Bounds for list-decoding and list-recovery of random linear codes
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866916317793091584 |
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| author | Guruswami, Venkatesan Li, Ray Mosheiff, Jonathan Resch, Nicolas Silas, Shashwat Wootters, Mary |
| author_facet | Guruswami, Venkatesan Li, Ray Mosheiff, Jonathan Resch, Nicolas Silas, Shashwat Wootters, Mary |
| contents | A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family.
In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $ε> 0$ is the gap to capacity).
(1) A random linear code of rate $1 - \log_q(\ell) - ε$ requires list size $L \ge \ell^{Ω(1/ε)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/ε)$ suffices w.h.p.
(2) A random linear code of rate $1 - h_q(p) - ε$ requires list size $L \ge \lfloor h_q(p)/ε+0.99 \rfloor$ for list-decoding from error fraction $p$, when $ε$ is sufficiently small.
(3) A random binary linear code of rate $1 - h_2(p) - ε$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/ε\rfloor + 2$.
The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_13247 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Bounds for list-decoding and list-recovery of random linear codes Guruswami, Venkatesan Li, Ray Mosheiff, Jonathan Resch, Nicolas Silas, Shashwat Wootters, Mary Information Theory Probability A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $ε> 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - ε$ requires list size $L \ge \ell^{Ω(1/ε)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/ε)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - ε$ requires list size $L \ge \lfloor h_q(p)/ε+0.99 \rfloor$ for list-decoding from error fraction $p$, when $ε$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - ε$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/ε\rfloor + 2$. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values. |
| title | Bounds for list-decoding and list-recovery of random linear codes |
| topic | Information Theory Probability |
| url | https://arxiv.org/abs/2004.13247 |