AI 中文总结
研究二元线性码在坐标划分下条件解码矩阵的热带分解秩与网格状态复杂度关系，证明其最小加分解秩及热带秩为\(2^{s}\)，给出上界及匹配下界，还探讨了与伊辛模型、图族的联系，指出该秩衡量表示不可压缩性。
AI 中文摘要
设\(C\subseteq\F_2^m\)为二元线性码，\([m]=L\sqcup R\)是其坐标的一个划分。\(C\)在此划分下的条件解码矩阵\(W\)由\(\F_2^{L}\times\F_2^{R}\)索引，其元素\(W(x_L,x_R)\)是陪集首重量\(d((x_L,x_R),C)\)，即从\((x_L,x_R)\)到码的最小汉明距离。我们证明\(W\)的最小加分解秩（Barvinok秩）及其热带秩恰好等于\(2^{s}\)，其中\(s = \dim C - \dim C_L - \dim C_R\)是\(C\)在该划分下最小网格的经典状态复杂度。上界是最小网格上维特比解码的两方读取；贡献是匹配的下界，它适用于任意最小加分解而非仅顺序网格实现，且从由码字横截构建的显式\(2^{s}\times 2^{s}\)热带非奇异子矩阵得出。将\(C\)专门化为图的割空间可将\(W\)与伊辛符号的条件基态能量（受挫指数）等同，并产生条件矩阵的最小加秩在顶点数上呈指数增长的自然图族；对于这些族，我们还记录了对比的局部陈述，即符号的所有有界半径视图都是切换平凡的，所以指数秩完全由非局部结构承载。我们明确指出，此秩衡量的是表示不可压缩性，而非计算硬度：平面族达到相同的指数秩，但其基态可在多项式时间内计算。
英文摘要
Let $C\subseteq\F_2^m$ be a binary linear code and let $[m]=L\sqcup R$ 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 $d\bigl((x_L,x_R),C\bigr)$, 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.
