Recently, symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One main task in symbol-pair coding theory is to design codes with large minimum pair distance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense they attain maximal minimum pair distance. In this paper, based on constacyclic codes, we construct some new MDS symbol-pair codes with minimum pair-distance five and six. Compared with classical q-ary MDS codes, the constructed MDS symbol-pair codes have length up to q2+q+1.

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.

Using Reed-Solomon codes and cyclic MDS codes of length 2m+ 1 over F2m, we obtain several classes of negacyclic MDS codes over GR(2a, m) of lengths 2m- 1 and 2m+ 1 as analogues of Hensel lifts of these codes.