In this work we show that given a connectivity graph $G$ of a $[[n,k,d]]$ quantum code, there exists $\{K_i\}_i, K_i \subset G$, such that $\sum_i |K_i|\in \Omega(k), \ |K_i| \in \Omega(d)$, and the $K_i$'s are $\tilde{\Omega}( \sqrt{{k}/{n}})$-expander. If the codes are classical we show instead that the $K_i$'s are $\tilde{\Omega}\left({{k}/{n}}\right)$-expander. We also show converses to these bounds. In particular, we show that the BPT bound for classical codes is tight in all Euclidean dimensions. Finally, we prove structural theorems for graphs with no "dense" subgraphs which might be of independent interest.

翻译：本文表明，给定一个[[n,k,d]]量子码的连通图$G$，存在$\{K_i\}_i, K_i \subset G$，使得$\sum_i |K_i|\in \Omega(k), \ |K_i| \in \Omega(d)$，且每个$K_i$为$\tilde{\Omega}( \sqrt{{k}/{n}})$-扩张子。若为经典码，则相应$K_i$为$\tilde{\Omega}\left({{k}/{n}}\right)$-扩张子。同时给出这些界逆命题的证明。特别地，我们证明经典码的BPT界在所有欧几里得维度下均为紧界。最后，我们证明不含"稠密"子图的图的结构性定理，该结果可能具有独立学术价值。