Non-Binary Quantum Synchronizable Codes From Repeated-Root Cyclic Codes

Abstract

In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths $p^{s}$ and $lp^{s}$ over $\mathbb {F}_{q}$ , where $s\geq 1~and l\geq 2$ are integers, and $p\geq 3$ is the odd characteristic. Within some loose limitations, these synchronizable codes can possess the best possible capability in synchronization recovery, and therefore, enriches the variety of good quantum synchronizable codes. Furthermore, by using known techniques in classical coding theory which convert the computation of the minimum distance of a repeated-root cyclic code to that of a shorter simple-root cyclic code, we prove that the repeated-root cyclic codes of lengths $p^{s}$ and $lp^{s}$ are in general better than narrow-sense BCH codes of close lengths in terms of minimum distances, and thereby enable the obtained synchronizable codes to correct more Pauli errors.