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List-decodability of Reed-Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form r=1−ε for ε tending to zero. Our main result states that there exist Reed-Solomon codes with rate Ω(ε) which are (1−ε,O(1/ε)) -list-decodable, meaning that any Hamming ball of radius 1−ε contains at most O(1/ε) codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance.
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Bounds on the List Size of Successive Cancellation List Decoding
Successive cancellation list decoding of polar codes provides very good performance for short to moderate block lengths. However, the list size required to approach the performance of maximum-likelihood decoding is still not well understood theoretically. This work identifies information-theoretic quantities that are closely related to this required list size. It also provides a natural approximation for these quantities that can be computed efficiently even for very long codes. Simulation results are provided for the binary erasure channel as well as the binary-input additive white Gaussian noise channel.
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- Award ID(s):
- 1718494
- PAR ID:
- 10303734
- Date Published:
- Journal Name:
- 2020 International Conference on Signal Processing and Communications (SPCOM)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/SPCOM50965.2020.9179593
