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are equivalent encoding operators. While W operates on the raw stabilizer generators and logcial operators, W ′ operates on the encoded stabilizer generators and logical operators. Hence, we can also solve the optimization problem for an operator of the form U = V W ′ U$_{E}$ (which is what we actually do in practice, combining V W ′ into a single optimization). 

## 3.3 Results of the Encoding Optimization Procedure 

We apply the encoding optimization procedure to a [[7 , 1 , 3]] quantum BCH code [15, 16] and Shor’s [[9 , 1 , 3]] code [1] and the results are shown in Table 3 and Table 4, where d$_{opt}$ is the minimum distance of the optimized EAQEC codes, d$_{std}$ is the highest minimum distance of an [[ n + c, k ]] standard stabilizer code, and N$_{opt}$ is the number of encoding operators that give an EAQEC code with minimum distance d$_{opt}$ . 

Example 4. The check matrix of a standard [[7 , 1 , 3]] quantum BCH code adopted in the encoding optimization procedure is 

<img src='content_image/317488.jpg'>

As shown in Table 3, the parameters [[7 , 1 , 7; 6]], [[7 , 1 , 5; 3]] and [[7 , 1 , 5; 2]] achieve the quantum singleton bound for EAQECC (11) and are not equivalent to any standard quantum stabilizer code. We would like to compare these two EAQEC codes to a competing EAQEC code with n = 7 and d = 5 by the construction of [10] or the binary version (19). According to Grassl’s table [17], a classical linear code over GF (4) (or GF (2)) that meets our requirement is a [7 , 2 , 5] linear quaternary code, which can be used to construct a [[7 , 2 , 5; 5]] EAQEC 

