A Construction of New Quantum MDS Codes

It has been a great challenge to construct new quantum maximum-distance-separable (MDS) codes. In particular, it is very hard to construct the quantum MDS codes with relatively large minimum distance. So far, except for some sparse lengths, all known q-ary quantum MDS codes have minimum distance ≤q/2 + 1. In this paper, we provide a construction of the quantum MDS codes with minimum distance >q/2 + 1. In particular, we show the existence of the q-ary quantum MDS codes with length n = q² + 1 and minimum distance d for any d q + 1 (this result extends those given in the works of Guardia (2011), Jin et al. (2010), and Kai an Zhu (2012)); and with length (q² + 2)/3 and minimum distance d for any d (2q+2)/3 if 3|(q + 1). Our method is through Hermitian selforthogonal codes. The main idea of constructing the Hermitian self-orthogonal codes is based on the solvability in F_q of a system of homogenous equations over F_q².