This paper proposes a highly efficient global coded-multiplexing scheme, conceptualized as Orthogonal Frequency Division Multiplexing over a finite field (FF-OFDM), for reliable multiuser communications. By utilizing a prime length cyclic code and its Hadamard equivalents as algebraic subcarriers, independent data streams are globally multiplexed via a Galois Fourier Transform (GFT) without rate loss. We show that this finite-field synthesis intrinsically generates a global Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code over $\mathrm{GF}(2^s)$, whose parity-check matrix is governed by the structural rigor of partial geometries. At the receiver, supported by a binary decomposition theorem, the received nonbinary global codeword is jointly decoded using parallel binary iterative soft-decision algorithms prior to demultiplexing. This joint decoding enables seamless reliability information sharing across all user streams, achieving near-bound error performance, rapid convergence without error floors, and strictly linear amortized decoding complexity.

翻译：本文提出一种高效全局编码复用方案，概念上可视为有限域正交频分复用（FF-OFDM），旨在实现可靠的多用户通信。通过采用素数长度循环码及其Hadamard等价形式作为代数子载波，利用伽罗瓦傅里叶变换（GFT）实现独立数据流的全局复用且无速率损失。研究表明，这种有限域合成方式本质生成了一种定义在$\mathrm{GF}(2^s)$上的全局准循环低密度奇偶校验（QC-LDPC）码，其校验矩阵受局部几何结构严谨约束。在接收端，基于二进制分解定理，接收到的非二进制全局码字在解复用前采用并行二进制迭代软判决算法进行联合译码。该联合译码机制使得所有用户流间的可靠性信息实现无缝共享，从而获得接近性能界的差错性能、快速收敛（无错误平层）以及严格线性平摊译码复杂度。