A Global Coding Scheme for OFDM over Finite Fields
2026-05-11 • Information Theory
Information Theory
AI summaryⓘ
The authors present a new method called FF-OFDM that allows multiple users to send data reliably at the same time using a special kind of coding over finite fields. They use mathematical tools like prime length cyclic codes and the Galois Fourier Transform to mix data streams without losing any rate. This approach automatically creates a structured error-correcting code that can be efficiently decoded using parallel algorithms, improving reliability and speed. Their decoding method shared information across all users, reaching performance close to theoretical limits with manageable complexity.
Orthogonal Frequency Division Multiplexing (OFDM)Finite Fields (GF)Galois Fourier Transform (GFT)Low-Density Parity-Check (LDPC) CodesQuasi-Cyclic CodesCyclic CodesHadamard MatricesSoft-decision DecodingMultiuser Communication
Authors
Juane Li, Qi-yue Yu, Khaled Abdel-Ghaffar, Shu Lin
Abstract
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.