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C = BA T + M , determines a series of type 2 unitary row operators. Actually, unitary row operators of type 2 do not affect the set N ( S ′ ) − S ′ I or the error-correcting ability, and the symmetric matrix M can be eliminated. If a set T = { t$_{1}$, · · · , t$_{c}$ } of c < r ancilla qubits are replaced by ebits, it can be verified that N ( S ′ ) = 〈 g$_{j}$ : j / ∈ T, ¯ Z$_{1}$, · · · , ¯ Z$_{k}$, ¯ Z$_{1}$, · · · , ¯ Z$_{k}$ 〉 remains unchanged under the operation of type 3 and type 4 unitary row operators on the h ′ j for j / ∈ T . It suffices to assume that the operation W consists of type 3 and type 4 unitary row operators that operate only on the h ′ j for j ∈ T . To sum up, we have the following theorem. 

Theorem 5. The operation of W is equivalent to applying a series of type 4 unitary row operators, followed by a series of type 3 unitary row operators, on the h ′ j for j ∈ T . In addition, there are 2 2 ck distinct selection operators with 

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Combining the effects of the unitary row operation V with the selection operator W in the previous section, we can optimize an encoding operation of the form U = V U$_{E}$W over 

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possibilities. We call this the encoding optimization procedure for EAQEC codes. Note that we can find another unitary row operator W ′ corresponding to W such that W ′ U$_{E}$ and U$_{E}$W 

