On combinatorial structures in linear codes

TL;DR

This paper explores the combinatorial structures within linear codes, establishing bounds on expanders related to quantum and classical codes, and providing structural theorems for graphs lacking dense subgraphs.

Contribution

It introduces bounds on expanders in the connectivity graphs of quantum and classical codes and proves the tightness of the BPT bound for classical codes.

Findings

Existence of specific expanders in quantum code graphs with bounds related to code parameters.

Classical codes have different expander bounds, specifically (frac{k}{n})-expanders.

The BPT bound for classical codes is shown to be tight in all Euclidean dimensions.

Abstract

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.