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Runlength-Limited Sequences
Abstract
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 a magnetic recording system, for example, the ( , ∞)-RLL constraint on the data sequence (where the bit 1 corresponds to a voltage peak of high amplitude and the bit 0 corresponds to no peak) ensures that successive 1s are spaced far enough apart, so that there is little inter-symbol interference between the voltage responses corresponding to the magnetic transitions. Reference [3] contains many examples of ( , )-RLL codes used in practice in magnetic storage and recording. More recently, ( , )-RLL input constrained sequences have also been investigated for joint energy and information transfer performance [4]. ...
... Definition II.3. The noiseless capacity of the ( , ∞)-RLL constraint is defined as last equality follows from the subadditivity of the sequence log 2( ) ( ,∞) ≥1.2) Linear Codes Over BMS Channels: A binary-input channel is defined by the binary-input input alphabet= {0, 1}, the output alphabet ⊆ R, and the transition probabilities ( ( | ) : ∈ , ∈ ), where (·| ) is a density function with respect to the counting measure, if is discrete, and with respect to the Lebesgue measure, if = R and the output distribution is continuous. ...
The paper considers coding schemes derived from Reed-Muller (RM) codes, for transmission over input-constrained memoryless channels. Our focus is on the (, ∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least 0s. In our study, we first consider (, ∞)-RLL subcodes of RM codes, taking the coordinates of the RM codes to be in the standard lexicographic ordering. We show, via a simple construction, that RM codes of rate have linear (, ∞)-RLL subcodes of rate · 2 − log 2 (+1). We then show that our construction is essentially rate-optimal, by deriving an upper bound on the rates of linear (, ∞)-RLL subcodes of RM codes of rate. Next, for the special case when = 1, we prove the existence of potentially non-linear (1, ∞)-RLL subcodes that achieve a rate of max 0, − 3 8. This, for > 3/4, beats the /2 rate obtainable from linear subcodes. We further derive upper bounds on the rates of (1, ∞)-RLL subcodes, not necessarily linear, of a certain canonical sequence of RM codes of rate. We then shift our attention to settings where the coordinates of the RM code are not ordered according to the lexicographic ordering, and derive rate upper bounds for linear (, ∞)-RLL subcodes in these cases as well. Finally, we present a new two-stage constrained coding scheme, again using RM codes of rate , which outperforms any linear coding scheme using (, ∞)-RLL subcodes, for values of close to 1.
... The (ℓ, ∞)-RLL constraint is a special case of the (ℓ, ℎ)-RLL constraint (we avoid using the more common notation that is ( , )-RLL for such constraints, in order to avoid confusion with existing notation), which admits only binary sequences in which successive 1s are separated by at least ℓ 0s, and the length of any run of 0s is at most ℎ. Such constraints find application in magnetic and optical recording systems, where the (ℓ, ∞)-RLL constraint on the data sequence ensures little inter-symbol interference (ISI) between the voltage responses corresponding to the magnetic transitions (see [17]). We let (ℓ,∞) denote the set of (ℓ, ∞)-RLL constrained binary words of length . ...
In this paper, we present numerical upper bounds on the sizes of constrained codes with a prescribed minimum distance. We accomplish this by extending Delsarte's linear program (LP) (Delsarte (1973)) to the setting of constrained codes, with the value of optimal solutions to this LP giving us the desired upper bound, for a fixed constraint. We also describe an equivalent LP, with fewer variables and LP constraints, obtained by symmetrizing our LP. We observe that for different constraints of interest, our upper bounds beat the generalized sphere packing upper bounds of Fazeli, Vardy, and Yaakobi (2015).
... The ( , ∞)-RLL constraint is a special case of the ( , )-RLL constraint, which admits only binary sequences in which successive 1s are separated by at least 0s, and the length of any run of 0s is at most . Such constraints help alleviate inter-symbol interference (ISI) between voltage responses corresponding to the magnetic transitions, in magnetic recording systems (see [18]). We let denote the set of ( , ∞)-RLL constrained binary words of length . ...
In this paper, we consider the problem of computing the sizes of subcodes of binary linear codes, all of whose codewords need to satisfy an additional property, which we call a constraint. Using a simple identity from the Fourier analysis of Boolean functions, we transform our counting problem into a question about the structure of the dual code. We illustrate the utility of our method in providing explicit values or numerical algorithms for our counting problem, from the somewhat surprising observation that for different constraints of interest, the Fourier transform of the indicator function of the constraint is efficiently computable.
... The ( , ∞)-RLL constraint is a special case of the ( , )-RLL constraint, which admits only binary sequences in which successive 1s are separated by at least 0s, and the length of any run of 0s is at most . Such constraints help alleviate inter-symbol interference (ISI) between voltage responses corresponding to the magnetic transitions, in magnetic recording systems (see [38]). We let denote the set of ( , ∞)-RLL constrained binary words of length . ...
In this paper, we study binary constrained codes that are resilient to bit-flip errors and erasures. In our first approach, we compute the sizes of constrained subcodes of linear codes. Since there exist well-known linear codes that achieve vanishing probabilities of error over the binary symmetric channel (which causes bit-flip errors) and the binary erasure channel, constrained subcodes of such linear codes are also resilient to random bit-flip errors and erasures. We employ a simple identity from the Fourier analysis of Boolean functions, which transforms the problem of counting constrained codewords of linear codes to a question about the structure of the dual code. We illustrate the utility of our method in providing explicit values or efficient algorithms for our counting problem, by showing that the Fourier transform of the indicator function of the constraint is computable, for different constraints. Our second approach is to obtain good upper bounds, using an extension of Delsarte’s linear program (LP), on the largest sizes of constrained codes that can correct a fixed number of combinatorial errors or erasures. We observe that the numerical values of our LP-based upper bounds beat the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).
... These applications are subject to unique challenges and constraints that must be satisfied by the data encoding. For instance, magnetic storage constrains a bounded number of zeros between consecutive ones due to voltage interference in the sensing circuitry that may otherwise lead to errors [Imm90]. In flash memory, constrained codes are used in order to mitigate the inter-cell interference between the nearby wordlines and bitlines [QYS14,BB11]. ...
Constrained coding is a fundamental field in coding theory that tackles efficient communication through constrained channels. While channels with fixed constraints have a general optimal solution, there is increasing demand for parametric constraints that are dependent on the message length. Several works have tackled such parametric constraints through iterative algorithms, yet they require complex constructions specific to each constraint to guarantee convergence through monotonic progression. In this paper, we propose a universal framework for tackling any parametric constrained-channel problem through a novel simple iterative algorithm. By reducing an execution of this iterative algorithm to an acyclic graph traversal, we prove a surprising result that guarantees convergence with efficient average time complexity even without requiring any monotonic progression. We demonstrate the effectiveness of this universal framework by applying it to a variety of both local and global channel constraints. We begin by exploring the local constraints involving illegal substrings of variable length, where the universal construction essentially iteratively replaces forbidden windows. We apply this local algorithm to the minimal periodicity, minimal Hamming weight, local almost-balanced Hamming weight and the previously-unsolved minimal palindrome constraints. We then continue by exploring global constraints, and demonstrate the effectiveness of the proposed construction on the repeat-free encoding, reverse-complement encoding, and the open problem of global almost-balanced encoding. For reverse-complement, we also tackle a previously-unsolved version of the constraint that addresses overlapping windows. Overall, the proposed framework generates state-of-the-art constructions with significant ease while also enabling the simultaneous integration of multiple constraints for the first time.
... Such constraints find application in magnetic and optical recording systems, where the ( , ∞)-RLL constraint on the data sequence (with the bit 1 corresponding to a voltage peak of high amplitude and the bit 0 corresponding to no peak) ensures that successive 1s are spaced far enough apart, so that there is little inter-symbol interference (ISI) between the voltage responses corresponding to the magnetic transitions. Reference [37] contains many examples of ( , )-RLL codes used in practice in magnetic storage and recording. More recently, ( , )-RLL input constrained sequences have also been investigated for joint energy and information transfer performance [6]. ...
In this paper, we study binary constrained codes that are also resilient to bit-flip errors and erasures. In our first approach, we compute the sizes of constrained subcodes of linear codes. Since there exist well-known linear codes that achieve vanishing probabilities of error over the binary symmetric channel (which causes bit-flip errors) and the binary erasure channel, constrained subcodes of such linear codes are also resilient to random bit-flip errors and erasures. We employ a simple identity from the Fourier analysis of Boolean functions, which transforms the problem of counting constrained codewords of linear codes to a question about the structure of the dual code. Via examples of constraints, we illustrate the utility of our method in providing explicit values or efficient algorithms for our counting problem. Our second approach is to obtain good upper bounds on the sizes of the largest constrained codes that can correct a fixed number of combinatorial errors or erasures. We accomplish this using an extension of Delsarte's linear program (LP) to the setting of constrained systems. We observe that the numerical values of our LP-based upper bounds beat those obtained by using the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).
... In a magnetic recording system, for example, the ( , ∞)-RLL constraint on the data sequence (where the bit 1 corresponds to a voltage peak of high amplitude and the bit 0 corresponds to no peak) ensures that successive 1s are spaced far enough apart, so that there is little inter-symbol interference between the voltage responses corresponding to the magnetic transitions. Reference [3] contains many examples of ( , )-RLL codes used in practice in magnetic storage and recording. More recently, ( , )-RLL input constrained sequences have also been investigated for joint energy and information transfer performance [4]. ...
The paper considers coding schemes derived from Reed-Muller (RM) codes, for transmission over input-constrained memoryless channels. Our focus is on the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s be separated by at least $d$ $0$s.
In our study, we first consider $(d,\infty)$-RLL subcodes of RM codes, taking the coordinates of the RM codes to be in the standard lexicographic ordering. We show, via a simple construction, that RM codes of rate $R$ have linear $(d,\infty)$-RLL subcodes of rate $R\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$. We then show that our construction is essentially rate-optimal, by deriving an upper bound on the rates of linear $(d,\infty)$-RLL subcodes of RM codes of rate $R$. Next, for the special case when $d=1$, we prove the existence of potentially non-linear $(1,\infty)$-RLL subcodes that achieve a rate of $\max\left(0,R-\frac38\right)$. This, for $R > 3/4$, beats the $R/2$ rate obtainable from linear subcodes. We further derive upper bounds on the rates of $(1,\infty)$-RLL subcodes, not necessarily linear, of a certain canonical sequence of RM codes of rate $R$. We then shift our attention to settings where the coordinates of the RM code are not ordered according to the lexicographic ordering, and derive rate upper bounds for linear $(d,\infty)$-RLL subcodes in these cases as well.
Finally, we present a new two-stage constrained coding scheme, again using RM codes of rate $R$, which outperforms any linear coding scheme using $(d,\infty)$-RLL subcodes, for values of $R$ close to $1$.
In this work, we propose efficient constrained coding schemes to significantly reduce the sneak path interference (SPI), a fundamental and
challenging problem, in the crossbar resistive memory arrays. Particularly, we attempt to combat the sneak path effect locally as follows.
For arrays of size (n \times n), we study coding methods that enforce every sliding window of size m \times m (where each window refers to a subarray
consisting of consecutive rows and consecutive columns), for some m<n to be: (i) (m,\delta) bounded weight: the number of 1 in every sliding window is
at most m^2/2 - \delta for \delta >= 0, and this constraint is called the locally bounded weight constraint. (ii) Sneak path free: the written bits
in every window do not induce any sneak path to any cell, and this constraint is called the locally sneak path free constraint. In this work, we study the maximum information rate that can be achieved (or channel capacity) and design codes for each constraint. Particularly, for the first constraint, for arbitrary m and \delta<=m/2, we provide an efficient construction of codes with the code rate 1-2/m, while the highest rate found in the prior art is approximately 1-1/m, which was only applicable for m=n-o(n)) and \delta=0.
High temperatures in electronic devices may have a negative effect on their performance. Various techniques have been proposed and studied to address and combat this thermal challenge. To guarantee that the peak temperature of the devices
will be bounded by some maximum temperature, the transmitted signal has to satisfy some constraints. With this motivation, we study the constrained channel that only accepts sequences that satisfy prescribed thermal constraints.
The main goal in this paper is to compute the capacity of this channel. We provide the exact capacity of the channel with some certain parameters and we also present some bounds on the capacity in various cases. Finally, we consider the model that multiple wires are available to use and find out the smallest number of wires required to satisfy the thermal constraints.
The binary symbol-pair constrained codes that can enable simultaneous transfer of information and energy is the topic of interest in this paper. The construction and properties of such binary symbol-pair code using the sliding window constraint are discussed in this paper. The sliding window constraint ensures the presence of at least t weighted symbols within any prescribed window of l consecutive symbol-pairs. The information capacity of (l,t)-constrained sequences has been obtained and analyzed. This paper provides the block code construction of (l,t) symbol-pair constrained codes of length n without using a n-step finite-state machine. The information capacity obtained in this paper is better than the information capacity of (l,t)-constrained codes in Schouhamer Immink and Kui (IEEE Commun Lett 24(9):1890–1893, 2020) [16].
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.
A description is given of the eight to fourteen modulation system (EFM) designed for the Compact Disc Digital Audio System with optical read-out. EFM combines high information density and immunity to tolerances in the light path with low power at the low-frequency end of the modulation bit stream spectrum. In this modulation scheme, blocks of eight data input bits are transformed into fourteen channel bits, which follow certain minimum and maximum run-length constraints by using a code book. To prevent violation of the minimum and maximum run-length constraints a certain number of merging bits are needed to concatenate the blocks. There are cases where the merging bits are not uniquely determined by the concatenation rules. This freedom of choice thus created is used for minimizing the power of the modulated bit sequence at low frequencies. The paper presents the results of algorithms that were used to minimize this low-frequency content.
A system for block encoding words of a digital signal achieves a maximum of error compaction and ensures reliability of a self-clocking decoder, while minimizing
any DC in the encoded signal. Data words of m bits are translated into information blocks ofn1 bits (n1 >m) that satisfy a (d,k)-constraint in which at least d "0" bits, but no more than k "0" bits occur between consecutive "I" bits. The information blocks are concatenated by inserting separation blocks of n2 bits there between, selected so that the (d,k)-constraint is satisfied over the boundary between any two information words. For each information word, the separation block that will yield the lowest net digital sum value is selected. Then, the encoded signal is modulated as an NRZ-M signal in which a "1" becomes a transition and a "0" becomes an absence of a transition. A unique synchronizing block is inserted periodically. A decoder circuit, using the synchronizing blocks to control its timing, disregards the separation blocks, but detects the information blocks and translates them back into reconstituted data words of m bits. The foregoing technique can be used to advantage in recording digitized music on an optical disc.
An encoder and decoder for a (0,3) 8/9 code is described elsewhere. The present design for such encoders and decoders has two properties not found in the earlier embodiment of the code, while retaining all of the properties described previously. One new property is word parity. Each code word provides opposite parity relation with the corresponding data byte. Such relation can be used as a diagnostic check of encoder decoder and other related hardware. In magnetic tape application, a 9 multiplied by 9 encoded frame comprises 8 data and one parity byte in a set of 9 tracks. In every 9 multiplied by 9-byte encoded frame, the new code always produces an odd number of ones.
Digital processing of the audio signal and optical scanning in the Compact Disc system yield significant advantages: insensitivity to surface damage of the disk, compactness of disk and player, exellent signal-to-noise ratio and channel separation (both 90 db) and a flat response over a wide range of frequencies (up to 20,000 Hz). The Compact Disc, with a diameter of only 120 mm, gives a continuous playing time of an hour or more. The analog audio signal is converted into a digital signal suitable for transcription on the disk. After the digital signal has been read from the disk by an optical 'pick-up' the original audio signal is recreated in the player.
A systematic approach to the analysis and construction of channel codes for digital baseband transmission is presented. The structure of the codes is dominated by the set of requirements imposed by channel characteristics and system operation. These requirements may be translated into symbol sequence properties which, in turn, specify a set of permissible sequence states. State-dependent coding of both fixed and variable length is a direct result. Properties of such codes are discussed and two examples are presented.
A method for determining maximum-size block codes, with the property that no concatenation of codewords violates the input restrictions of a given channel, is presented. The class of channels considered is essentially that of Shannon (1948) in which input restrictions are represented through use of a finite-state machine. The principal results apply to channels of finite memory and codes of length greater than the channel memory but shorter codes and non-finite memory channels are discussed briefly.
A partial ordering is first defined over the set of states. On the basis of this ordering, complete terminal sets of states are determined. Use is then made of Mason's general gain formula to obtain a generating function for the size of the code which is associated with each complete terminal set. Comparison of coefficients for a particular block length establishes an optimum terminal set and codewords of the maximum-size code are then obtained directly.
Two important classes of binary channels are then considered. In the first class, an upper bound is placed on the separation of 1's during transmission while, in the second class, a lower bound is placed on this separation. Universal solutions are obtained for both classes.