Finite Field Multiple Access for Sourced Massive Random Access with Finite Blocklength

For binary source transmission, this paper introduces the concept of element-pair (EP) and establishes that when the Cartesian product of $J$ distinct EPs satisfies the unique sum-pattern mapping (USPM) structural property, these $J$ EPs can form a uniquely-decodable EP (UD-EP) code. EPs are treated as virtual resources allocated to different users in finite fields, serving to distinguish users. This approach enables the reordering of multiplexing and channel encoding modules, effectively addressing the finite blocklength (FBL) challenge in multiuser reliable transmission. Next, we introduce an orthogonal EP code $\Psi_{\mathrm{o},\mathrm{B}}$ constructed over an extension field GF $(2^{m})$. Using this EP code, we develop a time-division mode of finite-field multiple-access (FFMA) systems, consisting of sparse-form and diagonal-form structures. Based on the diagonal-form (DF) structure, we present a specific configuration, referred to as polarization-adjusted DF-FFMA, which can simultaneously obtain the power gain and coding gain from the entire blocklength. Simulation results demonstrate that the proposed FFMA systems significantly improve error performance over a Gaussian multiple-access channel, compared to a slotted ALOHA system.