On combinatorial structures in linear codes

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 ${\sum }_i\mathrm{\mid }{K}_i\mathrm{\mid }\in \mathrm{\Omega }(k),\mathrm{\mid }{K}_i\mathrm{\mid }\in \mathrm{\Omega }(d)$, and the $K_i$’s are $\tilde{\mathrm{\Omega }}(\sqrt{k\mathrm{/}n})$-expander. If the codes are classical we show instead that the $K_i$’s are $\tilde{\mathrm{\Omega }}\left(k\mathrm{/}n\right)$-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.