Due to the application of high density data storage systems, symbol-pair codes are proposed to combat errors of the overlapping symbol pairs output over symbol-pair read channels. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that they have the highest pair error-correcting capability. In this paper, we construct two new classes of MDS symbol-pair codes with minimum pair distance seven based on simple-root cyclic codes. Our technique is through the decomposition of cyclic codes and the dual of each component code.

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length ((qm-1)/λ) with designed distance 2 ≤ δ ≤ ((qΓ(m+1)/2⌉- 1)/(λ) + 1) is settled, where λ is any factor of (q - 1). 2) The weight distributions of two classes of narrow-sense BCH codes of length ((qm- 1)/2) with designed distance δ = (((q - 1)qm-1- q⌊(m-1)12⌋- 1)/2) and δ = (((q - 1)qm-1- q⌊(m+1)/2⌋- 1)/2) are determined. 3) The weight distribution of a class of BCH codes of length ((qm- 1)/(q - 1)) is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.

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.

Letqbe an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters [[q2+1,q2-2d+3,d]]qwhereq≡ 1 (mod 4) and 2 ≤d≤q+1 is even, and [[(q2+1)/2,(q2+1)/2-2d+2,d]]qwhere 3 ≤d≤qis odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones.

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.