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arXiv · 2606.23579

Optimal Small Set Expanders and Their Codes

Tristram Bogart·Marcelo Fiori·Pedro Raigorodsky·Mauricio Velasco
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Abstract

A left-regular bipartite graph $G$ of degree $d$ is called a $(t,\alpha)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $\alpha |X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.

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Tristram Bogart, Marcelo Fiori, Pedro Raigorodsky, Mauricio Velasco. 2026-06-22. Optimal Small Set Expanders and Their Codes. https://arxiv.org/abs/2606.23579

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