On the Intractability of the Minimum Distance Problem for Regular LDPC Codes

The minimum distance problem (MDP) for low-density parity-check (LDPC) codes is a central problem in coding theory and is closely related to the analysis of low-weight codewords and error-floor behavior. Although the unrestricted MDP is computationally intractable, its complexity under degree constraints that commonly occur in LDPC code design has remained less clear. In this paper, we study the MDP for left regular and biregular Tanner graphs. We prove that the problem is $\mathrm{NP}$-complete and $\mathrm{W}[1]$-complete for $J$-left regular Tanner graphs for every fixed $J\geq 3$, and also for $(3,3)$-regular bipartite graphs. We further establish $\mathrm{W}[1]$-completeness for $(J,K)$-regular instances for every fixed $J,K\geq 3$. The reductions are based on a degree-preserving transformation framework consisting of hyperedge decomposition, check node splitting, and controlled variable replication. These transformations transfer hardness between different degree distributions while preserving explicit bijections among nonzero codewords, even covers, and nonempty $(a,0)$-trapping sets. The results delineate the computational limits of exact LDPC code analysis under natural regularity constraints.