Symmetric Unique Neighbor Expanders and Good LDPC Codes

TL;DR

This paper constructs an infinite family of bounded-degree unique-neighbor expanders as Cayley graphs, answering an open question, and demonstrates that certain LDPC codes are symmetric under a transitive group action.

Contribution

It introduces a new family of Cayley graph expanders with unique-neighbor properties and shows the symmetry of specific LDPC codes under group actions.

Findings

Constructed an infinite family of Cayley graph expanders with unique-neighbor properties.

Proved that certain LDPC codes are symmetric under a simply transitive group action.

Answered an open question by Tali Kaufman regarding Cayley graph expanders.

Abstract

An infinite family of bounded-degree 'unique-neighbor' expanders was constructed explicitly by Alon and Capalbo (2002). We present an infinite family F of bounded-degree unique-neighbor expanders with the additional property that every graph in the family F is a Cayley graph. This answers a question raised by Tali Kaufman. Using the same methods, we show that the symmetric LDPC codes constructed by Kaufman and Lubotzky (2012) are in fact symmetric under a simply transitive group action on coordinates.