Algebraic Quantum Synchronizable Codes

TL;DR

This paper introduces new quantum synchronizable codes constructed from cyclic, BCH, duadic, and repeated root cyclic codes, expanding the methods for quantum error correction with practical examples.

Contribution

It generalizes the construction of quantum synchronizable codes to repeated root cyclic codes and introduces new methods using code intersections, sums, and products.

Findings

Infinite families of QSCs from BCH and duadic codes

Extension of Fujiwara's work to repeated root cyclic codes

Construction of QSCs from cyclic code products

Abstract

In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara~\cite{fujiwara1} can be generalized to repeated root cyclic codes (RRCCs) such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulness of this extension is illustrated via examples of infinite families of QSCs from repeated root duadic codes. Finally, QSCs are constructed from the product of cyclic codes.