Threshold rates for properties of random codes

TL;DR

This work establishes a precise threshold-rate framework for symmetric properties of random codes, showing that the threshold $R^*$ equals the maximal entropy bound within a convex type-approximation set and can be attained via a first-moment analysis. It applies this framework to list-recovery, deriving sharp thresholds $R^* = 1 - \beta(p,\ell,L)/L$ with $\beta$ the maximal entropy over histogram-type distributions, and provides efficient algorithms to compute these thresholds. The authors obtain exact thresholds for list-decoding in the random-code model (notably $L=3$) and reveal a separation between random codes and random linear codes in this regime, while also determining the zero-error hashing threshold. They further situate the results within the broader threshold phenomenon landscape and outline directions to extend the framework to other code ensembles and properties, aiming to deepen the understanding of when simple first-moment bounds are tight and how to compute thresholds efficiently.

Abstract

Suppose that $P$ is a property that may be satisfied by a random code $C \subset Σ^n$. For example, for some $p \in (0,1)$, ${P}$ 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.

Figures (3)

• Figure 1: The threshold rate $R_{RC}$ (red) for $(p,3)$-list-decodability of random codes, and the threshold rate $R_{RLC}$ (blue, dashed) for $(p,3)$-list-decodability of random linear codes. Note that, uniformly over $p$, random linear codes have the greater threshold rate.
• Figure 2: The value $R^{\text{theorem}}$ that Theorem \ref{['thm:threshold']} would predict for $\mathcal{P}^{\text{toy}}$, and the value $R^\dagger$ that is a lower bound on the actual value of $R^*$, for $p \in (0,0.5)$.
• Figure 3: Plots of $\frac{H(\tau_{n,i})}{\dim(\tau_{n,i})}$ for each $i \in \{0,1,2,3,4\}$, ignoring $o_{n\to\infty}(1)$ terms. One can see that, uniformly over $p \in (0,1/4)$, the minimum is obtained by $\frac{H(\tau_{n,1})}{\dim(\tau_{n,1})}$.

Theorems & Definitions (69)

• Theorem 1.1: Informal; see Theorem \ref{['thm:threshold']} for the formal version
• Definition 2.1: Matrices with distinct columns
• Definition 2.2: Subsets as matrices
• Definition 2.3: Random code
• Definition 2.5: Noisy list-recovery
• Definition 2.6: Monotone-increasing property
• Definition 2.7: Minimal-set
• Remark 2.8
• Definition 2.9: Sharpness for random codes
• Definition 2.10: Row-permutation invariant collection of matrices