Structure of CSS and CSS-T Quantum Codes
Elena Berardini, Alessio Caminata, Alberto Ravagnani
TL;DR
The paper analyzes CSS and CSS-T quantum codes, showing that random CSS constructions over large fields yield codes with strong correction capabilities, while CSS-T codes face inherent limits preventing simultaneous high rate and high relative distance. It establishes a general density result for CSS codes and derives stringent bounds for CSS-T code parameters, highlighting that asymptotically good CSS-T families are unlikely. A Hermitian-curve construction is proposed to achieve bounds for CSS-T codes, yielding explicit parameter examples such as $\llbracket 8,3\rrbracket_2$ and $\llbracket 64,31\rrbracket_4$. Overall, the work provides a concise bridge between classical coding theory and quantum error correction, clarifying existence, limits, and explicit constructions for CSS and CSS-T codes.
Abstract
We investigate CSS and CSS-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a CSS code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that CSS-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of CSS-T codes. We conclude with a simple construction of CSS-T codes from Hermitian curves. The paper also offers a concise introduction to CSS and CSS-T codes from the point of view of classical coding theory.