An infinite class of quantum codes derived from duadic constacyclic codes
Reza Dastbasteh, Josu Etxezarreta Martinez, Andrew Nemec, Antonio deMarti iOlius, Pedro Crespo Bofill
TL;DR
The authors construct a broad family of binary quantum stabilizer codes from Hermitian dual-containing duadic constacyclic codes over $\mathbb{F}_4$, yielding variable-dimensional codes with a square-root type lower bound on the minimum distance. They develop an extended-splitting technique based on the multiplier $\mu_{-2}$ to analyze dual containment, distance bounds, and degeneracy, producing both non-degenerate and infinite families of degenerate codes, including codes of growing distance for fixed dimension. The work provides theoretical distance bounds for odd-like weights via extended splittings and validates the constructions with extensive numerical results, showing near-best known codes at short lengths. The results bridge classical algebraic coding theory and quantum error correction, offering practical code designs and new tools for distance analysis. Open questions include the asymptotic performance of these codes and the exact distance for broader families.
Abstract
We present a family of quantum stabilizer codes using the structure of duadic constacyclic codes over $\mathbb{F}_4$. Within this family, quantum codes can possess varying dimensions, and their minimum distances are lower bounded by a square root bound. For each fixed dimension, this allows us to construct an infinite sequence of binary quantum codes with a growing minimum distance. Additionally, we prove that this family of quantum codes includes an infinite subclass of degenerate codes. We also introduce a technique for extending splittings of duadic constacyclic codes, providing new insights into the minimum distance and minimum odd-like weight of specific duadic constacyclic codes. Finally, we provide numerical examples of some quantum codes with short lengths within this family.