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# Runlength-Limited Sequences

Authors:
• Turing Machines Inc

## 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. ...
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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 . ...
Conference Paper
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 . ...
Conference Paper
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 . ...
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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]. ...
Preprint
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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]. ...
Preprint
Full-text available
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]. ...
Preprint
Full-text available
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 $1$.
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