and and the g ′ $_{i}$s and h ′ $_{j}$s will be called the simplified generators of the stabilizer group. Similarly, we denote ¯ Z ′ i = U$_{E}$Z$_{r}$$_{+}$$_{i}$U † $_{E}$, ¯ X ′ j = U$_{E}$X$_{r}$$_{+}$$_{j}$U † $_{E}$. It is obvious that { g ′ $_{1}$, · · · , g ′ $_{r}$, h ′$_{1}$ , · · · , h ′ c } satisfy the commutative relations (3)-(6), and g ′ t$_{i}$ and h ′ i are a symplectic pair. Let S ′ = 〈 g ′ $_{1}$, · · · , g ′ $_{r}$, h ′$_{1}$ , · · · , h ′ c 〉 , and S ′ I = 〈 g$_{j}$ : j / ∈ T 〉 . Hence the normalizer group of S ′ is h ′ j = U$_{E}$X$_{t}$$_{j}$U † $_{E}$, N ( S ′ ) = 〈 g$_{i}$ : i / ∈ T, ¯ Z ′ $_{1}$, · · · , ¯ Z ′ $_{k}$, ¯ X ′ $_{1}$, · · · , ¯ X ′ k 〉 with dimension 2 n − ( r + c ) = 2 k + r − c. The minimum distance d of the EAQEC code defined by S is the minimum weight of an element in N ( S ′ ) − S ′ $_{I}$. In particular, when c = r , we have S ′ I =  and An [[ n, k, d ; c ]] EAQEC code must satisfy the quantum singleton bound for EAQEC codes [10]: N ( S ′ ) = 〈 ¯ Z ′ $_{1}$, · · · , ¯ Z ′ $_{k}$, ¯ X ′ $_{1}$, · · · , ¯ X ′ k 〉 . n + c − k ≥ 2( d − 1) . (11) We define a simplified check matrix H ′ as a binary ( r + c ) × 2 n matrix such that the r + c row vectors of H ′ are ϕ ( g ′ $_{i}$) for i = 1 , · · · , r and ϕ ( h ′ $_{j}$) for j = 1 , · · · , c . For simplicity, we usually order the generators g ′ i and h ′ j so that ϕ ( g ′ $_{i}$) is the i − th row vector of H ′ for i = 1 , · · · , r , ϕ ( h ′ $_{j}$) is the ( j + r ) − th row vector of H ′ for j = 1 , · · · , c , and the j -th and ( j + r ) − th row vectors are a symplectic pair. H ′ must satisfy the commutation relations (7)-(10), and in the case c = r , H ′ Λ$_{2}$$_{n}$ H ′ T =    O$_{r}$ × r I$_{r}$ × r I$_{r}$ × r O$_{r}$ × r    . (12) 11