On codes decoding a constant fraction of errors on the BSC

Hązła, Jan; Samorodnitsky, Alex; Sberlo, Ori

doi:10.1145/3406325.3451015

Cited by 7 publications

(17 citation statements)

References 16 publications

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“…This approach shows that RM codes must be close to a "twin code" that achieves capacity on BMS channels. Finally, the results of [10] very cleverly combine a number of earlier results (including the BEC result from [7]) to establish that the bit (and block) error probabilities of RM codes will vanish on general BMS channels but only for rates bounded away from capacity. For a good tutorial that covers RM codes and relevant prior work until 2020, we suggest [11].…”

Section: Introductionmentioning

confidence: 59%

“…To show this, one can apply Lemma 19 to H(t) − H(s) and then use Lemma 44 to verify that all terms in the resulting expansion are positive. Keeping only the first term in the expansion gives the bound in (10). In the case of the BSC, it can be verified that the factor 2 ln(2) is tight in the limit where the crossover probability approaches 1/2 (see Figure 3).…”

Section: A Bms Families Ordered By Degradationmentioning

confidence: 90%

“…For a detailed discussion of relevant prior work until 2017, see [7]. Since then, there have been a few papers that address this question directly or indirectly [8], [9], [10]. In [8], new bounds on RM weight enumerators are introduced and used to prove that low rate and high rate RM codes can correct a large number of erasures and errors.…”

Section: Introductionmentioning

confidence: 99%

“…Lastly, the new proof does not extend to all sequences of doubly-transitive codes nor does it imply that the block-error rate converges to 0. However, we are optimistic that an extension to block-error rate is possible, perhaps using techniques from [12], [13], [10].…”

Section: Introductionmentioning

confidence: 99%

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Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

Reeves

Pfister

2024

IEEE Trans. Inform. Theory

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This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing biterror probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

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Section: Introductionmentioning

confidence: 59%

Section: A Bms Families Ordered By Degradationmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

Reeves

Pfister

2024

IEEE Trans. Inform. Theory

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“…In particular, [Sam20] allows to leverage results for erasure channels to new bounds on the weight enumerator, via hypercontracivity arguments. Improvements on how to exploit the erasure duality were further developed in [HSS21] which achieves an important milestone: showing that RM codes can be decoded at constant rate over the BSC channel. While this does not give a capacity result, it settled an open problem from [ASW15b] as techniques therein did not seem able to capture this regime.…”

Section: Progress and Related Referencesmentioning

confidence: 99%

Reed-Muller Codes

Abbé¹,

Sberlo²,

Shpilka³

et al. 2023

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Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open.This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on 'subspace-sunflowers', handling their dependencies via an L p Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.

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A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

Abbe,

Sandon

2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

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On codes decoding a constant fraction of errors on the BSC

Cited by 7 publications

References 16 publications

Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

Reed-Muller Codes

A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

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