Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2p^s Over F _p^m

Abstract

Let p be an odd prime, and F p m is the finite field of p m elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2p s over F p m are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2p s over finite fields using the Calderbank-Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2p s over F p m. To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes. INDEX TERMS Cyclic codes, repeated-root codes, Hamming distance, MDS codes, quantum MDS codes, quantum synchronizable codes.

Some Classes of New Quantum MDS and Synchronizable Codes Constructed From Repeated-Root Cyclic Codes of Length 6ps

“…We also gave some QSCs constructed from cyclic and negacyclic codes of length 2p s over F p m . However, in [28], we did not find some QEC codes using the CSS and Steane's constructions. In this paper, we construct some new QEC codes from cyclic codes of length 6p s using the CSS and Steane's constructions and some new QSCs from cyclic codes of length 6p s .…”

“…In [28], we studied qMDS codes from negacyclic and cyclic codes of length 2p s over F p m . We also gave some QSCs constructed from cyclic and negacyclic codes of length 2p s over F p m .…”

On the complete weight distributions of quantum error-correcting codes

“…[36][37][38][39][40][41][42][43][44][45][46][47][48] provided explicit constructions of many new families of quantum MDS codes, whose weight distribution would satisfy Theorems 5 and 6. Quantum MDS codes from constacyclic codes [49,50] and repeated-root cyclic codes [51][52][53] are also presented. Here Theorems 5 and 6 apply to all quantum MDS codes, including those which are not quantum stabilizer codes.…”