Efficient and rate-optimal list-decoding in the presence of minimal feedback: Weldon and Slepian-Wolf in sheep's clothing

TL;DR

The paper tackles high-rate list-decoding over a $q$-ary channel with adversarial noise under minimal feedback. It introduces two core constructions: (i) a full-feedback Weldon-type scheme achieving rate $R=1-H_q(\\rho)-\\varepsilon$ with poly-time encoding/decoding and manageable list size, and (ii) a minimal-feedback framework that leverages Slepian-Wolf hashing, random permutations, and concatenated codes to approach the same rate with vanishing feedback. The results provide explicit tradeoffs: list size $\\exp(\\mathcal{O}(\\varepsilon^{-3/2}\\log^2(1/\\epsilon)))$, encoding/decoding time $n^{\\mathcal{O}(\\varepsilon^{-1}\\log(1/\\epsilon))}$, and error $\\mathcal{O}(n^{-\\eta})$ for arbitrary $\\eta>0$, with asymptotically $o(n)$ total feedback. Together, the schemes push towards information-theoretic limits in both random and adversarial noise settings while maintaining practical complexity and low-feedback overhead, with rigorous synchronization and termination analyses for both full and partial feedback regimes.

Abstract

Given a channel with length-$n$ inputs and outputs over the alphabet $\{0,1,\ldots,q-1\}$, and of which a fraction $\varrho \in (0,1-1/q)$ of symbols can be arbitrarily corrupted by an adversary, a fundamental problem is that of communicating at rates close to the information-theoretically optimal values, while ensuring the receiver can infer that the transmitter's message is from a ``small" set. While the existence of such codes is known, and constructions with computationally tractable encoding/decoding procedures are known for large $q$, we provide the first schemes that attain this performance for any $q \geq 2$, as long as low-rate feedback (asymptotically negligible relative to the number of transmissions) from the receiver to the transmitter is available. For any sufficiently small $\varepsilon > 0$ and $\varrho \in (1-{1}/{q}-Θ(\sqrt{\varepsilon}))$ our minimal feedback scheme has the following parameters: Rate $1-H_q(\varrho) - \varepsilon$ (i.e., $\varepsilon$-close to information-theoretically optimal -- here $H_q(\varrho)$ is the $q$-ary entropy function), list-size $\exp\left(\mathcal{O}\left(\varepsilon^{-3/2}\log^2(1/\varepsilon)\right)\right)$, computational complexity of encoding/decoding $n^{\mathcal{O}(\varepsilon^{-1}\log(1/\varepsilon))}$, storage complexity $\mathcal{O}(n^{η+1}\log n)$ for a code design parameter $η>1$ that trades off storage complexity with the probability of error. The error probability is $\mathcal{O}(n^{-η})$, and the (vanishing) feedback rate is $\mathcal{O}({1}/{\sqrt{\log(n)}})$.

Figures (8)

• Figure 1: A depiction of the progression of the Weldon scheme for adversarial channels. James corrupts a fraction $p_i$ of transmissions in stage $i$, resulting in the length of the next stage $\ell_{i+1} \approx \ell_i H_q(p_i)$. The termination stage occurs at the end.
• Figure 2: The setup for \ref{['prb:random_errors']}. The vectors ${\underline{\mathbf{x}}}, {\underline{\mathbf{y}}}$ share a joint distribution. Alice must prepare ${\underline{\mathbf{z}}}$ without observing ${\underline{\mathbf{y}}}$. Bob receives ${\underline{\mathbf{z}}}$ noiselessly and must recover ${\underline{\mathbf{x}}}$ from ${\underline{\mathbf{y}}}$ and ${\underline{\mathbf{z}}}$.
• Figure 3: \ref{['prot:no-adv_ce']}: A concatenated coding scheme for the random noise (Slepian-Wolf) problem. A systematic R.S code acts on ${\underline{\mathbf{x}}}$ by treating each chunk of length $C_c\log(N)$ as an element from an alphabet of size $N^{C_c}$. The systematic chunks are compressed by a hash function and the redundancy chunks are sent as is. Breaking up the vector ${\underline{\mathbf{x}}}$ into chunks of length $\Theta(\log(N))$ is what allows us to achieve computational efficiency.
• Figure 4: The mapping between Slepian-Wolf source coding, and channel coding. We "replace" the distortion in the form of the joint distribution by an adversary James. This makes for a clean mapping to the channel setting, and a further mapping to the recursive Weldon-type scheme apparent. If we can demonstrate success of this toy example by receiving ${\underline{\mathbf{z}}}$ noiselessly, the same protocol can be applied iteratively.
• Figure 5: Alice applies a permutation to create her "shuffled" vector $\underline{\bm{\phi}}$

Theorems & Definitions (13)

• Theorem 3.1: Full feedback
• Theorem 3.2: Minimal feedback
• Lemma 1
• Remark 5.1
• Remark 5.2
• Remark 5.3
• proof
• Remark C.1
• Lemma 2
• Lemma 3