Coordinate-View Confusability Graphs and Matroid Rank Certificates

Abstract

A coordinate-view presentation specifies a large confusability graph by coordinates rather than by an edge list. The problem is to certify zero-error recovery and Shannon capacity from the succinct presentation, before the exponentially large graph is materialized. For affine presentations over $\mathbb F_q$, Gaussian elimination gives a polynomial-time rank upper certificate from $O(rd\log q+Ld)$ input bits for a graph with $q^r$ vertices. Exactness of that certificate is equivalent to positivity of a Grassmannian avoidance count $N_{t^*}$. Positivity of $N_k$ is NP-complete for every fixed $k$, already over $\mathbb F_2$, while the ambient-rank parameter gives a fixed-parameter algorithm. The rank-one reduction is parsimonious for $\#SAT$. For arbitrary $t$, the finite decoder for $N_t$ reads the signed rank profile of the represented forbidden-point matroid; the kernel-intersection lattice alone does not determine the count. On the positive side, kernel sections give the exact formula $\Theta(G)=\log\alpha(G)=\min_{S\in\mathcal V}t(S)\log q$, with a projection-equality matrix optimal for Haemers minrank. A finite-field blocking theorem gives exactness when $q\ge L$, and Reed-Solomon/MDS codes give exact all-$k$-view families beyond that regime. Full-tuple coordinate-view graphs also have a polynomial-time cofinal-antichain normal form; transitive confusability is exactly intersection closure.