Two new families of quantum synchronizable codes

Abstract

In this paper, we present two new ways of quantum synchronization coding based on the (u + v|u − v) construction and the product construction respectively, and greatly enrich the varieties of available quantum synchronizable codes. The circumstances where the maximum synchronization error tolerance can be reached are explained for both constructions. Furthermore, repeated-root cyclic codes derived from the (u + v|u − v) construction are shown to be able to provide better Pauli error-correcting capability than BCH codes.

On a Family of Quantum Synchronizable Codes Based on the $(\lambda(u + v)|u - v)$ Construction

“…In this paper, we expand the work of [9] and present a family of quantum synchronizable codes based on the (λ(u+v)|u−v) construction. This family of quantum synchronizable codes are derived from cyclic codes and constacyclic codes.…”

“…One is based on the (u+v|u−v) construction from cyclic codes and negacyclic codes, and the other is to exploit the product construction to produce new cyclic codes from two cyclic codes with coprime lengths. In the former case, the obtained quantum synchronizable codes were shown to be able to provide better performance in correcting Pauli errors than non-primitive, narrow-sense BCH codes [8], [9], and achieve the maximum tolerance against misalignment under certain condition.…”

“…Subsequently, quantum synchronizable codes were presented from finite geometric codes [5], quadratic residue codes [7] and repeated-root cyclic codes [8]. Furthermore, Luo et al [9] provided two new ways of constructing quantum synchronizable codes. One is based on the (u+v|u−v) construction from cyclic codes and negacyclic codes, and the other is to exploit the product construction to produce new cyclic codes from two cyclic codes with coprime lengths.…”

“…For an (a l , a r ) − [[n, k]] quantum synchronizable code, it can correct not only Pauli errors, but block misalignment to the left by at most a l qudits and to the right by at most a r qudits for a pair of nonnegative integers (a l , a r ). Luo et al [9] exploited the well-known (u + v|u − v) construction on cyclic codes and negacyclic codes to get new cyclic codes with twice the lengths. Then they provided a new way in regard to generating quantum synchronizable code.…”

Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2ps Over F pm

“…Later, the authors in [29], [61], [92], [93] showed that finite geometric codes, quadratic residue codes, duadic codes and repeatedroot codes can be applied in synchronization coding. In [61], [62], Luo et al proved that repeated-root cyclic codes are useful in QSCs with better parameters in correcting Pauli errors than non-primitive, narrow-sense BCH codes and other available QSCs.…”