Cofilling Shattering: A Syndrome-Support Hierarchy for Check Erasures
Information Theory
Summary
The authors study a way to measure how difficult it is to find certain error patterns (called syndromes) in binary linear codes based on their minimum weight and dimension. They introduce a new hierarchy, Shat, which captures the smallest support needed to hide all easy-to-find error patterns in a subspace of syndromes. They show how this hierarchy relates to classical code parameters like minimum distance and generalized Hamming weights, and how it depends on the choice of code representation. Their results also connect to topological notions in simplicial complexes and provide examples where codes with the same structure differ in this hierarchy.
Authors
Joshua Steier
Abstract
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.