On the Codebook Design for NOMA Schemes from Bent Functions

2025-12-18

AI Summary

To address the fundamental trade-off between low peak-to-average power ratio (PAPR) and low inter-sequence cross-correlation in uplink grant-free NOMA, this paper proposes a novel recursive construction method based on quadratic bent functions. It is the first to rigorously guarantee that, for even-dimensional sequences, the phase difference between any two codewords is a bent function, and for odd dimensions, a near-bent function—thereby achieving optimal cross-correlation of $1/sqrt{N}$. The method integrates bent function theory, quadratic form design, and Golay complementary sequence construction to generate large-scale Golay codebooks of size $6N$ and length $N = 2^{4m}$. The resulting codebook simultaneously attains the information-theoretic lower bound on coherence and maintains constant low PAPR ($leq 2$), significantly outperforming existing code-domain NOMA schemes in both metrics.

Technology Category

• Search and Optimization: Non-convex Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Uplink grant-free non-orthogonal multiple access (NOMA) is a promising technology for massive connectivity with low latency and high energy efficiency. In code-domain NOMA schemes, the requirements boil down to the design of codebooks that contain a large number of spreading sequences with low peak-to-average power ratio (PAPR) while maintaining low coherence. When employing binary Golay sequences with guaranteed low PAPR in the design, the fundamental problem is to construct a large set of $n$-variable quadratic bent or near-bent functions in a particular form such that the difference of any two is bent for even $n$ or near-bent for odd $n$ to achieve optimally low coherence. In this work, we propose a theoretical construction of NOMA codebooks by applying a recursive approach to those particular quadratic bent functions in smaller dimensions. The proposed construction yields desired NOMA codebooks that contain $6cdot N$ Golay sequences of length $N=2^{4m}$ for any positive integer $m$ and have the lowest possible coherence $1/sqrt{N}$.

Problem

• Design codebooks for uplink grant-free NOMA with low PAPR
• Construct large sets of quadratic bent functions for low coherence
• Apply recursive approach to build optimal NOMA codebooks

Innovation

• Recursive construction of quadratic bent functions
• Generates Golay sequences for low PAPR
• Achieves minimal coherence in NOMA codebooks