New quantum MDS codes derived from constacyclic codes

Quantum error-correcting codes play an important role in both quantum communication and quantum computation. It has experienced a great progress since the establishment of the connections between quantum codes and classical codes (see [4]). It was shown that the construction of quantum codes can be reduced to that of classical linear error-correcting codes with certain self-orthogonality properties. Let q be a prime power. A q-ary quantum code Q of length n and size K is a K-dimensional subspace of a q-dimensional Hilbert space H = C n = C ⊗ · · · ⊗ C. The error correction and deletion capabilities of a quantum error-correcting code are the most crucial aspects of the code. If a quantum code has minimum distance d, then it can detect any d − 1 and correct any ⌊(d − 1)/2⌋ quantum errors. Let k = logqK. We use [[n, k, d]]q to denote a q-ary quantum code of length n with size q k and minimum distance d. One of the principal problems in quantum error-correction is to construct quantum codes with the best possible minimum distance. It is well known that quantum codes with parameters [[n, k, d]]q must satisfy the quantum Singleton bound: k ≤ n− 2d+ 2 (see [13] and [14]). A quantum code achieving this bound is called a quantum maximum-distance-separable (MDS) code. Quantum MDS codes are the most useful and interesting quantum codes in quantum error correction. In recent years, constructing quantum MDS codes has become one of the central topics for quantum codes. Several families of quantum MDS codes have been constructed (see [3],[5], [6], [7], [8], [9], [10], [16], [18], [19]). As we know, if the classical MDS conjecture holds, the length of nontrivial q-ary quantum MDS codes cannot exceed q + 1 (see [13]). The problem of constructing quantum MDS codes with n ≤ q+1 has been completely solved (see [6] and [7]). Many quantum MDS codes of length between q+1 and q +1 have also been constructed (see [3], [9], [10], [16], [17], [18]). Although so, there are still a lot of quantum MDS codes difficult to be constructed. It is a great challenge to construct new quantum MDS codes and a even more challenge to construct quantum MDS codes with relatively large minimum distance. As mentioned in [10], except for some sparse lengths, almost all known q-ary quantum MDS codes have minimum distance less than or equal to q/2 + 1. Recently, Kai and Zhu constructed two new classes of quantum MDS codes based on classical negacyclic codes (see [11]) and six new classes of quantum MDS codes based on classical constacyclic codes (see [12]). These codes have minimum distance larger than q/2 + 1 in general. Two classes of the quantum MDS codes constructed in [12] are