On the Codebook Design for NOMA Schemes from Bent Functions

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 $6\cdot N$ Golay sequences of length $N=2^{4m}$ for any positive integer $m$ and have the lowest possible coherence $1/\sqrt{N}$.

💡 Research Summary

The paper addresses a fundamental design problem in uplink grant‑free non‑orthogonal multiple access (NOMA) systems: how to construct a spreading‑code codebook that simultaneously guarantees low peak‑to‑average power ratio (PAPR) and low column‑wise coherence. Low PAPR is essential for efficient power‑amplifier operation in OFDM‑based transmissions, while low coherence reduces multi‑user interference and improves the performance of compressed‑sensing based joint channel estimation and multi‑user detection.

The authors focus on binary Golay complementary sequences, which are known to have PAPR bounded by 2. Davis and Jedwab showed that every binary Golay pair of length 2ⁿ can be generated from a quadratic Boolean function of the form

 Q_π(x) = Σ_{i=1}^{n‑1} x_{π(i)} x_{π(i+1)}

where π is a permutation of {1,…,n}. Adding a linear term L_c(x)=Σ c_i x_i yields the full function f_{c,π}(x)=Q_π(x)+L_c(x). Mapping the truth table of f_{c,π} to ±1 produces a Golay‑Davis‑Jedwab (GDJ) sequence.

In a code‑domain NOMA setting, a codebook is built by stacking GDJ sequences as columns of a spreading matrix Φ. The coherence μ(Φ) can be expressed in terms of the Walsh–Hadamard spectrum of the pairwise sums Q_{π₁}+Q_{π₂}. Specifically, if B_{ℓ₁,ℓ₂} denotes the symplectic matrix that records the quadratic cross‑terms of Q_{π_{ℓ₁}}+Q_{π_{ℓ₂}}, then

 μ(Φ)=1/√(2·r_min), r_min = min_{ℓ₁≠ℓ₂} rank(B_{ℓ₁,ℓ₂}).

For even n, a quadratic bent function has Walsh values ±2^{n/2}; for odd n, a near‑bent function has values in {0,±2^{(n+1)/2}}. Consequently, the theoretical lower bound on coherence is 1/√N (N=2ⁿ). Achieving this bound requires that every pairwise sum Q_{π₁}+Q_{π₂} be bent (or near‑bent) so that rank(B_{ℓ₁,ℓ₂})=n (or n‑1).

Previous works obtained such “compatible” permutations by exhaustive search (feasible only for n≤8), by graph‑theoretic constructions (yielding a fixed small set of size L=4), or by using quadratic Gold functions, which give a large set only when n+1 (or n) has a large prime factor. These approaches either do not scale or produce a small overloading factor L, limiting the number of users that can be supported.

The main contribution of this paper is a recursive construction that yields, for any integer m≥1, a set of 6·2^{4m} mutually compatible permutations of size N=2^{4m}. The construction proceeds in two steps:

1. Base case (dimension 4). The authors exhaustively analyze all permutations of {1,2,3,4}. They identify a maximal compatible set I_{S₄} of size six, each of which satisfies the Walsh‑Hadamard condition (WHC) on every pair of coordinates. This set forms the seed for the recursion.

2. Recursive step. Suppose we have a compatible set for dimension n and another for dimension m. For any π∈I_{S_n} and ρ∈I_{S_m}, define a new permutation σ∈S_{n+m} by concatenating the two permutations and appropriately relabeling indices (the paper denotes this operation as π⊕ρ). The authors prove that the quadratic function Q_{σ}(x) decomposes as Q_π(x₁,…,x_n)+Q_ρ(x_{n+1},…,x_{n+m}). Because the Walsh spectrum of a sum of independent bent (or near‑bent) functions is the product of their spectra, the resulting Q_{σ} retains the bent (or near‑bent) property, and the associated symplectic matrix B_{σ,τ} has rank equal to the sum of the ranks of the constituent blocks. Consequently, the minimum rank across all pairs remains maximal (n+m for even total dimension).

Applying this recursion repeatedly with the base case n=4 yields compatible sets for dimensions 8,12,16,… i.e., for any N=2^{4m}. The resulting codebook Φ consists of N×(6·N) columns, each column being a GDJ sequence derived from one of the compatible permutations. All columns have unit modulus, guaranteeing PAPR ≤2, and the coherence attains the lower bound μ=1/√N = 2^{-2m}.

The paper provides explicit permutation lists for m=1 (N=16) and m=2 (N=256) and verifies through direct computation that the coherence and PAPR meet the theoretical predictions. Compared with the Gold‑function based method, which achieves an overloading factor of roughly (p−1)/2 where p is the smallest prime divisor of n+1 (or n), the recursive construction delivers a fixed overloading factor of 6·N for any N of the form 2^{4m}. This eliminates the dependence on number‑theoretic properties of n and offers a scalable, deterministic way to generate large NOMA codebooks.

In summary, the authors present a mathematically rigorous, recursive scheme for constructing large families of quadratic bent (or near‑bent) functions whose pairwise differences remain bent. By translating these functions into Golay‑Davis‑Jedwab sequences, they obtain NOMA spreading matrices that simultaneously achieve the minimum possible coherence and a guaranteed low PAPR. The work bridges Boolean function theory, Walsh‑Hadamard analysis, and practical NOMA system design, providing a concrete pathway to support massive connectivity with minimal interference and energy consumption.