Quantum CSS Duadic and Triadic Codes: New Insights and Properties
Reza Dastbasteh, Olatz Sanz Larrarte, Josu Etxezarreta Martinez, Antonio deMarti iOlius, Javier Oliva del Moral, Pedro Crespo Bofill
TL;DR
The paper develops binary quantum CSS codes from duadic and triadic structures, focusing on extending small splittings to larger, potentially degenerate codes and deriving practical distance bounds. It introduces extended splittings to build higher-dimensional, degenerate quantum duadic codes and provides a BCH-based framework to bound their true minimum distances. A new family of quantum CSS triadic codes is presented, achieving a nontrivial asymptotic rate of at least $1/3$. The results yield numerous explicit code parameters and a scalable method for generating high-distance, resource-efficient quantum codes suitable for future quantum communications and processing.
Abstract
In this study, we investigate the construction of quantum CSS duadic codes with dimensions greater than one. We introduce a method for extending smaller splittings of quantum duadic codes to create larger, potentially degenerate quantum duadic codes. Furthermore, we present a technique for computing or bounding the minimum distances of quantum codes constructed through this approach. Additionally, we introduce quantum CSS triadic codes, a family of quantum codes with a rate of at least $\frac{1}{3}$.