Linear and Non-Linear Run Length Limited Codes

Lechner, Gottfried; Land, Ingmar; Grant, Alex

doi:10.1109/lcomm.2015.2425408

Cited by 5 publications

(4 citation statements)

References 5 publications

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“…Thus, Theorem III.2 shows that the sequence of linear subcodes {C (𝑑,∞) 𝑚 (𝑅)} 𝑚≥1 , in equation ( 4), is rate-optimal whenever 𝑑 + 1 is a power of 2, in that it achieves the rate upper bound of 𝑅/(𝑑 + 1). We remark here that the problem of identifying linear codes that are subsets of the set of (𝑑, ∞)-RLL sequences of a fixed length, has been studied [13]. The results therein show that the largest linear code within 𝑆 (𝑚) (𝑑,∞) has rate no larger than 1 𝑑+1 , as 𝑚 → ∞.…”

Section: Resultsmentioning

confidence: 97%

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

Rameshwar¹,

Kashyap²

2022

Preprint

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In this work, we address the question of the largest rate of linear subcodes of Reed-Muller (RM) codes, all of whose codewords respect a runlength-limited (RLL) constraint. Our interest is in the (𝑑, ∞)-RLL constraint, which mandates that every pair of successive 1s be separated by at least 𝑑 0s. Consider any sequence {C 𝑚 } 𝑚≥1 of RM codes with increasing blocklength, whose rates approach 𝑅, in the limit as the blocklength goes to infinity. We show that for any linear (𝑑, ∞)-RLL subcode, Ĉ𝑚 , of the code C 𝑚 , it holds that the rate of Ĉ𝑚 is at most 𝑅 𝑑+1 , in the limit as the blocklength goes to infinity. We also consider scenarios where the coordinates of the RM codes are not ordered according to the standard lexicographic ordering, and derive rate upper bounds for linear (𝑑, ∞)-RLL subcodes, in those cases as well. Next, for the setting of a (𝑑, ∞)-RLL input-constrained binary memoryless symmetric (BMS) channel, we devise a new coding scheme, based on cosets of RM codes. Again, in the limit of blocklength going to infinity, this code outperforms any linear subcode of an RM code, in terms of rate, for low noise regimes of the channel.

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Section: Resultsmentioning

confidence: 97%

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

Rameshwar¹,

Kashyap²

2022

Preprint

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“…Secondly, in this work, we are concerned with (𝑑, ∞)or (0, 𝑘)-RLL subcodes of a linear (Reed-Muller) code. We point out that the dual problem of identifying linear codes that are subsets of the set of (𝑑, ∞)or (0, 𝑘)-RLL sequences of a fixed length, has also been studied [20]. The results there can be suitably extended to show that the rate of the largest linear code within the set of (𝑑, ∞)or (0, 𝑘)-RLL sequences of length 𝑛, equal, respectively, 1 𝑑+1 and 𝑘 𝑘+1 , as 𝑛 → ∞.…”

Section: Resultsmentioning

confidence: 99%

On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

Rameshwar¹,

Kashyap²

2022

Preprint

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This paper considers the input-constrained binary memoryless symmetric (BMS) channel, without feedback. The channel input sequence respects the (𝑑, ∞)-runlength limited (RLL) constraint, which mandates that any pair of successive 1s be separated by at least 𝑑 0s. 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, ∞)-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•2 − log 2 (𝑑+1) , where 𝐶 is the capacity of the BMS channel. Finally, we also introduce techniques for upper bounding the rate of any (1, ∞)-RLL subcode of a specific capacity-achieving sequence of RM codes.

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“…Thus, Theorem III.2 shows that the sequence of linear subcodes {C ( ,∞) ( )} ≥1 , in Theorem III.3, achieves the rate upper bound of /( + 1), when + 1 is a power of 2. We remark here that the results in [13] show that the largest linear code within the set of ( , ∞)-RLL constrained sequences of length , has rate no larger than 1 +1 , as → ∞. However, such a result offers no insight into rates achievable over BMS channels.…”

mentioning

confidence: 79%

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

Rameshwar

Kashyap

2022

2022 IEEE Information Theory Workshop (ITW)

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In this work, we address the question of the largest rate of linear subcodes of Reed-Muller (RM) codes, all of whose codewords respect a runlength-limited (RLL) constraint. Our interest is in the ( , ∞)-RLL constraint, which mandates that every pair of successive 1s be separated by at least 0s. Consider any sequence {C } ≥1 of RM codes with increasing blocklength, whose rates approach , in the limit as the blocklength goes to infinity. We show that for any linear ( , ∞)-RLL subcode, Ĉ , of the code C , it holds that the rate of Ĉ is at most +1 , in the limit as the blocklength goes to infinity. We also consider scenarios where the coordinates of the RM codes are not ordered according to the standard lexicographic ordering, and derive rate upper bounds for linear ( , ∞)-RLL subcodes, in those cases as well. Next, for the setting of a ( , ∞)-RLL input-constrained binary memoryless symmetric (BMS) channel, we devise a new coding scheme, based on cosets of RM codes. Again, in the limit of blocklength going to infinity, this code outperforms any linear subcode of an RM code, in terms of rate, for low noise regimes of the channel.

show abstract

Linear and Non-Linear Run Length Limited Codes

Cited by 5 publications

References 5 publications

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

On the Performance of Reed-Muller Codes Over $(d,\infty)$-RLL Input-Constrained BMS Channels

Linear Runlength-Limited Subcodes of Reed-Muller Codes and Coding Schemes for Input-Constrained BMS Channels

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