Hermitian Hulls of Constacyclic Codes and A New Family of Entanglement-Assisted Quantum MDS Codes

Abstract

Let q be an odd prime power and n = (q² + 1)/2. Through a new idea to present the defining set and a new proof technology with induction and skew asymmetric cosets pairs, we construct a family of maximum distance separable (MDS) codes of length n via constacyclic codes, and determine the dimensions of their Hermitian hulls. Furthermore, from these MDS codes we obtain some new entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes with flexible parameters [[(n,n − 2d + c + 2,d;c]], where d = d₁(q − 1) + 2d₀ + 3, $c=2d_{1}(d_{1}-1)+4\delta _{d_{1},d_{0}}$, 0 ≤ d₁ ≤ (q − 1)/2, 0 ≤ d₀ ≤ (q − 3)/2, $\delta _{d_{1},d_{0}}=d_{0}+1$ if d₁ ≥ d₀ + 2 and $\delta _{d_{1},d_{0}}=d_{1}$ if d₁ ≤ d₀ + 1. The EAQMDS codes have new parameters with odd k,d and even c, which differ completely from all the previous [[n,k,d;c]] EAQMDS codes (even k,d and odd c) with the same length n = (q² + 1)/2. Specially, when d₁ = 0 and d = 2d₀ + 3, the above codes are quantum maximum-distance-separable (QMDS) codes with parameters [[(n,n − 2d + 2,d]] and odd 3 ≤ d ≤ q, which are equivalent to the QMDS codes constructed by Kai and Zhu from negacyclic codes (IEEE Trans. Inf. Theory. 59(2), 1193–1197 38). So, in the sense of equivalence, the QMDS codes constructed by Kai and Zhu are special cases of our result.