Application of Constacyclic Codes to Quantum MDS Codes

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over F_q2 satisfying C^⊥H⊆ C, where C^⊥H denotes the Hermitian dual code of C. For a linear code C of length n over F_q2, we say that C is a dual-containing code if C^⊥H⊆ C and C≠ F_q2ⁿ. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.