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# Construction of Maximum Run-Length Limited Codes Using Sequence Replacement Techniques

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## Abstract

The sequence replacement technique converts an input sequence into a constrained sequence in which a prescribed subsequence is forbidden to occur. Several coding algorithms are presented that use this technique for the construction of maximum run-length limited sequences. The proposed algorithms show how all forbidden subsequences can be successively or iteratively removed to obtain a constrained sequence and how special subsequences can be inserted at predefined positions in the constrained sequence to represent the indices of the positions where the forbidden subsequences were removed. Several modifications are presented to reduce the impact of transmission errors on the decoding operation, and schemes to provide error control are discussed as well. The proposed algorithms can be implemented efficiently, and the rates of the constructed codes are close to their theoretical maximum. As such, the proposed algorithms are of interest for storage systems and data networks.

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... Then, in Section III, we present the main contribution of this work, algorithms for translating arbitrary binary source data into k-constrained q-ary data. Among the three code design methods we describe, the second method removes forbidden substrings of q-ary sequences by using a recursive, 'sequence replacement', method yielding a significant improvement in coding redundancy than the prior art binary sequence replacement method [9]. In the third method, standard binary maximum runlength limited sequences are transformed into maximum runlength limited q-ary sequences using two simple steps of precoding, which opens the door to using the vast prior art binary code constructions to DNA-based storage. ...
... The three sequence replacement techniques published by Wijngaarden et al. [9] are recursive methods for removing forbidden substrings from a binary source word. The encoder removes the forbidden substrings, and the positions of the forbidden substrings are encoded as binary pointer words, and subsequently inserted at predefined positions of the codeword. ...
... Very efficient constructions of binary k ′ -constrained codes that avoid long repetitions of a 'zero' have been published in the literature, see, for example, the survey in [9]. We show that after applying a simple coding step to a k ′ -constrained binary sequence, we obtain a strand of nucleotides, where the length of a homopolymer run is at most m = ...
Article
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We consider coding techniques that limit the lengths of homopolymer runs in strands of nucleotides used in DNA-based mass data storage systems. We compute the maximum number of user bits that can be stored per nucleotide when a maximum homopolymer runlength constraint is imposed. We describe simple and efficient implementations of coding techniques that avoid the occurrence of long homopolymers, and the rates of the constructed codes are close to the theoretical maximum. The proposed sequence replacement method for k-constrained q-ary data yields a significant improvement in coding redundancy than the prior art sequence replacement method for the k-constrained binary data. Using a simple transformation, standard binary maximum runlength limited sequences can be transformed into maximum runlength limited q-ary sequences, which opens the door to applying the vast prior art binary code constructions to DNA-based storage.
... RLL codes have been applied in practice to various data storage devices, including virtually all magnetic and optical disc recording systems [14]. Over the years, different construction schemes for RLL codes, with varied enhancements, have been proposed and analyzed [15]- [18]. Study of RLL codes has continued to be an important research topic, and recent work includes its application to high density data storage [19], DNA-based storage [20], and visible light communication [21]. ...
... and w = ⌈LB⌉ belongs to the feasible set A Emax SEC (B), and the proof is complete using (18). ...
... Over the years, different construction approaches with varied enhancements have been proposed for RLL codes [15]- [18]. We remark here that SEC codes are also amenable to efficient implementation via concatenation [35], where the inner code is a heavy weight code [36] and the outer code is a high rate code over large alphabet, such as a Reed-Solomon code [37]. ...
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Run-length limited (RLL) codes are a well-studied class of constrained codes having application in diverse areas such as optical and magnetic data recording systems, DNA-based storage, and visible light communication. RLL codes have also been proposed for the emerging area of simultaneous energy and information transfer, where the receiver uses the received signal for decoding information as well as for harvesting energy to run its circuitry. In this paper, we show that RLL codes are not the best codes for simultaneous energy and information transfer, in terms of the maximum number of codewords which avoid energy outage, i.e., outage-constrained capacity. Specifically, we show that sliding window constrained (SWC) codes and subblock energy constrained (SEC) codes have significantly higher outage-constrained capacities than RLL codes.
... RLL codes have been applied in practice to various data storage devices, including virtually all magnetic and optical disc recording systems [14]. Over the years, different construction schemes for RLL codes, with varied enhancements, have been proposed and analyzed [15]- [18]. Study of RLL codes has continued to be an important research topic, and recent work includes its application to high density data storage [19], DNA-based storage [20], and visible light communication [21]. ...
... and w = ⌈LB⌉ belongs to the feasible set A Emax SEC (B), and the proof is complete using (18). ...
... Over the years, different construction approaches with varied enhancements have been proposed for RLL codes [15]- [18]. We remark here that SEC codes are also amenable to efficient implementation via concatenation [35], where the inner code is a heavy weight code [36] and the outer code is a high rate code over large alphabet, such as a Reed-Solomon code [37]. ...
Article
Full-text available
Run-length limited (RLL) codes are a well-studied class of constrained codes having application in diverse areas, such as optical and magnetic data recording systems, DNA-based storage, and visible light communication. RLL codes have also been proposed for the emerging area of simultaneous energy and information transfer, where the receiver uses the received signal for decoding information as well as for harvesting energy to run its circuitry. In this paper, we show that RLL codes are not the best codes for simultaneous energy and information transfer, in terms of the maximum number of codewords which avoid energy outage, i.e., outage-constrained capacity. Specifically, we show that sliding window constrained (SWC) codes and sub-block energy constrained (SEC) codes have significantly higher outage-constrained capacities than RLL codes for moderate to large energy buffer sizes.
... In many cases, the proposed techniques require careful selection of the order of repeat removals and involve a special encoding process for the repeats. Other replacement techniques were investigated in [16], [30], with the goal of imposing runlength or balancing constraints on a string. In these scenarios, removing offending substrings does not cause the introduction of other offending substrings, which makes the underlying problem solution simpler than repeat replacement. ...
... Next, we turn to the problem of designing an efficient encoder for an L-reconstruction code. Our constructive approach is inspired by techniques described in [25] and [30] for removing runs of zeros exceeding a certain length from arbitrary strings. Unlike the known runlength replacement strategy, our approach -repeat replacement -is iterative and it may lead to the creation of new repeats in already processed substrings. ...
Article
The problem of reconstructing strings from their substring spectra has a long history and in its most simple incarnation asks for determining under which conditions the spectrum uniquely determines the string. We study the problem of coded string reconstruction from multiset substring spectra, where the strings are restricted to lie in some codebook. In particular, we consider binary codebooks that allow for unique string reconstruction and propose a new method, termed repeat replacement, to create the codebook. Our contributions include algorithmic solutions for repeat replacement and constructive redundancy bounds for the underlying coding schemes. We also consider extensions of the problem to noisy settings in which substrings are compromised by burst and random errors. The study is motivated by applications in DNA-based data storage systems that use high throughput readout sequencers.
... The sequence replacement technique has been widely applied in the literature [9], [20]- [22]. This is an efficient method for removing forbidden substrings from a source word. ...
Article
We propose coding techniques that simultaneously limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given ℓ, ϵ > 0, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy all of the following properties: • Runlength constraint: the maximum homopolymer run in each codeword is at most ℓ, • GC-content constraint: the GC-content of each codeword is within [0.5 - ϵ; 0.5 + ϵ], • Error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. While various combinations of these properties have been considered in the literature, this work provides generalizations of codes constructions that satisfy all the properties with arbitrary parameters of ℓ and ϵ. Furthermore, for practical values of ℓ and ϵ, we show that our encoders achieve higher rates than existing results in the literature and approach capacity. Our methods have low encoding/decoding complexity and limited error propagation.
... II. DESIGN OF HIGH-RATE K CONSTRAINED CODES Among many works available in the literature on the design of k constrained codes [9], [10], [11], [12], [13], the nibble replacement method recently proposed in [14] achieves code rates higher than most of the literature, with simple encoders and decoders and limited error propagation. However, the nibble replacement method designs codes in the NRZI format whereas a change in the state of the recording medium corresponds to a channel bit '1', and no change corresponds to a '0'. ...
Article
This paper proposes systematic code design methods for constructing efficient spectrum shaping codes with the maximum runlength limited constraint k, which are widely used in data storage systems for digital consumer electronics products. Through shaping the spectrum of the input user data sequence, the codes can effectively circumvent the interaction between the data signal and servo signal in high-density data storage systems. In particular, we first propose novel methods to design high-rate k constrained codes in the non-return-to-zero (NRZ) format, which can not only facilitate timing recovery of the storage system, but also avoid error propagation during decoding and reduce the system complexity. We further propose to combine the Guided Scrambling (GS) technique with the k constrained code design methods to construct highly efficient spectrum shaping k constrained codes. Simulation results demonstrate that the designed codes can achieve significant spectrum shaping effect with only around 1% code rate loss and reasonable computational complexity.
... Then, the polarities of the extrinsic LLRs of the bit positions corresponding to the non-zero bits in q are reversed in the LLR adjuster before being passed to the decoder and the equalizer. Specifically, 6 6.2 6.4 6.6 6. (A1) Non-flipped system using a non-reset LDPC decoder; (A2) Non-flipped system using a reset LDPC decoder; (B1) Flipped system using the Soft-O method and a non-reset LDPC decoder; (B2) Flipped system using the Soft-O method and a reset LDPC decoder; (C1) Flipped system using the Soft-I method and a non-reset LDPC decoder; (C2) Flipped system using the Soft-I method and a reset LDPC decoder; (D1) Flipped system using the Soft-II method and a non-reset LDPC decoder; (D2) Flipped system using the Soft-II method and a reset LDPC decoder; (E1) Flipped system using the Soft-III method and a non-reset LDPC decoder; (E2) Flipped system using the Soft-III method and a reset LDPC decoder. ...
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In this paper, a low-density parity-check (LDPC) coded recording system is investigated, for which the run-length-limited (RLL) constraint is satisfied by deliberate flipping at the write side and by estimating the flipped bits at the read side. Two approaches are proposed for enhancing the error performance of such a system. The first approach is to alleviate the negative effect of incorrect estimation of the flipped bits by adjusting the soft information. The second approach is to increase the likelihood of the correct detection of flipped bits by designing a flipped-bit detection algorithm that utilizes both the RLL constraint and the parity-check constraint of the LDPC code. These two approaches can be combined to obtain significant improvement in performance over previously proposed methods.
... Kautz [4] was probably the first to present a simple algorithmic method, called enumerative encoding, for translating user words into -constrained codewords and vice versa. Wijngaarden and Immink presented various codes of rate , where subsequences that violate the maximum runlength are iteratively removed to obtain a -constrained sequence [5]. ...
Article
In this paper, we will present coding techniques for the character-constrained channel, where information is conveyed using q-bit characters (nibbles), and where w prescribed characters are disallowed. Using codes for the character-constrained channel, we present simple and systematic constructions of high-rate binary maximum runlength constrained codes. The new constructions have the virtue that large lookup tables for encoding and decoding are not required. We will compare the error propagation performance of codes based on the new construction with that of prior art codes.
... Example 1: Using the binary sequence replacement technique [23], we encode binary source sequences of length n ≤ 65 into binary codewords of length n + 1 that have a maximum runlength m = 6. Using a prefix of two bits plus one interfix bit, we balance the m-constrained word into nearly balanced (m = 6)-constrained word of length n + 4. ...
Preprint
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We describe properties and constructions of constraint-based codes for DNA-based data storage which accounts for the maximum repetition length and AT balance. We present algorithms for computing the number of sequences with maximum repetition length and AT balance constraint. We present efficient routines for translating binary runlength limited and/or balanced strings into DNA strands. We show that the implementation of AT-balanced codes is straightforwardly accomplished with binary balanced codes. We present codes that accounts for both the maximum repetition length and AT balance.
... One of the interesting approaches is the maximum RLL coding. Codes from this group are basically RLL(0,k) techniques which eliminate predefined, unwanted sequences from the output stream of symbols (Van Wijngaarden and Immink, 2010). (1,7) has coding rate of R = 2/3 and achieves better density rate of DR = 1.33. ...
Article
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Comprehensive (d,k) sequences study is presented, complemented with the design of a new, efficient, Run-Length Limited (RLL) code. The new code belongs to group of constrained coding schemas with a coding rate of R = 2/5 and with the minimum run length between two successive transitions equal to 4. Presented RLL (4, ∞) code uses channel capacity highly efficiently, with 98.7% and consequently it achieves a high-density rate of DR = 2.0. It is implying that two bits can be recorded, or transmitted with one transition. Coding techniques based on the presented constraints and the selected coding rate have better efficiency than many other currently used codes for high density optical recording and transmission.
... Hence, x ∈ B 4k (n). Now, we may modify the sequence replacement techniques [5], [20] to encode for the restricted-sum-balanced constraint. ...
Preprint
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An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this paper, we investigate codes that combat either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with log n + O(log log n) redundancy bits, while the other corrects a single indel with ⌈log n⌉ + 2 redundant bits. These two encoders are order-optimal. The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the GC-balanced constraint and require that exactly half of the symbols of any DNA codeword to be either C or G. In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into GC-balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of GC-balanced codes that correct a single indel or a single edit.
... Hence, x ∈ B 4k (n). Now, we may modify the sequence replacement techniques [5], [20] to encode for the restricted-sum-balanced constraint. ...
Preprint
Full-text available
An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this paper, we investigate codes that combat either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with log n + O(log log n) redundancy bits, while the other corrects a single indel with log n + 2 redundant bits. These two encoders are order-optimal. The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the GC-balanced constraint and require that exactly half of the symbols of any DNA codeword to be either C or G. In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into GC-balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of GC-balanced codes that correct a single indel or a single edit.
... The sequence replacement technique has been widely used in the literature [8], [15]- [17]. This is an efficient method for removing forbidden substrings from a source word. ...
Preprint
Full-text available
We propose coding techniques that limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given $\ell, {\epsilon} > 0$, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy the following properties: (i) Runlength constraint: the maximum homopolymer run in each codeword is at most $\ell$, (ii) GC-content constraint: the GC-content of each codeword is within $[0.5-{\epsilon}, 0.5+{\epsilon}]$, (iii) Error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. For practical values of $\ell$ and ${\epsilon}$, we show that our encoders achieve much higher rates than existing results in the literature and approach the capacity. Our methods have low encoding/decoding complexity and limited error propagation.
... We now use the sequence replacement technique to construct W(m, L, [p 1 L, p 2 L]) where L (1/c 2 ) log e n and c = min{1/2 − p 1 , p 2 − 1/2}. The sequence replacement technique has been widely used in the literature [17]- [20]. This is an efficient method for removing forbidden substrings from a source word. ...
Preprint
The subblock energy-constrained codes (SECCs) have recently attracted attention due to various applications in communication systems such as simultaneous energy and information transfer. In a SECC, each codeword is divided into smaller subblocks, and every subblock is constrained to carry sufficient energy. In this work, we study SECCs under more general constraints, namely bounded SECCs and sliding-window constrained codes (SWCCs), and propose two methods to construct such codes with low redundancy and linear-time complexity, based on Knuth’s balancing technique and sequence replacement technique. For certain codes parameters, our methods incur only one redundant bit.
... A weaker bound than the one in Theorem 8 for σ = 2 was given in [13] (Theorem 13). Finally, to encode the (b, k)-constrained de Bruijn code efficiently with only a single symbol of redundancy, we may use sequence replacement techniques [62]. ...
Preprint
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The de Bruijn graph, its sequences, and their various generalizations, have found many applications in information theory, including many new ones in the last decade. In this paper, motivated by a coding problem for emerging memory technologies, a set of sequences which generalize sequences in the de Bruijn graph are defined. These sequences can be also defined and viewed as constrained sequences. Hence, they will be called constrained de Bruijn sequences and a set of such sequences will be called a constrained de Bruijn code. Several properties and alternative definitions for such codes are examined and they are analyzed as generalized sequences in the de Bruijn graph (and its generalization) and as constrained sequences. Various enumeration techniques are used to compute the total number of sequences for any given set of parameters. A construction method of such codes from the theory of shift-register sequences is proposed. Finally, we show how these constrained de Bruijn sequences and codes can be applied in constructions of codes for correcting synchronization errors in the $\ell$-symbol read channel and in the racetrack memory channel. For this purpose, these codes are superior in their size on previously known codes.
... Kautz [4] was probably the first to present a simple algorithmic method, called enumerative encoding, for translating user words into -constrained codewords and vice versa. Wijngaarden and Immink presented various codes of rate , where subsequences that violate the maximum runlength are iteratively removed to obtain a -constrained sequence [5]. ...
Article
Full-text available
In this paper, we will present coding techniques for the character-constrained channel, where information is conveyed using q-bit characters (nibbles), and where w prescribed characters are disallowed. Using codes for the character-constrained channel, we present simple and systematic constructions of high-rate binary maximum runlength constrained codes. The new constructions have the virtue that large lookup tables for encoding and decoding are not required. We will compare the error propagation performance of codes based on the new construction with that of prior art codes.
... Our method is based on the sequence replacement technique. The sequence replacement technique has been widely used in the literature [23]- [26]. It is an efficient method for removing forbidden windows from a source word. ...
Preprint
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The subblock energy-constrained codes (SECCs) and sliding window-constrained codes (SWCCs) have recently attracted attention due to various applications in communcation systems such as simultaneous energy and information transfer. In a SECC, each codewod is divided into smaller non-overlapping windows, called subblocks, and every subblock is constrained to carry sufficient energy. In a SWCC, the energy constraint is enforced over every window. In this work, we focus on the binary channel, where sufficient energy is achieved theoretically by using relatively high weight codes, and study SECCs and SWCCs under more general constraints, namely bounded SECCs and bounded SWCCs. We propose two methods to construct such codes with low redundancy and linear-time complexity, based on Knuth's balancing technique and sequence replacement technique. For certain codes parameters, our methods incur only one redundant bit. We also impose the minimum distance constraint for error correction capability of the designed codes, which helps to reduce the error propagation during decoding as well.
... The decoding algorithm is described in [27]. ...
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Thesis (doctoral)--Universität Essen, 1998.
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We present new methods to protect maximum run-length constrained sequences against random and burst errors and to avoid error propagation. Specific parallel conversion techniques and enumerative coding algorithms for the transformation of binary user information into constrained codewords are proposed. The new schemes are simple and very efficient. The methods can be used for synchronization in communication systems and for modulation coding in magnetic and optical recording systems
Conference Paper
New combinatorial construction techniques are proposed which convert binary user information into a (0,k) constrained sequence having the virtue that at most k zeroes' between logical ones' will occur. In this way sequences are constructed which have a limited runlength. These codes find application in optical and magnetic recording systems. The new construction methods provide efficient, high rate codes with a low complexity. The low complex combinatorial structure of the encoder and the decoder ensure a very fast and efficient parallel conversion of binary information to codewords and vice versa. Specifically, we present the combinatorial structures to convert 16 data bits into a 17 bit constrained sequence to obtain an optimum (0,4) code, a (0,6) code with at most one byte error propagation, and a (0,6/6)-code, respectively. Serious error propagation is avoided by using constrained codes with several unconstrained positions, which are reserved to store the parity bits of an error control code which protects the constrained codeword
Article
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
Article
Many of the types of modulation codes designed for use in storage devices using magnetic recording are discussed. The codes are intended to minimize the negative effects of intersymbol interference. The channel model is first presented. The peak detection systems used in most commercial disk drives are described, as are the run length-limited (d,k) codes they use. Recently introduced recording channel technology based on sampling detection-partial-response (or PRML) is then considered. Several examples are given to illustrate that the introduction of partial response equalization, sampling detection, and digital signal processing has set the stage for the invention and application of advanced modulation and coding techniques in future storage products.< >
Article
We introduce the fixed-rate bit stuff (FRB) algorithm for efficiently encoding and decoding maximum-runlength-limited (MRL) sequences. Our approach is based on a simple, variable-rate technique called bit stuffing . Bit stuffing produces near-capacity achieving codes for a wide range of constraints, but encoding is variable-rate, which is unacceptable in most applications. In this work, we design near-capacity fixed-rate codes using a three-step procedure. The fixed-length input data block first undergoes iterative preprocessing, followed by variable-rate bit stuffing, and finally dummy-bit padding to a fixed output length. The iterative preprocessing is key to achieving high encoding rates. We discuss rate computation for the proposed FRB algorithm and show that the asymptotic (in input block length) encoding rate is close to the average rate of the variable-rate bit stuff code. Then, we proceed to explore the effect of decreasing/increasing the number of preprocessing iterations. Finally, we derive a lower bound on the encoding rate with finite-length input blocks and tabulate the parameters required to design FRB codes with rate close to 100/101 and 200/201.
Article
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 .
Article
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