{ ˜ 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