Quantum codes from affine variety codes and their subfield-subcodes
Carlos Galindo, Fernando Hernando
TL;DR
The paper addresses constructing quantum stabilizer codes with good parameters by leveraging affine variety codes and their subfield-subcodes. It combines CSS code construction with a detailed analysis of subfield-subcodes via the $\mathcal{T}$ map and cyclotomic-set structure to obtain stable codes over small fields while exploiting larger evaluation fields. The authors provide a constructive basis and dimension formula for subfield-subcodes, establish a self-orthogonality criterion using cyclotomic complements, and derive explicit stabilizer-code parameters; several new codes exceed the Feng-Ma quantum Gilbert-Varshamov bound and improve on prior results. This yields practical families of quantum codes with strong error-correction capabilities over smaller alphabets, expanding options for quantum error correction in finite-field settings.
Abstract
We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction. With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.