Finite Field Multiple Access II:from Symbol-wise to Codeword-wise

A finite-field multiple-access (FFMA) system separates users within a finite field by utilizing different element-pairs (EPs) as virtual resources. The Cartesian product of distinct EPs forms an EP code, which serves as the input to a finite-field multiplexing module (FF-MUX). This allows the FFMA technique to reorder the channel coding and multiplexing modules, enabling the superimposed signals to function as codewords that can be decoded by a channel code. This flexibility allows the FFMA system to efficiently support a large number of users with short packet traffic, addressing the finite blocklength (FBL) challenge in multiuser reliable transmission. Designing EP codes is a central challenge in FFMA systems. In this paper, we construct EP codes based on a bit(s)-to-codeword transformation approach and define the corresponding EP code as a codeword-wise EP (CWEP) code. We then investigate the encoding process of EP codes, and propose unique sum-pattern mapping (USPM) structural property constraints to design uniquely decodable CWEP codes. Next, we present the $\kappa$-fold ternary orthogonal matrix ${\bf T}_{\rm o}(2^{\kappa}, 2^{\kappa})$ over GF$(3^m)$, where $m = 2^{\kappa}$, and the ternary non-orthogonal matrix ${\bf T}_{\rm no}(M,m)$ over GF$(3^m)$, for constructing specific CWEP codes. Based on the proposed CWEP codes, we introduce three FFMA modes: channel codeword multiple access (FF-CCMA), code division multiple access (FF-CDMA), and non-orthogonal multiple access (FF-NOMA). Simulation results demonstrate that all three modes effectively support massive user transmissions with well-behaved error performance.