Sharp threshold rates for random codes

by   Venkatesan Guruswami, et al.

Suppose that 𝒫 is a property that may be satisfied by a random code C ⊂Σ^n. For example, for some p ∈ (0,1), 𝒫 might be the property that there exist three elements of C that lie in some Hamming ball of radius pn. We say that R^* is the threshold rate for 𝒫 if a random code of rate R^* + ε is very likely to satisfy 𝒫, while a random code of rate R^* - ε is very unlikely to satisfy 𝒫. While random codes are well-studied in coding theory, even the threshold rates for relatively simple properties like the one above are not well understood. We characterize threshold rates for a rich class of properties. These properties, like the example above, are defined by the inclusion of specific sets of codewords which are also suitably "symmetric." For properties in this class, we show that the threshold rate is in fact equal to the lower bound that a simple first-moment calculation obtains. Our techniques not only pin down the threshold rate for the property 𝒫 above, they give sharp bounds on the threshold rate for list-recovery in several parameter regimes, as well as an efficient algorithm for estimating the threshold rates for list-recovery in general.



page 1

page 2

page 3

page 4


Threshold Rates of Codes Ensembles: Linear is Best

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

LDPC Codes Achieve List Decoding Capacity

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

Punctured Large Distance Codes, and Many Reed-Solomon Codes, Achieve List-Decoding Capacity

We prove the existence of Reed-Solomon codes of any desired rate R ∈ (0,...

List-decodability with large radius for Reed-Solomon codes

List-decodability of Reed-Solomon codes has received a lot of attention ...

Numerical and analytical bounds on threshold error rates for hypergraph-product codes

We study analytically and numerically decoding properties of finite rate...

A Geometric View of the Service Rates of Codes Problem and its Application to the Service Rate of the First Order Reed-Muller Codes

Service rate is an important, recently introduced, performance metric as...

A Study of the Separating Property in Reed-Solomon Codes by Bounding the Minimum Distance

According to their strength, the tracing properties of a code can be cat...
This week in AI

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