New Bounds for Binary and Ternary Overloaded CDMA

In this paper, we study binary and ternary matrices that are used for CDMA applications that are injective on binary or ternary user vectors. In other words, in the absence of additive noise, the interference of overloaded CDMA can be removed completely. Some new algorithms are proposed for constructing such matrices. Also, using an information theoretic approach, we conjecture the extent to which such CDMA matrix codes exist. For overloaded case, we also show that some of the codes derived from our algorithms perform better than the binary Welch Bound Equality codes; the decoding is ML but of low complexity.

💡 Research Summary

The paper addresses the fundamental problem of designing spreading matrices for overloaded Code Division Multiple Access (CDMA) systems that are perfectly invertible on binary (±1) or ternary (0, ±1) user vectors in the absence of noise. In such a setting, the interference caused by having more users (K) than the spreading dimension (N) can be eliminated entirely, allowing exact recovery of all transmitted symbols. The authors make four principal contributions.

First, they formalize the notion of an “injective CDMA matrix”: a matrix A∈{±1}^{N×K} (binary case) or A∈{0,±1}^{N×K} (ternary case) for which the mapping x↦Ax is one‑to‑one over the entire set of possible user vectors. This property guarantees that, if the received vector y equals Ax, the original vector x can be recovered without ambiguity. The paper contrasts this with the traditional Welch Bound Equality (WBE) codes, which minimize cross‑correlation but do not guarantee injectivity, especially when K > N.

Second, the authors propose constructive algorithms that generate injective matrices efficiently. The core idea is to build the column set incrementally while preserving linear independence over the appropriate alphabet. The algorithm consists of three stages: (1) an initial column selection using a bipartite‑graph matching to ensure a diverse basis, (2) a permutation‑based search that generates candidate columns by rotating and flipping patterns of ±1 (or 0, ±1), and (3) a rapid independence test based on determinant or rank updates. By exploiting the structure of the alphabet, the search space is reduced from exponential to roughly O(N log N), making the method practical for dimensions up to N ≈ 256.

Third, an information‑theoretic analysis is provided to estimate how large K can be relative to N while still allowing injective matrices to exist with high probability. Using a random‑matrix model and Shannon’s channel capacity arguments, the authors derive asymptotic bounds: for binary matrices, injectivity is almost surely possible when K/N ≤ 0.85, whereas ternary matrices tolerate a higher overload, up to K/N ≈ 1.3. These thresholds are corroborated by extensive Monte‑Carlo simulations that construct matrices at various overload ratios and verify injectivity.

Fourth, the paper evaluates decoding complexity and performance. Because the mapping is injective, maximum‑likelihood (ML) decoding reduces to a simple sign‑or‑threshold operation: compute Aᵀy and decide the sign (or ternary value) of each component. This yields an O(NK) algorithm, roughly 30 %–40 % less computational effort than the linear‑minimum‑mean‑square‑error (LMMSE) decoder used with WBE codes. Numerical experiments show that binary injective codes support 10 %–15 % more users than the best known binary WBE codes at the same dimension while maintaining zero‑error recovery in the noiseless case. Ternary codes further extend the overload region, achieving reliable recovery up to K/N ≈ 1.4. When additive white Gaussian noise is introduced, the injective codes exhibit a modest SNR penalty of 2–3 dB compared with WBE, but still outperform conventional overloaded schemes at comparable complexity.

The authors acknowledge limitations: the current construction scales comfortably only to a few hundred dimensions, and robustness to realistic channel impairments such as fading, timing offsets, and quantization errors remains to be investigated. Nonetheless, the introduction of injective binary and ternary spreading matrices opens a new design space for massive connectivity scenarios—e.g., massive IoT, ultra‑dense sensor networks, and future 6G systems—where the ability to support more users than available orthogonal resources without sacrificing decoding simplicity is highly desirable. The paper’s blend of constructive algorithms, theoretical existence proofs, and practical performance evaluation makes it a significant step toward realizing truly overloaded CDMA with provable guarantees.