Threshold Rates of Codes Ensembles: Linear is Best

TL;DR

This paper establishes new bounds on the list-recoverability and list-decoding rates of random linear codes, showing they may outperform random codes and highlighting linear codes' optimality.

Contribution

It provides the first lower bounds on list-recoverability for random linear codes and extends list-decoding bounds, demonstrating linear codes' superior performance in certain regimes.

Findings

Lower bounds on list-size for list-recovery close to capacity

New rate bounds for list-of-3 decoding over f2

Linear codes outperform random codes in list-decoding rates

Abstract

In this work, we prove new results concerning the combinatorial properties of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over $\mathbb F_q$ $\varepsilon$-close to capacity to list-recover with error radius $\rho$ and input lists of size $\ell$. We show that the list-size $L$ must be at least $\frac{\log_q\binom{q}{\ell}-R}{\varepsilon}$, where $R$ is the rate of the random linear code. As a comparison, we also pin down the list size of random codes which is $\frac{\log_q\binom{q}{\ell}}{\varepsilon}$. This leaves open the possibility (that we consider likely) that random linear codes perform better than random codes for list-recoverability, which is in contrast to a recent gap shown for the case of list-recovery from erasures (Guruswami et al., IEEE TIT 2021B). Next, we consider list-decoding with constant list-sizes. Specifically, we obtain new lower bounds on the rate required for list-of-$3$ decodability of random linear codes over $\mathbb F_2$; and list-of-$2$ decodability of random linear codes over $\mathbb F_q$ (for any $q$). This expands upon Guruswami et al. (IEEE TIT 2021A) which only studied list-of-$2$ decodability of random linear codes over $\mathbb F_2$. Further, in both cases we are able to show that the rate is larger than that which is possible for uniformly random codes.