Unified Algebraic Absorption of Finite-Blocklength Penalties via Generalized Logarithmic Mapping

2026-03-22

AI Summary

This work addresses the challenge of uniformly modeling higher-order non-Gaussian penalty terms in finite-blocklength channel coding by introducing a novel framework based on generalized q-algebra and a dynamic scaling law \(1 - q_n = \alpha n^{-1}\). The approach embeds higher-order statistical biases directly into the algebraic structure of information density. Through the q-logarithmic mapping, it uniquely unifies the incorporation of higher-order moment effects at finite blocklengths without resorting to Edgeworth expansions or Hermite polynomial corrections. The method precisely recovers the third-order coding normal approximation, and its k-th order algebraic terms naturally correspond to Edgeworth correction terms of order \(O(n^{1 - k/2})\), thereby establishing a rigorous mathematical link between information-theoretic limits and generalized algebraic structures.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In finite-blocklength information theory, evaluating the fundamental limits of channel coding typically relies on normal approximations and Edgeworth expansions, which introduce additive polynomial corrections for skewness and higher-order moments. This paper proposes an alternative approach: rather than appending external error terms to a Gaussian baseline, we absorb these finite-length penalties using a generalized $q$-algebraic framework. By introducing a dynamic scaling law $1-q_n = αn^{-1}$ for the tuning parameter, we prove that the $q$-generalized information density corresponds to macroscopic higher-order fluctuations. Specifically, by setting this scaling constant to $α= T/(3V^2)$ (where $V$ is the varentropy and $T$ is the third central moment), our framework recovers the third-order coding limit, absorbing the $O(1)$ non-Gaussian penalty without relying on Hermite polynomials. Furthermore, we demonstrate that the $k$-th degree term of our algebraic expansion matches the $O(n^{1-k/2})$ asymptotic order of the $(k+1)$-th moment Edgeworth correction. This approach unifies classical probabilistic approximations within a single algebraic structure, establishing a mathematical connection between finite-blocklength analysis and generalized logarithmic mappings.

Problem

• finite-blocklength
• channel coding
• non-Gaussian penalties
• higher-order moments
• information density

Innovation

• q-algebra
• finite-blocklength
• generalized logarithm
• Edgeworth expansion
• information density