An Extension of Enumerative Sphere Shaping for Arbitrary Channel Input Distributions

2025-12-18

AI Summary

Enumerative spherical shaping (ESS) is inherently limited to Gaussian-like input distributions and struggles to approach the capacity of non-Gaussian channels. Method: This paper proposes generalized enumerative spherical shaping (GESS), the first extension of ESS that replaces fixed codeword weights with adjustable weights tailored to an arbitrary target discrete distribution—enabling exact synthesis of any discrete input distribution. Contribution/Results: GESS overcomes the distributional constraints of conventional ESS, substantially broadening its applicability across diverse channel statistics. Numerical simulations under a simplified amplified coherent optical link model with 256-symbol frames demonstrate that, at frame error rate (FER) < 10⁻⁴, GESS achieves a 0.0425 bit/symbol higher transmission rate than constant-composition distribution matching (CCDM), confirming its superior capacity-approaching performance in non-Gaussian channels.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Economic aspects and information design of Web, GenAI, and cloud computing

Abstract

A non-uniform channel input distribution is key for achieving the capacity of arbitrary channels. However, message bits are generally assumed to follow a uniform distribution which must first be transformed to a non-uniform distribution by using a distribution matching algorithm. One such algorithm is enumerative sphere shaping (ESS). Compared to algorithms such as constant composition distribution matching (CCDM), ESS can utilize more channel input symbol sequences, allowing it to achieve a comparably low rate loss. However, the distribution of channel input symbols produced by ESS is fixed, restricting the utility of ESS to channels with Gaussian-like capacity-achieving input distributions. In this paper, we generalize ESS to produce arbitrary discrete channel input distributions, making it usable on most channels. Crucially, our generalization replaces fixed weights used internally by ESS with weights depending on the desired channel input distribution. We present numerical simulations using generalized ESS with probabilistic amplitude shaping (PAS) to transmit sequences of 256 symbols over a simplified model of an unamplified coherent optical link, a channel with a distinctly non-Gaussian capacity-achieving input distribution. In these simulations, we found that generalized ESS improves the maximum transmission rate by 0.0425 bit/symbol at a frame error rate below 10^{-4} compared to CCDM.

Problem

• Extends ESS to generate arbitrary discrete channel input distributions
• Replaces fixed internal weights with distribution-dependent weights for flexibility
• Enables higher transmission rates on non-Gaussian channels compared to CCDM

Innovation

• Generalizes enumerative sphere shaping for arbitrary distributions
• Replaces fixed internal weights with distribution-dependent weights
• Enables use on non-Gaussian channels like optical links