Finite Field Multiple Access

In the past several decades, various multiple-access (MA) techniques have been developed and used. These MA techniques are carried out in complex-field domain to separate the outputs of the users. It becomes problematic to find new resources from the physical world. It is desirable to find new resources, physical or virtual, to confront the fast development of MA systems. In this paper, an algebraic virtual resource is proposed to support multiuser transmission. For binary transmission systems, the algebraic virtual resource is based on assigning each user an element pair (EP) from a finite field GF($p^m$). The output bit from each user is mapped into an element in its assigned EP, called the output symbol. For a downlink MA system, the output symbols from the users are jointly multiplexed into a unique symbol in the same field GF($p^m$) for further physical-layer transmission. The EPs assigned to the users are said to form a multiuser algebraic uniquely decodable (UD) code. Using EPs over a finite field, a network, a downlink, and an uplink orthogonal/non-orthogonal MA systems are proposed, which are called finite-field MA (FFMA) systems. Methods for constructing algebraic UD codes for FFMA systems are presented. An FFMA system can be designed in conjunction with the classical complex-field MA techniques to provide more flexibility and varieties.