SC-LDPC Codes Over $\mathbb{F}_q$: Minimum Distance, Decoding Analysis and Threshold Saturation

We investigate random spatially coupled low-density parity-check (SC-LDPC) code ensembles over finite fields. Under different variable-node edge-spreading rules, the random Tanner graphs of several coupled ensembles are defined by multiple independent, uniformly random monomial maps. The two main coupled ensembles considered are referred to as the standard coupled ensemble and the improved coupled ensemble. We prove that both coupled ensembles exhibit asymptotically good minimum distance and minimum stopping set size. Theoretical and numerical results show that the improved coupled ensemble can achieve better distance performance than the standard coupled ensemble. We introduce the essential preliminaries and analytical tools needed to analyze the iterative decoding threshold of coupled ensembles over any finite field. We consider a class of memoryless channels with special symmetry, termed q-ary input memoryless symmetric channels (QMSCs), and show that, for these channels, the distribution of channel messages (in form of probability vectors) likewise exhibits this symmetry. Consequently, we define symmetric probability measures and their reference measures on a finite-dimensional probability simplex, analyze their foundational properties and those of their linear functionals, endow their respective spaces with metric topologies, and conduct an in-depth study of their degradation theory. Based on our analytical framework, we establish a universal threshold saturation result for both of the coupled ensembles over a q-ary finite field on QMSCs. Specifically, as the coupling parameters increase, the belief-propagation threshold of a coupled system saturates to a well-defined threshold that depends only on the underlying ensemble and the channel family.