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Encoding of dklr-sequences using one weight set

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Abstract

Traditional schemes for encoding and decoding runlength-constrained sequences using the enumeration principle require two sets of weighting coefficients. A new enumeration is presented requiring only one set of coefficients

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... Severe error propagation resulting from the use of long codewords can be avoided by reversing the conventional hierarchy 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 integervalued weights used in forming this sum are a function of the channel constraints in force. ...
... 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 index in set . The inverse rank has the virtue that the 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. ...
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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
... This includes all fixed-rate codes with finite input and output block lengths. Two existing near-optimal, fixed-rate encoding techniques are based on enumerative coding [10],[19],[43] and arithmetic coding [31],[58] . Some capacity-achieving variablerate codes are outlined in [3],[6],[27]. ...
... Chap.6] for a summary,[10],[43],[19] for more details), which has been shown to achieve very high encoding rates that approach capacity with increasing codeword length n. However, the disadvantages of enumerative coding are bitwise encoding and decoding ; and additions and comparisons with pre-stored, n-bit weighting coefficients. ...
Article
Run-Length-Limited (RLL) channels are found in digital recording systems like the Hard Disk Drive (HDD), Compact Disc (CD), and Digital Versatile Disc (DVD). This thesis presents novel encoding algorithms for RLL channels based on a simple technique called bit stuffing. First, two new capacity-achieving variable-rate code constructions are proposed for (d,k) constraints. The variable-rate encoding ideas are then extended to (0,G/I) and other RLL constraints. Since variable-rate codes are of limited practical value, the second half of this thesis focuses on fixed-rate codes. The fixed-rate bit stuff (FRB) algorithm is proposed for the design of simple, high-rate (0,k) codes. The key to achieving high encoding rates with the FRB algorithm lies in a novel, iterative pre-processing of the fixed-length input sequence prior to bit stuffing. Detailed rate analysis for the proposed FRB algorithm is presented, and upper and lower bounds on the asymptotic (in input block length) encoding rate are derived. Several system-level issues of the proposed FRB codes are addressed, and FRB code parameters required to design rate 100/101 and rate 200/201 (0,k) codes are tabulated. Finally, the proposed fixed-rate encoding is extended to (0,G/I) constraints. Ph.D. Committee Chair: McLaughlin, Steven; Committee Member: Barnwell, Thomas; Committee Member: Barry, John; Committee Member: Fekri, Faramarz; Committee Member: Tetali, Prasad
... An alternative of the above enumeration scheme can be given by counting the number of elements of that have a lexicographic index higher than , the inverse rank of . The inverse rank of is given by (2) where , the complement of In the next section, taken from [16], we will apply the above theory to the enumeration of sequences. We will first give an algorithm for enumerating -constrained sequences for which the length of the leading zero-run is not constrained. ...
... [18] we know that the sequence of the mantissa of will ultimately become (and remain) periodic. That is, there are integers and such that (15) In other words, per-cycle period of length , the number of sequences increases with a fixed factor, which is equal to a power of two From the above it is immediate that (16) The theory of feedback registers [18] stipulates that the cycle period must be smaller than As this number is huge in the range of parameters of practical interest, we are inclined to believe that Proposition 2 is not of great practical interest. However, results of a computer search, which are listed in Table II, reveal that relatively small cycle periods are surprisingly frequent. ...
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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
... Enumerative coding makes it possible to translate source words into codewords and vice versa by invoking an algorithm rather than performing the translation with a look-up table. In [5], a method is described that requires storage capacity of a bank of n integer coefficients. The algorithm itself is conceptually very simple. ...
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Runlength-limited (RLL) codes have found widespread usage in optical and magnetic recording products. Specifically, the RLL codes EFM and its successor, EFMPlus, are used in the compact discs (CD) and the digital versatile discs (DVD), respectively. EFMPlus offers a 6% increase in storage capacity with respect to EFM. The work reports on the feasibility and limits of EFM like codes that offer an even larger capacity. To this end, we provide an overview of the various limiting factors, such as runlength constraint, dc-content, and code complexity, and outline their relative effect on the code rate. In the second part of the article we show how the performance predicted by the tenets of information theory can be realized in practice. A worked example of a code whose rate is 7.5% larger than EFMPlus, namely a rate 256/476, (d=2, k=15) code, showing a 13 dB attenuation at f<sub>b</sub>=10<sup>-3 </sup>, is given to illustrate the theory
... Extremely large values of p and q are encountered in the design of high-rate codes, or when the efficiency of the code should be very high. In such cases, methods as guided scrambling [38], or a promising variant of enumerative encoding ( [39], [40], see also [34], Chapter 1) should be considered. Here, the extremely large block length imposes a special system architecture to limit error propagation. ...
Chapter
Modulation codes such as runlength-limited codes have been widely employed in magnetic and optical data storage systems. We review the main techniques involved in the design and use of these codes: the maximal code rate or capacity, graphical presentations of constraints, encoders and decoders, and code construction methods such as the ACH state-splitting algorithm. We conclude this survey by discussing some recent developments and research trends.
... The construction is designed to preserve the advantages of fixed-length RLL codes such as: a) the translation of source words into codewords can be accomplished with a single look-up table; b) the cascading of codewords can be done with a simple merging rule. Although Immink [5] has presented a new version of the coding-decoding algorithm that uses a single new set of weighting coefficients n s , he did not provide also a method for determining the value of those integer coefficients. The proposed algorithm could be used for the computing the n s coefficients. ...
Article
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The construction procedure for (d,k,l,r)-sequences by traditional methods (based on the enumeration principle) requires two sets of weighting coefficients. Based on a set of parameters and recursive relationships, the proposed algorithm with just one set of weighting coefficients is presented. A new formula to determine the number of the messages permitted on constrained channels is introduced.
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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
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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
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Introduction In channel coding schemes we are usually faced with the problem of translating a given source word into another codeword and vice versa that satisfies some prescribed constraints. In the absence of an algorithmic rule defining the relationship between the source word and codeword, the translation operation will be simple look-up tables. As hardware grows with the number of codewords used, i.e. exponentially with the codeword length, there is a technological limit to the length of the words that can be translated using such a simple look-up table. A preferable alternative technique, called enumerative coding, makes it possible to perform the translation byinvoking an algorithmic procedure [1]. Essentially,enumerative decoding is accomplished by forming the weighted sum of the codeword received. The integer-valued weights used in forming the sum are a function of the channel constraints in force. Encoding is done by a method which is
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Many modulation systems used in magnetic and optical recording are based on binary run-length-limited codes. We generalize the concept of dk -limited sequences of length n introduced by Tang and Bald by imposing constraints on the maximum number of consecutive zeros at the beginning and the end of the sequences. It is shown that the encoding and decoding procedures are similar to those of Tang and Bald. The additional constraints allow a more efficient merging of the sequences. We demonstrate two constructions of run-length-limited codes with merging rules of increasing complexity and efficiency and compare them to Tang and Bahl's method.
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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.
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The paper describes a technique for constructing fixed-length block codes for (d, k)-constrained channels. The codes described are of the simplest variety-codes for which the encoder restricted to any particular channel state is a one-to-one mapping and which is not permitted to “look ahead” to future messages. Such codes can be decoded with no memory and no anticipation and are thus an example of what Schouhamer Immink (1992) has referred to as block-decodable. For a given blocklength n and given values of (d, k), the procedure constructs a code with the highest possible rate among all such block codes, and it does so without the iterative search that is typically used (i.e., Franaszek's recursive elimination algorithm). The technique used is similar to Beenker and Immink's (1983) “Construction 2” in that every message is associated with a (d, k, l, r) sequence of length n-d; however the values used in the present approach are l=k-d and r=k-1, as opposed to Beenker and Schouhamer Immink's values of l=r=k-d. Thus the present approach demonstrates that “Construction 2” is optimal for d=1 but is suboptimal for d>1. Furthermore, the structure of the present codes permits enumerative coding techniques to simplify encoding and decoding