arXiv2026
Certifying that a parity-check matrix has no nonempty set of at most t variable nodes leaving at most b checks unsatisfied arises in minimum-distance and trapping-set analysis. Such absence is the infeasibility of a polynomial system, provable by an independently checkable Nullstellensatz certificate. The degree of these certificates depends strongly on how the two cardinality constraints are encoded, although the encodings have identical solutions. For a local encoding with inclusion-exclusion and cardinality obstructions, we construct explicit certificates of degree at most t + b + K + 1, with K logarithmic in the maximum check degree; this bound is independent of the block length and cannot be improved in general. For a standard counting encoding, a generalized planted-knapsack theorem, which pulls back a knapsack functional with an arbitrary target, gives lower bounds linear in the block length for cycle codes, regular codes of girth at least six near the feasibility boundary, array-based quasi-cyclic codes and minimum-distance certificates for arrays of circulant permutation matrices, where the local degree stays bounded. On array codes and regular codes of girth at least eight, planted column pairs extend these bounds to odd column weight with no unsatisfied checks. The planting argument is optimal among such pullbacks. All computed degrees and certificates are verified in exact rational arithmetic; bounds are distinguished from exact minimum degrees. Local certificates for quasi-cyclic matrices, including a lifted fifth-generation New Radio base graph, pass two independent verifiers.