A Global Coding Scheme for OFDM over Finite Fields

TL;DR

This paper introduces a novel finite-field OFDM scheme that multiplexes multiple data streams efficiently, enabling joint decoding with near-error-free performance and linear complexity.

Contribution

It presents a new global coded-multiplexing scheme over finite fields using Galois Fourier Transform and algebraic codes, achieving high reliability and low complexity.

Findings

Achieves near-bound error performance with rapid convergence.

Supports joint decoding across all user streams.

Maintains strictly linear amortized decoding complexity.

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.