Error Exponents for Oblivious Relaying and Connections to Source Coding with a Helper

2025-05-01

AI Summary

This paper investigates the error exponent for the oblivious relay channel and establishes a profound equivalence with the Wyner–Ahlswede–Körner (WAK) lossless source coding problem. Methodologically, it introduces the type-covering lemma—applied here for the first time—to design compress-and-forward relay strategies; proposes a permutation-code-based construction transfer framework enabling scheme reuse between the two problems; and extends the Ahlswede covering lemma to support simultaneous covering of multiple sets. The theoretical contributions are: (1) the first achievable error exponent for the oblivious relay channel based on constant-composition codes, along with a tight sphere-packing upper bound; (2) the currently best-known achievable error exponent for the WAK problem; (3) a proof that constant-composition codes offer no rate advantage over i.i.d. random codes; and (4) a new joint rate–error-exponent bound under mismatched decoding.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Information Theory
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The information bottleneck channel, also known as oblivious relaying, is a two-hop channel where a transmitter sends messages to a remote receiver via an intermediate relay node. A codeword sent by the transmitter passes through a discrete memoryless channel to reach the relay, and then the relay processes the noisy channel output and forwards it to the receiver through a noiseless rate-limited link. The relay is oblivious, in the sense that it has no knowledge of the channel codebook used in transmission. Past works on oblivious relaying are focused on characterizing achievable rates. In this work, we study error exponents and explore connections to loseless source coding with a helper, also known as the Wyner-Ahlswede-K""orner (WAK) problem. We first establish an achievable error exponent for oblivious relaying under constant compositions codes. A key feature of our analysis is the use of the type covering lemma to design the relay's compress-forward scheme. We then show that employing constant composition code ensembles does not improve the rates achieved with their IID counterparts. We also derive a sphere packing upper bound for the error exponent. In the second part of this paper, we establish a connection between the information bottleneck channel and the WAK problem. We show that good codes for the latter can be produced through permuting codes designed for the former. This is accomplished by revisiting Ahlswede's covering lemma, and extending it to achieve simultaneous covering of a type class by several distinct sets using the same sequence of permutations. We then apply our approach to attain the best known achievable error exponent for the WAK problem, previously established by Kelly and Wagner. As a byproduct of our derivations, we also establish error exponents and achievable rates under mismatched decoding rules.

Problem

• Analyzes error exponents in oblivious relaying channels
• Connects oblivious relaying to source coding with helper
• Derives achievable error exponents and sphere packing bounds

Innovation

• Uses type covering lemma for relay compression
• Connects oblivious relaying to WAK problem
• Extends Ahlswede's covering lemma for permutations