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

2026-06-22

AI Summary

This study investigates the computational complexity of determining the minimum distance of regular LDPC codes. Focusing on left-regular (with variable-node degree $J \geq 3$) and bi-regular ($J, K \geq 3$) Tanner graph structures, the authors establish—through a novel reduction framework that preserves both node degrees and codeword structure—that the minimum distance problem is NP-complete and W[1]-complete under these regularity constraints. This framework leverages degree-preserving transformations, including hyperedge decomposition, check-node splitting, and controlled variable replication. The results rigorously characterize the parameterized intractability of exactly computing the minimum distance for any regular LDPC code family with fixed degrees at least three, thereby delineating fundamental theoretical limits on its exact solvability.

Abstract

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.

Problem

• minimum distance problem
• LDPC codes
• computational complexity
• regular Tanner graphs
• NP-completeness

Innovation

• minimum distance problem
• LDPC codes
• computational intractability
• regular Tanner graphs
• parameterized complexity