Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2<italic>p^s</italic> Over F_{<italic>p^m</italic>}

Abstract

Let p be an odd prime, and Fp(m) is the finite field of pm elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2ps over Fp(m) are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2ps 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 2ps over Fp(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.

Keywords

• Cyclic codes
• repeated-root codes
• Hamming distance
• MDS codes
• quantum MDS codes
• quantum synchronizable codes