Bounds for list-decoding and list-recovery of random linear codes

by   Venkatesan Guruswami, et al.

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 ℓ if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than ℓ 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 →∞, is known to be 1-h_q(p) for list-decoding, and 1-log_q ℓ 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(ℓ) - ϵ requires list size L >ℓ^Ω(1/ϵ) for list-recovery from input list size ℓ. This is surprisingly in contrast to completely random codes, where L = O(ℓ/ϵ) suffices w.h.p. (2) A random linear code of rate 1 - h_q(p) - ϵ requires list size L > h_q(p)/ϵ+0.99 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 ≤ h_2(p)/ϵ + 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.



page 1

page 2

page 3

page 4


LDPC Codes Achieve List Decoding Capacity

We show that Gallager's ensemble of Low-Density Parity Check (LDPC) code...

Threshold Rates of Codes Ensembles: Linear is Best

In this work, we prove new results concerning the combinatorial properti...

Optimally Resilient Codes for List-Decoding from Insertions and Deletions

We give a complete answer to the following basic question: "What is the ...

Synchronization Strings: List Decoding for Insertions and Deletions

We study codes that are list-decodable under insertions and deletions. S...

Ratio List Decoding

We extend the notion of list decoding to ratio list decoding which invo...

Singleton-type bounds for list-decoding and list-recovery, and related results

List-decoding and list-recovery are important generalizations of unique ...

Secure list decoding and its application to bit-string commitment

We propose a new concept of secure list decoding, which is related to bi...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.