# Low-Density Code-Domain NOMA: Better Be Regular

## TL;DR

This paper derives a spectral density expression for regular low-density code-domain NOMA, showing it outperforms irregular and dense NOMA in throughput, with feasible implementation prospects for 5G.

## Contribution

It provides a closed-form spectral density for regular LDCD-NOMA and demonstrates its superior throughput compared to irregular and dense NOMA schemes.

## Key findings

- Regular LDCD-NOMA outperforms irregular and dense NOMA in total throughput.
- Spectral density expressions are derived using graph theory and statistical physics methods.
- Regular LDCD-NOMA can be efficiently implemented with message-passing algorithms.

## Abstract

A closed-form analytical expression is derived for the limiting empirical squared singular value density of a spreading (signature) matrix corresponding to sparse low-density code-domain (LDCD) non-orthogonal multiple-access (NOMA) with regular random user-resource allocation. The derivation relies on associating the spreading matrix with the adjacency matrix of a large semiregular bipartite graph. For a simple repetition-based sparse spreading scheme, the result directly follows from a rigorous analysis of spectral measures of infinite graphs. Turning to random (sparse) binary spreading, we harness the cavity method from statistical physics, and show that the limiting spectral density coincides in both cases. Next, we use this density to compute the normalized input-output mutual information of the underlying vector channel in the large-system limit. The latter may be interpreted as the achievable total throughput per dimension with optimum processing in a corresponding multiple-access channel setting or, alternatively, in a fully-symmetric broadcast channel setting with full decoding capabilities at each receiver. Surprisingly, the total throughput of regular LDCD-NOMA is found to be not only superior to that achieved with irregular user-resource allocation, but also to the total throughput of dense randomly-spread NOMA, for which optimum processing is computationally intractable. In contrast, the superior performance of regular LDCD-NOMA can be potentially achieved with a feasible message-passing algorithm. This observation may advocate employing regular, rather than irregular, LDCD-NOMA in 5G cellular physical layer design.