Improved rate or distance for full-Clifford CSS families

Determine whether there exist CSS code families that transversally realize the full logical Clifford group and whose rate or minimum distance scales better than the square root of the physical blocklength.

Background

The paper describes existing full-logical-Clifford constructions whose number of logical qubits and minimum distance scale as the square root of the number of physical qubits. After identifying the goal of shorter single-block constructions, it explicitly isolates improving these asymptotic parameters as an open problem.

A positive result would improve the rate or error-correction performance of CSS families supporting transversal implementations of the full logical Clifford group beyond the currently discussed square-root scaling.