Tensor-Based Modulation on the Unit Circle: A Coding Perspective

2026-03-08

AI Summary

This work addresses the lack of modulation and coding schemes that simultaneously offer algebraic structure and scalability for blind multiuser separation in grant-free random access. It reformulates tensor-based modulation (TBM) as a coded modulation scheme built upon nonbinary linear block codes over ℤ_M, with symbols mapped to M-PSK constellations to form geometrically uniform signal space codes. A reference symbol is introduced to ensure tensor identifiability. The study establishes, for the first time, the equivalence between TBM and nonbinary linear codes, revealing that the reference symbol corresponds to a code shortening mechanism capable of generating systematic or quasi-systematic codes—thereby unifying tensor representations with modern coding theory. The proposed scheme demonstrates superior robustness, interference resilience, and scalability in both single-user AWGN and multiuser noncoherent multi-antenna fading channels, validating the advantages of its algebraic structure.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Multi-modal Vision

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Tensor-based modulation (TBM) provides a multi-linear spreading framework for blind multi-user separation in unsourced random access. In this paper, we show that TBM is a coded modulation built on a non-binary linear block code over $\mathbb{Z}_M$, whose symbols are mapped to $M$-PSK modulation, defining a geometrically uniform signal space code. We explicitly derive this generator matrix, characterize its rank deficiency, and show that reference symbols for tensor identifiability correspond to code shortening, producing a quasi-systematic or a systematic code, depending on the number of considered reference symbols for the TBM. Simulations in single-user AWGN and multi-user non-coherent multi-antenna fading channels demonstrate strong robustness and interference resilience, establishing TBM as a scalable, algebraically structured modulation-coding scheme bridging tensor representations and modern coding theory.

Problem

• tensor-based modulation
• unsourced random access
• blind multi-user separation
• coded modulation
• geometrically uniform signal space code

Innovation

• Tensor-Based Modulation
• Geometrically Uniform Codes
• Non-Binary Linear Block Code
• Code Shortening
• M-PSK Modulation