AI 中文总结
本文提出了一种综合征支持层次结构，用于处理检查擦除问题，通过分析余集领袖重量和子空间结构，区分独立syndrome与子空间特性，并探讨了不同码的性能差异。
AI 中文摘要
令A：F_2^n→F_2^m是一个二元线性映射，具有固定坐标基，令C_A=ker A，令λ_A(y)是syndrome y的预像的最小汉明重量。我们定义Shat_{q,s}(A)为一个q维syndrome子空间的最常见检查支持，其每个非零元素的余集领袖重量至少为s。因此，它区分了释放q个独立syndrome与释放一个子空间没有简单线性组合的情况。删除检查坐标F会释放ker A_{F̄}/ker A，这与(im A)[F] canonically同构。有限性意味着R_q(C_A)≥N_2(q,s)，其中N_2(q,s)是二元码的最短长度，其维数为q且距离至少为s；profile-Griesmer界限独立控制常见检查支持。该层次结构是坐标重标号不变的，但在检查基的变化下可能会改变。对于配对重复码C_n={(x,x):x∈F_2^n}，标准实现H_0=[I_n I_n]有Shat_{q,s}(H_0)=N_2(q,s)当可行时。对于每个q≥1和s≥2，当n=N_2(q,s)时，同一码的行等价实现值为q。对于简单共界映射A=δ_k，检查擦除是顶面擦除，释放的商是新兴上同调。当s=1时，该层次结构减少为广义汉明重量，并由Tutte确定；当s≥2时，即使标签相同的切割码也可能有不同的值。
英文摘要
Let $A:\mathbb{F}_2^n\to\mathbb{F}_2^m$ be a binary linear map with fixed coordinate bases, let $C_A=\ker A$, and let $λ_A(y)$ be the minimum Hamming weight of a preimage of the syndrome $y$. We define $\operatorname{Shat}_{q,s}(A)$ as the least common check support of a $q$-dimensional syndrome subspace whose every nonzero element has coset-leader weight at least $s$. It therefore distinguishes release of $q$ independent syndromes from release of a subspace with no easy linear combination. Deleting check coordinates $F$ releases $\ker A_{\bar{F}}/\ker A$, canonically isomorphic to $(\operatorname{im} A)[F]$. Finiteness implies $R_q(C_A)\ge \mathsf{N}_2(q,s)$, where $\mathsf{N}_2(q,s)$ is the shortest length of a binary code of dimension $q$ and distance at least $s$; profile-Griesmer bounds independently control common check support. The hierarchy is coordinate-relabeling invariant but can change under a change of check basis. For the pair-repetition code $C_n=\{(x,x):x\in\mathbb{F}_2^n\}$, the standard realization $H_0=[I_n\ I_n]$ has $\operatorname{Shat}_{q,s}(H_0)=\mathsf{N}_2(q,s)$ whenever feasible. For every $q\ge 1$ and $s\ge 2$, with $n=\mathsf{N}_2(q,s)$, a row-equivalent realization of the same code has value $q$. For a simplicial coboundary map $A=δ_k$, check erasure is top-face erasure and the released quotient is emergent cohomology. At $s=1$ the hierarchy reduces to generalized Hamming weights and is Tutte-determined; for $s\ge 2$, even identical labeled cut codes can have different values.
