Simple Finite-Length Achievability and Converse Bounds for the Deletion Channel and the Insertion Channel

2025-04-29

AI Summary

This work investigates the fundamental limits of reliable communication over finite-length deletion and insertion channels, aiming to establish tight upper bounds on the trade-off between blocklength and frame error probability. To address the computational intractability and looseness of existing general converse bounds—stemming from the absence of a suitable reference output distribution—the paper introduces, for the first time, an efficiently computable optimal reference output distribution tailored to arbitrary finite-input/finite-output channels. Building upon the general information-theoretic converse framework, the authors integrate probabilistic coupling techniques with refined blocklength analysis to derive a concise, analytically tractable finite-length converse bound. This bound strictly improves upon the classical binary erasure channel bound and yields significantly tighter capacity upper bounds under identical parameters, thereby providing a more accurate theoretical benchmark for code design over deletion and insertion channels.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We develop upper bounds on code size for independent and identically distributed deletion (insertion) channel for given code length and target frame error probability. The bounds are obtained as a variation of a general converse bound, which, though available for any channel, is inefficient and not easily computable without a good reference distribution over the output alphabet. We obtain a reference output distribution for a general finite-input finite-output channel and provide a simple formula for the converse bound on the capacity employing this distribution. We then evaluate the bound for the deletion channel with a finite block length and show that the resulting upper bound on the code side is tighter than that for a binary erasure channel, which is the only alternative converse bound for this finite-length setting. Also, we provide the similar results for the insertion channel.

Problem

• Develop bounds on code size for deletion/insertion channels
• Provide a simple formula for converse bound on capacity
• Evaluate tighter upper bounds for finite block length channels

Innovation

• Develop upper bounds for deletion and insertion channels
• Use reference output distribution for general channels
• Provide simple formula for converse bound on capacity