Quantum MDS and Synchronizable Codes from Cyclic and Negacyclic Codes of Length 2 p over F p

Abstract

© 2013 IEEE. Let p be an odd prime, and \mathbb 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 \mathbb 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 \mathbb 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.