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Galaxy Codes: Advancing Achievability for Deterministic Identification via Gaussian Channels

Holger Boche, Christian Deppe, Safieh Mahmoodi, Golamreza Omidi

TL;DR

The paper tackles deterministic identification over Gaussian channels with power constraints and introduces Galaxy codes, a fractal, multi-level scheme that leverages hierarchical projections on hyperspheres. By combining spherical codes with a depth-structured galaxy construction and careful projection-based decoding, it proves a new achievable DI rate of at least $\tfrac{3}{8}$ on the $L(n,R)=2^{nR\log n}$ scale, improving the prior $\tfrac{1}{4}$ bound. The analysis establishes both codeword separation and vanishing error probabilities (Types I and II) through detailed packing arguments and Gaussian tail bounds. This advance narrows the gap toward the true DI capacity and demonstrates the potential of geometry-inspired, hierarchical coding for deterministic message identification in continuous-channel settings.

Abstract

Deterministic identification offers an efficient solution for scenarios where decoding entire messages is unnecessary. It is commonly used in alarm systems and control systems. A key advantage of this approach is that the capacity for deterministic identification in Gaussian channels with power constraints grows superexponentially, unlike Shannon's transmission capacity. This allows for a significantly higher number of messages to be transmitted using this event-driven method. So far, only upper and lower bounds for deterministic identification capacity have been established. Our work introduces a novel construction: galaxy codes for deterministic identification. Using these codes, we demonstrate an improvement in the achievability bound of 1/4 to 3/8, representing a previously unknown advance that opens new possibilities for efficient communication.

Galaxy Codes: Advancing Achievability for Deterministic Identification via Gaussian Channels

TL;DR

The paper tackles deterministic identification over Gaussian channels with power constraints and introduces Galaxy codes, a fractal, multi-level scheme that leverages hierarchical projections on hyperspheres. By combining spherical codes with a depth-structured galaxy construction and careful projection-based decoding, it proves a new achievable DI rate of at least on the scale, improving the prior bound. The analysis establishes both codeword separation and vanishing error probabilities (Types I and II) through detailed packing arguments and Gaussian tail bounds. This advance narrows the gap toward the true DI capacity and demonstrates the potential of geometry-inspired, hierarchical coding for deterministic message identification in continuous-channel settings.

Abstract

Deterministic identification offers an efficient solution for scenarios where decoding entire messages is unnecessary. It is commonly used in alarm systems and control systems. A key advantage of this approach is that the capacity for deterministic identification in Gaussian channels with power constraints grows superexponentially, unlike Shannon's transmission capacity. This allows for a significantly higher number of messages to be transmitted using this event-driven method. So far, only upper and lower bounds for deterministic identification capacity have been established. Our work introduces a novel construction: galaxy codes for deterministic identification. Using these codes, we demonstrate an improvement in the achievability bound of 1/4 to 3/8, representing a previously unknown advance that opens new possibilities for efficient communication.
Paper Structure (10 sections, 13 theorems, 100 equations, 3 figures)

This paper contains 10 sections, 13 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Known results and main result
  4. Galaxy codes from spherical codes
  5. Properties of Galaxy codes
  6. Number of codewords
  7. Distances between codewords
  8. Error Analysis
  9. Proof of Theorem \ref{['mainresult']}
  10. Conclusions

Key Result

Theorem 1

The deterministic identification (DI) capacity of the Gaussian channel $\mathcal{G}$ with power constraints in the $2^{n \log n}$-scale, i.e., for $L(n, R) = 2^{(n \log n) R}$, is bounded by: As a result, the DI capacity is infinite in the exponential scale and zero in the double exponential scale. Formally,

Figures (3)

  • Figure 1: Projection of point and vector.
  • Figure 2: Illustration of a Galaxy of depth 1 and 2, when $M(n,\theta)=8$
  • Figure 3: Illustration of a Galaxy of depth 3, when $M(n,\theta)=8$

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Theorem 1: salariseddigh2021deterministic
  • Theorem 2: vorobyev2024deterministicidentificationcodesfading
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Theorem 5
  • Lemma 2
  • Theorem 6
  • ...and 6 more