Small quantum Tanner codes from left--right Cayley complexes

TL;DR

This paper introduces a construction of quantum Tanner codes from left--right Cayley complexes, characterizes their properties, computes their dimensions for specific cases, and identifies small instances with promising parameters.

Contribution

It provides a new framework for constructing quantum Tanner codes from Cayley complexes, including their characterization, dimension calculation, and small example instances.

Findings

Quantum Tanner codes have linear minimum distance and constant rate.

Explicit parameters for small quantum Tanner codes are identified.

The dimension of these codes is computed for right degree 2.

Abstract

Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters $[[144,12,11]]$, $[[432,20,\leq 22]]$ and $[[576,28,\leq 24]]$ for generators of weight 9.