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Runlength-Limited Sequences

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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.
... Below we discuss properties of Construction I-based codes for special cases of and t. 1) Case t = − 1: The ( , t = − 1) constraint is in the data storage world [16] also known as a minimum runlength constraint, or d-constraint, where d = t denotes the minimum zero runlength. Freiman and Wyner [7] presented maximal block codes whose codewords start with at least t one's; only n − t bits of the codeword require (de)coding. ...
... 2) Case t = 1: Codes with a maximum runlength of 0's, or k-constraint, where k denotes the maximum zero runlength, have found widespread application in data storage devices [16]. Note that an ( = k + 1, t = 1)-constrained sequence can be obtained by inverting the binary symbols of a k-constrained sequence. ...
... For the case = 4 and t = 2, using (3), (16), and (17), we find A 4,2 ≈ 0.6636 and λ ≈ 1.7143, see Section II, so that η ≈ 1 − 0.76/n. ...
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Energy-harvesting sliding-window constrained block codes guarantee that within any prescribed window of ℓ consecutive bits the constrained sequence has at least t, t≥1, 1’s. Prior art code design methods build upon the finite-state machine description of the (ℓ,t) constraint, but as the number of states equals ℓ choose t, a code design becomes prohibitively complex for mounting ℓ and t. We present a new block code construction that circumvents the enumeration of codewords using a finite-state description of the (ℓ,t)-constraint. The codewords of the block code are encoded and decoded using a single look-up table. For (ℓ=4,t=2), the new block codes are maximal, that is, they have the largest possible number of codewords for its parameters.
... Figure 2 shows a state transition graph that represents the constraint. This constraint is a special case of the ( , )-RLL constraint, which admits only binary sequences with at least and at most 0s between successive 1s (see [3] for examples of ( , )-RLL codes used in practice). input sequences that respect the ( , ∞)-RLL constraint, passed through the classical BMS channel, which is a special case of the discrete memoryless channel (DMC), introduced by Shannon in [4]. ...
... This constraint is a special case of the ( , )-RLL constraint, which admits only binary sequences with at least and at most 0s between successive 1s. Reference [3] includes examples of ( , )-RLL codes used in practice in magnetic storage and recording. The system model under investigation in this paper considers input sequences that respect the ( , ∞)-RLL constraint, passed through the classical BMS channel, which is a special case of the discrete memoryless channel (DMC), introduced by Shannon in [4]. ...
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This paper considers the input-constrained binary memoryless symmetric (BMS) channel, without feedback. The channel input sequence respects the $(d,\infty)$-runlength limited (RLL) constraint, which mandates that any pair of successive $1$s be separated by at least $d$ $0$s. We consider the problem of designing explicit codes for such channels. In particular, we work with the Reed-Muller (RM) family of codes, which were shown by Reeves and Pfister (2021) to achieve the capacity of any unconstrained BMS channel, under bit-MAP decoding. We show that it is possible to pick $(d,\infty)$-RLL subcodes of a capacity-achieving (over the unconstrained BMS channel) sequence of RM codes such that the subcodes achieve, under bit-MAP decoding, rates of $C\cdot{2^{-\left \lceil \log_2(d+1)\right \rceil}}$, where $C$ is the capacity of the BMS channel. Finally, we also introduce techniques for upper bounding the rate of any $(1,\infty)$-RLL subcode of a specific capacity-achieving sequence of RM codes.
... One-dimensional (1D) run-length limited (RLL) code was previously proposed for magnetic and optical recording systems [4][5][6] to specify the minimum and maximum run-length of bit 1's that may occur in a recorded data sequence. In practice, the RLL code is normally defined by two parameters that are d and k, where the parameter d is used for controlling the highest transition frequency that means it can avoid the ISI effect, while the parameter k is used to ensure that the transition of the 978-1-7281-6486-1/20/$31.00 ©2020 IEEE recorded bit data has properly appeared, which can help the synchronization process to precisely recover the sampled data sequence [7]. ... Conference Paper The severity of inter-symbol interference (ISI) and inter-track interference (ITI) effects caused by the reduction of the spacing between bit islands in the along- and across-track directions to acquire ultra-high areal densities (ADs) is the main problem in bit-patterned magnetic recording (BPMR) systems. We call these two effects as two-dimensional (2D) interference, which can degrade the overall system performance. Practically, a one-dimensional (1D) modulation code, e.g., run-length limited (RLL) code, is essentially utilized to prevent the severe ISI effect. The use of 2D detectors is also one of the important ways to cope with the severe ITI; however, its complexity is quite high. To avoid the severe ITI effect and complex 2D detectors; therefore, we propose to apply the 1D RLL code as the 2D modulation code. Then, the 2D modulation encoding constraint is employed to reduce the number of states and branches in the trellis of the proposed modified 2D Viterbi detector. Simulation results reveal that the proposed detector not only delivers a lower complexity but also provides superior bit error rate gain, especially at ultra-high ADs and/or large location fluctuation. Article To achieve ultra-high areal densities in future magnetic recording technologies such as bit-patterned magnetic recording (BPMR) systems, each bit-island must be closely moved. However, the reduction of bit island spacing always leads to both inter-symbol interference (ISI) and inter-track interference (ITI), which can degenerate system performance. Previously, a one-dimensional (1D) run-length limited (RLL) code was employed to efficiently avoid the ISI effect, while a two-dimensional (2D) detector is widely utilized to deal with the ITI effect as well. However, its complexity is higher than the traditional 1D detector. Therefore, to avoid the fatal data patterns that lead to severe ITI effect, we first present the application of 1D RLL code to be as the 2D modulation code in the multi-head multi-track BPMR systems. Moreover, we also present the modified 2D Viterbi detector that is designed based on our proposed encoding condition. The numbers of state and branch in the traditional trellis’s structure are properly reduced according to encoding condition, thus it does not only provide a lower complexity but also operate better than the traditional 2D Viterbi detector. Results reveal that our proposed coding scheme and modified 2D Viterbi detector can efficiently protect and cope with the ITI effect. Article Constrained coding is used widely in digital communication and storage systems. In this article, we study a generalized sliding window constraint called the skip-sliding window. A skip-sliding window (SSW) code is defined in terms of the length$L$of a sliding window, skip length$J$, and cost constraint$E$in each sliding window. Each valid codeword of length$L + kJ$is determined by$k+1$windows of length$L$where window$i$starts at$(iJ + 1)$th symbol for all non-negative integers$i$such that$i \leq k$; and the cost constraint$E\$ in each window must be satisfied. SSW coding constraints naturally arise in applications such as simultaneous energy and information transfer, and SSW codes are also potential candidates for visible light communications. In this work, two methods are given to enumerate the size of SSW codes and further refinements are made to reduce the enumeration complexity. Using the proposed enumeration methods, the noiseless capacity of binary SSW codes is determined and some useful observations are made, such as the fact that SSW codes provide greater capacity than certain related classes of constrained codes. Moreover, we provide noisy capacity bounds for SSW codes.
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