  In the past several decades, various techniques have been developed and used
for multiple-access (MA) communications. With the new applications for 6G, it
is desirable to find new resources, physical or virtual, to confront the fast
development of MA communication systems. For binary source transmission, this
paper proposes an element-pair (EP) coding scheme for supporting massive users
with short packet traffic, which solves the finite blocklength (FBL) of
multiuser reliability transmission problem. Each user is assigned a unique EP,
and the collection of EPs assigned to the users possesses the unique
sum-pattern mapping (USPM) structural property. We present methods for
constructing symbol-wise EP codes with USPM structural property based on the
prime field and extension field of prime field, respectively. Based on the
orthogonal EP code constructed over GF($2^m$), we propose finite-field MA
(FFMA) systems over a Gaussian multiple-access channel (GMAC), including both
the sparse-form and diagonal-from structures. The proposed FFMA is then applied
to network layer and forms network FFMA systems for pure digital networks, in
which an EP is viewed as a virtual resource block (VRB). Simulation results
show that the error performance of the proposed FFMA over a GMAC can approach
the error performance as that of the single-user transmission.
