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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.

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... The code also needs to be efficient in the sense that it is easy to code a source word and easy to decode a coded word back to its source word. The information presented in this section was taken mainly from [4] and [5]. ...

... Of course, it could also be interesting to discuss other aspects of the signal such as its spectral content. For example, the effect of line coding on the spectral content of a signal is discussed in [5]. This however, is out of the scope of this paper. ...

This paper describes the techniques used in demod-ulation and error correction. It presents where demodulation and error correction occur in the transmission-reception chain. It will also explain techniques used for forward error correction. A brief note about hardware implementation of these codes will also be made.

... If p value is small, it means that system works well and synchronous ability is powerful [15] [16] . ...

In digital storage system, the aim of channel coding is to improve efficiency and reliability of the channel. In order to improve the capacity and quality for optical storage system, the study on the modulation code is significant. A new RLL (2, 12; 8, 15) run-length limited code is presented. The construction method of the code is discussed and the procedures in encoding and decoding of the code are also given. The major characteristic parameters of different run-length limited codes are compared; the decoder is implemented based on FPGA. The RLL(2, 12; 8, 15) code with high modulation rate, namely high encoding efficient, is fit for high density optical storage systems. The study of modulation code for high-density optical discs with independent intellectual property rights is also of great significance.

... 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 with probability less than . ...

We report on a class of high-rate de-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.

Balanced sequences and balanced codes have attracted a lot of research in the last seventy years due to their diverse applications in information theory as well as other areas of computer science and engineering. There have been some methods to classify balanced sequences. This work suggests two new different hierarchies to classify these sequences. The first one is based on the largest
$\ell $
for which each
$\ell $
-tuple is contained the same amount of times in the sequence. This property is a generalization for the property required for de Bruijn sequences. The second hierarchy is based on the number of balanced derivatives of the sequence. Enumeration for each such family of sequences and efficient encoding and decoding algorithms are provided in this paper.

We present and analyze a new construction of bipolar balanced codes where each codeword contains equally many -1’s and +1’s. The new code is minimally modified as the number of symbol changes made to the source word for translating it into a balanced codeword is as small as possible. The balanced codes feature low redundancy and time complexity. Large look-up tables are avoided.

We present and analyze a new systematic construction of bipolar balanced codes where each code word contains equally many −1's and +1's. The new code is minimally modified as the number of symbol changes made to the source word for translating it into a balanced code word is as small as possible. The balanced codes feature low redundancy and time complexity. Large look-up tables are avoided.

Our research deals with the encoding and decoding of balanced sequences using Gray
codes. Given that any non-binary sequence, can always be balanced through certain
algorithms, we show that the encoding and decoding of a balanced sequence can be
performed through a simple and efficient method where the prefix is a Gray code. Our
balancing scheme makes use of a generalization of Knuth's balancing algorithm, per-
formed on the overall sequence length which includes the information sequence as well
as the designed prefix. Our proposed method was firstly applied to certain information
source lengths and then generalized for any length.
We conclude with a detailed complexity and redundancy analysis for our balancing
algorithm.

Summary One of the simplest means of inducing a spectral null at DC is the use of a polarity bit code (PBC), as devised by Bowers and Carter. (See [I] for details.) In polarity bit encoding k = n - 1 bits enter the encoder and a single “1” is appended to the string; then either the resulting n-tuple OT its complement is transmitted. The n-tuple selected is the one that will result in the smallest absolute charge accumulation after selection. (Recall: The accumulated charge at a point in a binary sequence is the total number of 1’s that precede the point minus the total number of 0’s; it is also referred to as the running digital sum (RDS).) It is easy to see that this technique bounds the charge’s absolute value to no more than 3n/2 and so induces the spectral null. (At the decoder the receiver knows whether or not to invert the received n-tuple by looking at the last bit.) The polarity bit code is a simple example of what we call a trellis constrained DC-free block code. In such a code the state of the encoder is equal to the running digital sum prior to codeword selection; the codebook associated with a transition from any state is designed so as to bound the RDS. For the polarity bit code there are two codebooks: Those binary n-tuples with Hamming weight less than n/2 and those with Hamming weight more than n/2. (When n is even, half of the n-tuples with weight n/2 go into each codebook.) Every transition from a state representing a positive RDS uses the first codebook, while every transition from a state corresponding to a negative RDS takes a codeword from the second codebook. The polarity bit code was designed solely to provide a bounded RDS, so its error correcting capabilities are minimal.

Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature. I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to
avoid introducing new errors in the Second Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink
Rotterdam, November 2004

Since the early 1980s we have witnessed the digital audio and video revolution: the Compact Disc (CD) has become a commodity audio system. CD-ROM and DVD-ROM have become the de facto standard for the storage of large computer programs and files. Growing fast in popularity are the digital audio and video recording systems called DVD and BluRay Disc. The above mass storage products, which form the backbone of modern electronic entertainment industry, would have been impossible without the usage of advanced coding systems.
Pulse Code Modulation (PCM) is a process in which an analogue, audio or video, signal is encoded into a digital bit stream. The analogue signal is sampled, quantized and finally encoded into a bit stream. The origins of digital audio can be traced as far back as 1937, when Alec H. Reeves, a British scientist, invented pulse code modulation \cite{Ree}. The advantages of digital audio and video recording have been known and appreciated for a long time. The principal advantage that digital implementation confers over analog systems is that in a well-engineered digital recording system the sole significant degradation takes place at the initial digitization, and the quality lasts until the point of ultimate failure. In an analog system, quality is diminished at each stage of signal processing and the number of recording generations is limited. The quality of analog recordings, like the proverbial 'old soldier', just fades away. The advent of ever-cheaper and faster digital circuitry has made feasible the creation of high-end digital video and audio recorders, an impracticable possibility using previous generations of conventional analog hardware.
The general subject of coding for digital recorders is very broad, with its roots deep set in history. In digital recording (and transmission) systems, channel encoding is employed to improve the efficiency and reliability of the channel. Channel coding is commonly accomplished in two successive steps: (a) error-correction code followed by (b) recording (or modulation) code. Error-correction control is realized by adding extra symbols to the conveyed message. These extra symbols make it possible for the receiver to correct errors that may occur in the received message.
In the second coding step, the input data are translated into a sequence with special properties that comply with the given "physical nature" of the recorder. Of course, it is very difficult to define precisely the area of recording codes and it is even more difficult to be in any sense comprehensive. The special attributes that the recorded sequences should have to render it compatible with the physical characteristics of the available transmission channel are called channel constraints. For instance, in optical recording a '1' is recorded as pit and a '0' is recorded as land. For physical reasons, the pits or lands should neither be too long or too short. Thus, one records only those messages that satisfy a run-length-limited constraint. This requires the construction of a code which translates arbitrary source data into sequences that obey the given constraints. Many commercial recorder products, such as Compact Disc and DVD, use an RLL code.
The main part of this book is concerned with the theoretical and practical aspects of coding techniques intended to improve the reliability and efficiency of mass recording systems as a whole. The successful operation of any recording code is crucially dependent upon specific properties of the various subsystems of the recorder. There are no techniques, other than experimental ones, available to assess the suitability of a specific coding technique. It is therefore not possible to provide a cookbook approach for the selection of the 'best' recording code.
In this book, theory has been blended with practice to show how theoretical principles are applied to design encoders and decoders. The practitioner's view will predominate: we shall not be content with proving that a particular code exists and ignore the practical detail that the decoder complexity is only a billion times more complex than the largest existing computer. The ultimate goal of all work, application, is never once lost from sight. Much effort has been gone into the presentation of advanced topics such as in-depth treatments of code design techniques, hardware consequences, and applications. The list of references (including many US Patents) has been made as complete as possible and suggestions for 'further reading' have been included for those who wish to pursue specific topics in more detail.
The decision to update Coding Techniques for Digital Recorders, published by Prentice-Hall (UK) in 1991, was made in Singapore during my stay in the winter of 1998. The principal reason for this decision was that during the last ten years or so, we have witnessed a success story of coding for constrained channels. The topic of this book, once the province of industrial research, has become an active research field in academia as well. During the IEEE International Symposia on Information Theory (ISIT and the IEEE International Conference on Communications (ICC), for example, there are now usually three sessions entirely devoted to aspects of constrained coding. As a result, very exciting new material, in the form of (conference) articles and theses, has become available, and an update became a necessity.
The author is indebted to the Institute for Experimental Mathematics, University of Duisburg-Essen, Germany, the Data Storage Institute (DSI) and National University of Singapore (NUS), both in Singapore, and Princeton University, US, for the opportunity offered to write this book. Among the many people who helped me with this project, I like to thank Dr. Ludo Tolhuizen, Philips Research Eindhoven, for reading and providing useful comments and additions to the manuscript.
Preface to the Second Edition
About five years after the publication of the first edition, it was felt that an update of this text would be inescapable as so many relevant publications, including patents and survey papers, have been published. The author's principal aim in writing the second edition is to add the newly published coding methods, and discuss them in the context of the prior art. As a result about 150 new references, including many patents and patent applications, most of them younger than five years old, have been added to the former list of references. Fortunately, the US Patent Office now follows the European Patent Office in publishing a patent application after eighteen months of its first application, and this policy clearly adds to the rapid access to this important part of the technical literature.
I am grateful to many readers who have helped me to correct (clerical) errors in the first edition and also to those who brought new and exciting material to my attention. I have tried to correct every error that I found or was brought to my attention by attentive readers, and seriously tried to avoid introducing new errors in the Second
Edition.
China is becoming a major player in the art of constructing, designing, and basic research of electronic storage systems. A Chinese translation of the first edition has been published early 2004. The author is indebted to prof. Xu, Tsinghua University, Beijing, for taking the initiative for this Chinese version, and also to Mr. Zhijun Lei, Tsinghua University, for undertaking the arduous task of translating this book from English to Chinese. Clearly, this translation makes it possible that a billion more people will now have access to it.
Kees A. Schouhamer Immink, Rotterdam, November 2004

Codes C(m, r) of length 2
m
over 1, -1 are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of C(m, r) all have an rth order spectral null at zero frequency. Establishing the connection between C(m, r) and the parity-check matrix of Reed-Muller codes, the minimum distance of C(m, r) is obtained along with upper bounds on the redundancy of C(m, r). An efficient algorithm is presented for encoding unconstrained binary sequences into C(m, 2).

Codes \({\mathcal{C}}(m,r)\) of length 2m
over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of \({\mathcal{C}}(m,r)\) all have an rth order spectral null at zero frequency. Establishing the connection between \({\mathcal{C}}(m,r)\) and the parity-check matrix of Reed-Muller codes, the minimum distance of \({\mathcal{C}}(m,r)\) is obtained along with upper bounds on the redundancy of \({\mathcal{C}}(m,r)\). An efficient algorithm is presented for encoding unconstrained binary sequences into \({\mathcal{C}}(m,r)\).

Constant-weight binary sequences with constrained run lengths of zeros and ones are introduced. These run-length constraints
are separate and independent. Using the Babkin-Cover enumerative scheme, the number of these sequences is found. Then enumeration-based
encoding and decoding procedures are constructed.

Typescript (photocopy). Thesis (M.S.)--Oregon State University, 2001. Includes bibliographical references (leaves 41-43).

Let T(n,k) denote the set of all words of length n over the alphabet {+1, -1}, having a kth order spectral-null at zero frequency. A subset of T(n,k) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of T(n,k) are derived. In particular we prove that (k - 1) log2 (n/k) less-than-or-equal-to n - log2 \T(n,k)\ less-than-or-equal-to O(2k log2 n) for infinitely many values of n. On the other hand, we show that T(n,k) is empty unless n is divisible by 2m, where m = left-perpendicularlog2 kright-perpendicular + 1. Furthermore, bounds on the minimum Hamming distance d of T(n,k) are provided, showing that 2k less-than-or-equal-to d less-than-or-equal-to k(k - 1) + 2 for infinitely many n. We also investigate the minimum number of sign changes in a word x is-an-element-of T(n,k) and provide an equivalent definition of T(n,k) in terms of the positions of these sign changes. An efficient algorithm for encoding arbitrary information sequences into a second-order spectral-null code of redundancy 3 log2 n + O(log log n) is presented. Furthermore, we prove that the first nonzero moment of any word in T(n,k) is divisible by k! and then show how to construct a word with a spectral null of order k whose first nonzero moment is any even multiple of k!. This leads to an encoding scheme for spectral-null codes of length n and any fixed order k, with rate approaching unity as n --> infinity.

Volume holographic memories (VHM) are page-oriented optical
storage systems whose pages commonly contain on the order of one million
pixels. Typically, each stored data page is composed of an equal number
of binary pixels in either a low-contrast (“off”) state or a
high-contrast (“on”) state. By increasing the number of
“off” pixels and decreasing the number of “on”
pixels per page, there is an associated gain in VHM system storage
capacity. When grayscale pixels are used, a further gain is possible by
similarly controlling the fraction of pixels at each gray level. This
paper introduces a constant-weight, nonbinary, shortened enumerative
permutation modulation block code to produce pages that exploit the
proposed capacity advantage. In addition to the code description, we
present an encoder and a low-complexity maximum-likelihood (ML) decoder
for the shortened permutation code. A proof verifies our claim of ML
decoding. Applying this class of code to VHMs predicts a 49% increase in
storage capacity when recording modulation coded 3-bit (eight gray
level) pixels compared with a VHM using a binary signaling alphabet and
equal-probable (unbiased) data

A new construction of direct current (DC)-free error-correcting
codes based on convolutional codes is proposed. The new code is
constructed by selecting a proper subcode from a convolutional code
composed of two different component codes. The encoder employs a Viterbi
algorithm as the codeword selector so that the selected code sequences
satisfy the DC constraint. A lower bound on the free distance of such
codes is proposed, and a procedure for obtaining this bound is
presented. A sufficient condition for these codes to have a bounded
running digital sum (RDS) is proposed. Under the assumption of a
simplified codeword selection algorithm, we present an upper bound on
the maximum absolute value of the RDS and derive the sum variance for a
given code. A new construction of standard DC-free codes, i.e., DC-free
codes without error-correcting capability, is also proposed. These codes
have the property that the decoder can be implemented by simple
symbol-by-symbol hard decisions. Finally, under the new construction, we
propose several codes that are suitable for the systems that require
small sum variance and good error-correction capability

The sound from a Compact Disc system encoded into data bits and modulated into channel bits is sent along the 'transmission channel' consisting of write laser - master disk - user disk - optical pick-up. The maximum information density on the disk is determined by the diameter d of the laser light spot on the disk and the 'number of data bits per light spot'. The effect of making d smaller is to greatly reduce the manufacturing tolerances for the player and the disk. The compromise adopted is d approximately equals 1 mu m, giving very small tolerances for objective disk tilt, disk thickness and defocusing.

In digital transmission it is sometimes desirable for the channel stream to have low power near zero frequency. Suppression of the low-frequency components is achieved by constraining the unbalance of the transmitted positive and negative pulses. Rate and spectral properties of unbalance constrained codes with binary symbols based on simple bi-mode coding schemes are calculated.

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.

The encoding of independent data symbols as a sequence of discrete amplitude, real variables with given power spectrum is considered. The maximum rate of such an encoding is determined by the achievable entropy of the discrete sequence with the given constraints. An upper bound to this entropy is expressed in terms of the rate distortion function for a memoryless finite alphabet source and mean-square error distortion measure. A class of simple dc-free power spectra is considered in detail, and a method for constructing Markov sources with such spectra is derived. It is found that these sequences have greater entropies than most codes with similar spectra that have been suggested earlier, and that they often come close to the upper bound. When the constraint on the power spectrum is replaced by a constraint On the variance of the sum of the encoded symbols, a stronger upper bound to the rate of dc-free codes is obtained. Finally, the optimality of the binary biphase code and of the ternary bipolar code is decided.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

The stochastic process appearing at the output of a digital encoder is investigated. Based upon the statistics of the code being employed, a systematic procedure is developed by means of which the average power spectral density of the process can be determined. The method is readily programmed on the digital computer, facilitating the calculation of the spectral densities for large numbers of codes. As an example of its use, the procedure is applied in the case of a specific multi-alphabet, multi-level code.

The pages of this expensive but invaluable reference work are dense with formulae of stupefying complexity. Chapters 1 and 2 treat definite/indefinite integral properties of a great variety of special functions, Chapters 3 and 4 (which are relatively brief) treat definite integrals of some piece-wi

The role of line coding is to convert source data to a digital form resistant to noise in combination with such other impairments as a specific medium may suffer (notably intersymbol interference, digit timing jitter and carrier phase error), while being reasonably economical in the use of bandwidth. This paper discusses the nature and role of various constraints on code words and word sequences, including those commonly used on metallic lines, optical fibres, carrier channels and radio links ; and gives some examples from each of these applications. It should serve both as a general review of the subject and as an introduction to the companion papers on specific topics.

Block codes are multilevel codes of constant length whose codewords are the channel encodings of binary sequences of constant length. The spectral density calculation of pulse trains resulting from the use of these codes is dealt with. Under the assumption that the encoding be represented by a finite-state synchronous sequential machine, by means of homogenous Markov chain theory, the spectral density is evaluated in closed form. Both the continuous and the discrete spectral components are easily obtained as soon as the encoder functions are specified. The result obtained is particularly attractive since the finite-state machine model can be used to represent a broad class of encoding schemes of engineering interest. As an example, the spectral density is calculated in the case of the Franaszek MS-43 code.

In digital transmission of binary (+1,-1) signals it is desirable that the stream of pulses which constitutes the signal have no dc, that is, that the power spectrum go to zero at zero frequency. It is desirable that, for a given efficiency or entropy, the spectrum rise slowly with increasing frequency. We have obtained the spectrum for selected blocks with equal numbers of plus ones and minus ones. For a given efficiency, this is better than the spectrum obtained by Rice, using the Monte Carlo method, for block encoding using polarity pulses. An algorithm given by Schalwijk should allow simple encoding into selected blocks.

It is difficult to record DC and low-frequency signals with the magnetic recording method. Furthermore, in high-density magnetic recording, the signal-to-noise ratio and the high-frequency output level are low and low-frequency crosstalk noise from the adjacent tracks is relatively high. A modulation code for high-density digital magnetic recording must have a large Tw and a DC-free characteristic. When we developed the R-DAT system, we developed two run-length limited 8/10 conversion rate modulation codes for use with the R-DAT. This paper discusses various possible modulation codes and confirms the superiority of one particular 8/10 modulation code.

When alphabets of digital symbols are used to represent information for data processing, storage, and transmission, redundancy in the alphabets is traditionally used for the purpose of error compensation. This paper deals with alphabets of redundant codes, both binary and higher level, where the emphasis is on using redundancy to produce code alphabets with unique properties in their frequency spectra that can be exploited in the design of the system in which they are used. In particular, techniques are presented for synthesizing alphabets that produce spectral nulls at frequencies 1/kT, where T is the duration of a word element. Some of the interesting alphabets are a 10-word, 5-bit alphabet with spectrum zero at 1/2T; a 10-word, 6-bit alphabet with spectrum zero at 1/3T; a 36-word, 8-bit alphabet with zero at 1/4T; and a 36-word, 8-bit alphabet with zeros at both 0 and 1/2T.

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).

Coding schemes in which each codeword contains equally many zeros and ones are constructed in such a way that they can be efficiently encoded and decoded.

A practical method is described for encoding an unrestricted binary signal into a form suitable for transmission through a binary regenerated signal path while incurring only a small increase in modulation rate.

The coding system described minimises the d.c. component in the transmitted binary signal by inverting characters when this will reduce the accumulated mark and space disparity. An added digit indicates when inversion has taken place. Conversion to and from unrestricted binary coding is relatively simple.

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.

In pulse-amplitude modulation (PAM) digital transmission systems line encoding is used for shaping the spectrum of the encoded symbol sequence to suit the frequency characteristics of the transmission channel. In particular, it is often required that the encoded symbol sequence have a zero mean and spectral density vanishing at zero frequency. We show that the finite running digital sum condition is a necessary and sufficient condition for this to occur. The result holds in particular for alphabetic codes, which are the most widely used line codes.

Let S be a given subset of binary n-sequences. We provide an explicit scheme for calculating the index of any sequence in S according to its position in the lexicographic ordering of S . A simple inverse algorithm is also given. Particularly nice formulas arise when S is the set of all n -sequences of weight k and also when S is the set of all sequences having a given empirical Markov property. Schalkwijk and Lynch have investigated the former case. The envisioned use of this indexing scheme is to transmit or store the index rather than the sequence, thus resulting in a data compression of (logmidSmid)/n .

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

For n >0, d ⩾0, n ≡ d
(mod 2), let K ( n , d ) denote the minimal
cardinality of a family V of ±1 vectors of dimension
n , such that for any ±1 vector w of dimension
n there is a v ∈ V such that | v -
w |⩽ d , where v - w is the usual
scalar product of v and w . A generalization of a
simple construction due to D.E. Knuth (1986) shows that K ( n
, d )⩽[ n /( d +1)]. A linear algebra
proof is given here that this construction is optimal, so that
K ( n , d )-[ n /( d +1)] for all
n ≡ d (mod 2). This construction and its
extensions have applications to communication theory, especially to the
construction of signal sets for optical data links

Brychkov, and 0. I. Marichev, Integrals and series

- A P Prudnikov

A. P. Prudnikov, Yu. A. Brychkov, and 0. I. Marichev, Integrals and series. Gordon and Breach Science Publishers, vol. 1, no. 19, p. 613.

Efficient balanced codes Zero disparity coding system

- D E Knuth

D. E. Knuth, " Efficient balanced codes, " IEEE Trans. Inform. Theory, vol. IT-32, no. 1, pp. 51-53, Jan. 1986. See also P. S. Henry, " Zero disparity coding system, " US. Patent 4,309,694, Jan. 1982.

Balancing sets of vectors IEEE Tram Performance of simple binary dc-constrained codes

- N Alon
- E E Bergmann
- D Coppersmith
- A M Odlyzko

N. Alon, E. E. Bergmann, D. Coppersmith, and A. M. Odlyzko, " Balancing sets of vectors, " IEEE Tram. Inform. Theory, vol. 34, no. 1, pp. 128-130, Jan. 1988. [I21 K. A. S. Immink, " Performance of simple binary dc-constrained codes, " Philips J. Res., vol. 40, pp. 1-21, 1985.

Zero disparity coding system

- P S Henry

Software for the development of sliding block encoders for modulation codes

- J Tang

J. Tang, "Software for the development of sliding block encoders
for modulation codes." MS. thesis. Univ. of California. San Diego, -,
1987.

Method and apparatus for generating a noiseless sliding block code for a (1,7) channel with rate 2/3

- R Adler
- M Hassner
- J Moussouris

R. Adler, M. Hassner, and J. 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.

Data encoding method and system employing two-thirds rate code with full word look-ahead

- G Jacoby
- M Cohn
- A Bates

G. Jacoby, M. Cohn, and A. Bates, "Data encoding method and
system employing two-thirds rate code with full word look-ahead,"
U.S. patent 4,337,458, 1982.

Constrained codes for PRML

- B Marcus
- P Siegel

B. Marcus and P. Siegel, "Constrained codes for PRML," IBM
Res. Rep. RJ 4371, July 1984.