List Decoding Bounds for Binary Codes with Noiseless Feedback

TL;DR

The paper studies list-decoding radii for binary error-correcting codes with noiseless feedback. It establishes an unconditional upper bound of $r<\tfrac{1}{2}-\tfrac{1}{2^{\ell+2}-2}$ for any list size $\ell$, and proves that the Spencer–Winkler construction achieves $(2,\tfrac{3}{7}-\varepsilon)$-list decodability, matching the upper bound for $\ell=2$. It also shows that this approach fails to reach the bound for $\ell=3$, via a concrete adversarial strategy that reduces decodability to at most $(3,\tfrac{31}{67}+\varepsilon)$, highlighting a gap and prompting questions about the true asymptotics. The work uses a coin-game reformulation to translate feedback communication into combinatorial dynamics, enabling unconditional bounds and constructive protocols. Overall, the results separate the feedback-list-decoding landscape from the non-feedback case (where the radius approaches $1/2$ more slowly) and open avenues for exploring exponential versus subexponential improvements in the radius with increasing list size.

Abstract

In an error-correcting code, a sender encodes a message $x \in \{ 0, 1 \}^k$ such that it is still decodable by a receiver on the other end of a noisy channel. In the setting of \emph{error-correcting codes with feedback}, after sending each bit, the sender learns what was received at the other end and can tailor future messages accordingly. While the unique decoding radius of feedback codes has long been known to be $\frac13$, the list decoding capabilities of feedback codes is not well understood. In this paper, we provide the first nontrivial bounds on the list decoding radius of feedback codes for lists of size $\ell$. For $\ell = 2$, we fully determine the $2$-list decoding radius to be $\frac37$. For larger values of $\ell$, we show an upper bound of $\frac12 - \frac{1}{2^{\ell + 2} - 2}$, and show that the same techniques for the $\ell = 2$ case cannot match this upper bound in general.

Figures (1)

• Figure 3: An upper bound on the positions of the first 5 coins after $m$ rounds. To each listed position bound, additionally add $+\varepsilon q$. Eve's selection of odd or even is also noted. Symbols in the cells are referenced in the proofs below.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1: $(\ell, r)$-List Decodable Feedback Code
• Definition 2.2: $(\ell, r; K, n)$-Coin Game
• Theorem 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof