagate. It is demonstrated that decreasing the X -percent bandwidth using these codes leads to a rapid increase in the difference waveform bandwidth, and hence in the channel bandwidth necessary to maintain error rate performance. Thus, these codes are bandwidth expansion codes in disguise. Signal-to-noise ratio and channel bandwidth requirements for these codes are discussed and compared with those of M -level codes [pulse-amplitude modulation (PAM)] for two kinds of receivers.

The familiar error-correction codes allow a reduction in the required signal-to-noise ratio at the expense of an increase in bandwidth. Here we reverse the problem and investigate codes that permit a reduced bandwidth at the expense of an increase in the required signal-to-noise ratio. Theoretical properties of these bandwidth compaction codes have been published previously. This paper emphasizes the tradeoff between bandwidth and signal-to-noise ratio when the codes are used. The inherent error detection and error correction properties of the codes are also explored.

Methods are presented for the encoding of information into binary sequences in which the number of ZEROS occurring between each pair of successive ONES has both an upper and a lower bound. The techniques, based on the state structure of the constraints, permit the construction of short, efficient codes with favorable error-propagation-limiting properties.

A method of analyzing the correctable errors in disk files is presented. It allows one to infer the most probable error in the encoded-data stream given only the unencoded readback and error-correction information. This method is applied to the errors observed in seven months of operation of four IBM 3380 head-disk assemblies. It is shown that nearly all the observed errors can be explained as single-bit errors at the input to the channel decoder. About 90 percent of the errors were related to imperfections in the disk surfaces. The remaining 10 percent were mostly due to heads which were unusually susceptible to random noise-induced errors.

A special case with binary sequences was presented at the IEEE 1969 International Symposium on Information Theory in a paper titled “Run-Length-Limited Codes.

This paper provides a tutorial introduction to recording codes for magnetic disk storage devices and a review of progress in code construction algorithms. Topics covered include: a brief description of typical magnetic recording channels; motivation for use of recording codes; methods of selecting codes to maximize data density and reliability; and techniques for code design and implementation.

A new 2/3-rate run-length limited code with d = 1 and k = 7 is described in this paper. It is a state dependent, look-ahead code that has advantages over the MFM and 3PM (2, 7) codes. Compared to MFM the advantages are an increase of 33% in the data rate and a 33% “increase in the detection window and the minimum time between transitions (Tmin). Compared to the 3PM (2, 7) code, the window is increased by 33%, while Tmin has been reduced by 11 %. Additionally, the wavelength ratio [Tmax/Tmin) has been increased by 50% with respect to the 3PM (2, 7) code. The main parameters of the new code are shown in Fig. 12. The 33% increase in the detection window of the ISS-2/3 code allows for higher noise levels, while the slightly smaller value of Tmin somewhat increases the, effect of intersymbol interference. The net result is to allow for about a 10% increase in the data rate, compared to the 3PM code. The system has been implemented in the ISS-8470 high density disk file, featuring 4418 bits/cm data density, 2.097 MBytes/sec data rate and 683 MBytes capacity.

A computer model of a peak detecting magnetic recording channel has been implemented and used for channel design and performance evaluation. The model predicts raw error rate, ontrack and off-track, as a function of linear density, run-length-limited (RLL) modulation code, write precompensation rules, and tapped-delay-line (TDL) equalizer. It assumes noise additivity and validity of linear superposition. and it bases calculations on a measured disk/electronics noise spectrum and digitized isolated transition readback signals from the data track and adjacent tracks. Details of the model are described, and illustrative applications to RLL (d,k) code selection and pulse slimming equalizer design for a specific channel are discussed.

The power spectral density (PSD) is the average power per unit frequency of encoded random data transmitted over a perfect channel. The one-sided PSDs of a number of channel codes of recent interest in digital magnetic recording are calculated from codeword dictionaries and state diagrams. Given here are:

A few new modulation codes(or run-length-limited codes) in digital magnetic recording is presented. Linear density limits of the new codes and the existing codes are evaluated for a typical recording channel. One of the new codes shows the highest linear density limit among the existing codes and is improved over, for example, MFM and 4/5-rate-NRZI codes by about 20% to 30% in the density limit for the channel. A method for constructing the new codes is presented. Copyright © 1976 by The Institute of Electrical and Electronics Engineers, Inc.

It is proven that 100-percent efficient fixed-rate codes for run-length-limited (RLL) (d,k) and RLL charge-constrained (d, k; c) channels are possible in only two eases, namely (d,k; c)=(0,1;1) and (1,3;3) . Specifically, the binary Shannon capacity of RLL (d, k) constrained systems is shown to be irrational for all values of (d, k),0 leq d < k . For RLL charge-constrained systems with parameters (d, k;c) , the binary capacity is irrational for all values of (d, k; c),0 leq d < k,2c geq k + 1 , except (0,1; 1) and (1,3;3) , which both have binary capacity 1/2 .

Method and Apparatus for Generating a Noiseless Sliding Block Code for a (1,7) Channel with Rate 2/3

- R Adler
- M Hassner
- . T Moussouris

R. Adler, M. Hassner, and. T. Moussouris, "Method and Apparatus for Generating a Noiseless Sliding Block Code for a (1,7) Channel with Rate 2/3." U.S. Patent 4,413,251, 1982.

Run-Length-Limited Codes

- K Norris

K. Norris, "Run-Length-Limited Codes." Xerox Disclosure J. 5, 647-648(1980).

Sequential Encoding and Decoding of Variable Length. Fixed Rate Data Codes

- J Eggenberger
- P Hodges

J. Eggenberger and P, Hodges, "Sequential Encoding and Decoding of Variable Length. Fixed Rate Data Codes," U.S. Patent 4,115,768, 1978.

Data Encoding Method and System Employing Two-Thirds Rate Code with Full Word Look-Ahead

- M Cohn
- G Jacoby
- A Bates

M. Cohn, G. Jacoby, and A. Bates III, "Data Encoding Method and System Employing Two-Thirds Rate Code with Full Word Look-Ahead," U.S. Patent 4,337,458, 1982.

Efficient Code for Digital Magnetic Recording

- P Franaszek

P. Franaszek, "Efficient Code for Digital Magnetic Recording," IBM Tech. Disclosure Bull. 23, 4375-4378 (1981).

Run-Length-Limited Variable Length Coding with Error Propagation Limitation

- P Franaszek

P. Franaszek, "Run-Length-Limited Variable Length Coding with Error Propagation Limitation," U.S. Patent 3,689,899, 1972.

Algorithms for Sliding Block Codes Theory <b>IT-29<

- R Adler
- D Coppersmith
- M Hassner