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introduced in Section 2. We discuss the encoding optimization procedure by first considering the case of maximal entanglement and the selection operator in Section 3.1, including the construction of the family of [[ n, 1 , n ; n − 1]] EAQEC codes for n odd. We then generalize to adding arbitrary amounts of entanglement. The effect of unitary row operations are discussed in Section 3.2, which completes the encoding optimization procedure. The results of applying the encoding optimization procedure to some standard quantum stabilizer codes are provided in Section 3.3. Then we propose the random search algorithm for EAQEC codes in Section 4, together with several examples. The circulant construction of EAQEC codes is described in Section 5, followed by the discussion section. 

## 2 Preliminaries 

## 2.1 Stabilizer Codes 

Suppose S is an abelian subgroup of Pauli group G n that does not include − I , with a set of r ≡ n − k independent generators { g$_{1}$, g$_{2}$, · · · , g$_{r}$ } . An [[ n, k, d ]] quantum stabilizer code C ( S ) corresponding to the stabilizer group S is the 2 $^{k}$-dimensional subspace of the n − qubit state space fixed by S . The minimum distance d is the minimum weight of an element in N ( S ) − S , where N ( S ) is the normalizer group of S . An element g = i $^{m}$M$_{1}$ ⊗ M$_{2}$ ⊗ . . . ⊗ M$_{n}$ in G $_{n}$, where M$_{i}$ ∈ { I, X, Y, Z } and m ∈ { 0 , 1 , 2 , 3 } , can be expressed as g = i m $^{′}$X$_{α}$ Z$_{β}$ with α, β two binary n -tuples and m ′ ∈ { 0 , 1 , 2 , 3 } . In this expression, if M$_{j}$ = I, X , Z , or Y , then the j − th bits of α and β are ( α$_{j}$, β$_{j}$ ) = (0 , 0), (1 , 0), (0 , 1), or (1 , 1), respectively, and m ′ ≡ m + l (mod 4), where l is the number of M$_{j}$ ’s equal to Y . We define a homomorphism ϕ : G n ↦→ Z 2 n by 

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