{ ˜ g$_{4}$, ˜ h$_{3}$ } = 0. Let h ′ 1 = ˜ h$_{2}$ ˜ h$_{4}$ = IXXXX , h ′ 2 = ˜ h$_{1}$ , h ′ 3 = ˜ h$_{4}$ , and h ′ 4 = ˜ h$_{1}$ ˜ h$_{3}$ = XXXXI . Then we have [ g ′ $_{i}$, g ′ $_{j}$] = 0 , for all i, j, [ f ′ $_{i}$, f ′ $_{j}$] = 0 , for all i, j, { g ′ $_{i}$, f ′ i } = 0 , for all i, [ g ′ $_{i}$, f ′ $_{j}$] = 0 , for i = j. Hence we have introduced 4 simplified generators such that there are 4 symplectic pairs. Observe that if any one of the simplified generators h ′ $_{1}$, h ′ $_{2}$, h ′ $_{3}$, h ′ 4 is removed and c = 3, the minimum distance instantly drops to 2. If two simplified generators h ′ $_{1}$, h ′ 2 or h ′ $_{1}$, h ′ 4 or h ′ $_{2}$, h ′ 3 are removed and c = 2, the minimum distance further decreases to 1. ✷ According to [10], given a parity-check matrix ˆ H of an [ n, k, d ] classical binary linear code, an [[ n, 2 k + c − n, d ; c ]] EAQEC code can be constructed from a simplified check matrix H ′ , defined as where the number of ebits c required for this EAQEC code is given by (14). The family of EAQEC codes in Theorem 2 can also be obtained by this construction. When c = n − k, the quantum singleton bound (11) becomes H ′ =   $^{O}$ ˆ H ˆ H O    , (19) n − k ≥ d − 1 , which is exactly the same as the classical singleton bound. However, there are no nontrivial classical binary codes achieve the singleton bound from [14]. 20