Quantum MDS and synchronizable codes from cyclic and negacyclic codes of length 4 p^s over F^pm

Abstract

In this paper, we establish all quantum error-correcting codes (briefly, QEC codes) from cyclic and negacyclic codes of length 4 ps over Fpm using CSS and Steane’s constructions. Some new QEC codes are given in the sense that their parameters are different from all the previous constructions. In addition, if we fix the length and the Hamming distance of a QEC code, some QEC codes are better than all QEC codes in the sense that their dimensions are larger than the dimensions of all the previous QEC codes. We also investigate quantum maximum distance separable (briefly, qMDS) codes from cyclic and negacyclic codes of length 4 ps over finite fields using the CSS and Hermitian constructions. We provide all qMDS codes constructed from dual codes of cyclic and negacyclic codes of length 4 ps over finite fields using the Hermitian construction. We also construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 4 ps over Fpm. 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 non-primitive narrow-sense BCH codes.