content_image/313535.jpg content_image/313533.jpg content_image/313534.jpg
and define a symplectic inner product ⊙ between two elements ( α$_{1}$, β$_{1}$ ) and ( α$_{2}$, β$_{2}$ ) in Z 2 n by 

<img src='content_image/313535.jpg'>

where · is the usual inner product in Z n $_{2}$. Thus, two elements g, h in G n commute if and only if the symplectic inner product ϕ ( g ) ⊙ ϕ ( h ) is zero. Then a check matrix H corresponding to the stabilizer S is defined as a binary r × 2 n matrix such that the the i − th row vector of H is ϕ ( g$_{i}$ ). For convenience, H is denoted by 

<img src='content_image/313533.jpg'>

where H$_{X}$ and H$_{Z}$ are two r × n binary matrices. The check matrix H must satisfy the following commutative condition, 

<img src='content_image/313534.jpg'>

where Λ$_{2}$$_{n}$ =   $^{O$_{n}$}$ × n I$_{n}$ × n I$_{n}$ × n O$_{n}$ × n    , O$_{i}$ × j is an i × j zero matrix, and I$_{r}$ × r is an r -dimensional identity matrix. The i -th column of H$_{X}$ is the error syndrome of the error operator Z$_{i}$ and the j -th column of H$_{Z}$ is the error syndrome of the error operator X$_{j}$ . The error syndrome of Y$_{l}$ is the sum of the l -th column of H$_{X}$ and the l -th column of H$_{Z}$ . In general, the error syndrome of g = i $^{m}$M$_{1}$ ⊗ M$_{2}$ ⊗ . . . ⊗ M$_{n}$ in G $_{n}$, where M$_{i}$ ∈ { I, X, Y, Z } , is the linear combination of the error syndromes of each factor M$_{i}$ . For a code with minimum distance d , if the error syndromes of error operators of weight smaller than or equal to ⌊ $^{d}$− 1 2 ⌋ are distinct, we call that code nondegenerate. Otherwise, it is degenerate. 

