Associations between intrinsic matroid structure and minimum finite field requirements

Finite field representability connects matroid theory with linear network coding, yet variation in minimum field size across matroids lacks a structural explanation. This study examines the association between intrinsic structure and minimum field size. It introduces a duality invariant Matroid Structural Signature (V5-MSS) and a Finite Field Requirement Correlation Index (FRCI). The signature combines dual rank imbalance, intrinsic dependence, circuit and cocircuit coupling, and local geometric capacity. FRCI is an explanatory association index, not a predictor of minimum field size. A SageMath data set contains 52 matroids, including 47 representable cases. Spearman correlation, baseline comparisons, ablation, bootstrap intervals, permutation tests, and leave one family out analyses evaluate the index. Across representable cases, FRCI correlates positively with minimum field size (Spearman ρ = 0.8527, p = 2.81 × 10⁻¹⁴). Element count, corank, and cocircuit count show no significant association; rank is negatively associated, and circuit count is only weakly positive. Dependence and coupling contribute most to the observed association. These findings offer a structural account of field size variation relevant to matroidal network coding, while remaining limited to the studied sample.