Trellis State Complexity as an Exact Tropical Factorization Rank

Abstract

Let $C\subseteq\F_2^m$ be a binary linear code and let $[m]=L\bigsqcup R$ be a bipartition of its coordinates. The \emph{conditional decoding matrix} of $C$ at this cut is the matrix $W$ indexed by $\F_2^{L}\times\F_2^{R}$ whose entry $W(x_L,x_R)$ is the coset-leader weight $d((x_L,x_R),C)$, the minimum Hamming distance from the word $(x_L,x_R)$ to the code. We prove that the min-plus factorization rank (Barvinok rank) of $W$, and likewise its tropical rank, equal $2^s$ exactly, where $s=\dim C-\dim C_L-\dim C_R$ is the classical state complexity of the minimal trellis of $C$ at the cut. The upper bound is a two-party reading of Viterbi decoding on the minimal trellis; the contribution is the matching lower bound, which holds against arbitrary min-plus factorizations rather than only sequential trellis realizations, and is obtained from an explicit $2^s\times 2^s$ tropically nonsingular submatrix built from a transversal of codewords. Specializing $C$ to the cut space of a graph identifies $W$ with the conditional ground-state energy of Ising signings (the frustration index), and yields natural graph families whose conditional matrices have min-plus rank exponential in the number of vertices; for these families we also record the contrasting local statement that all bounded-radius views of a signing are switching-trivial, so the exponential rank is carried entirely by non-local structure. We note explicitly that this rank measures representational incompressibility, not computational hardness: planar families attain the same exponential rank while their ground states are computable in polynomial time.