Figure 24.6 Cosets of C2 = (7,3) in C1 = (7,4). The codewords of C2 are listed in the top row, while the codewords of C1 (called x = 0, 1, 2, 15) are listed in the left column. The elements of C1 that are in C2 are highlighted with a dark background. The two rightmost columns show the coset names and identify their equivalences, the only two different cosets being captured in 0 + C2 and 2 + C2. the code C2 has the same correction capability as C1, namely t = (dmin - 1)/2 = 1 bit (see Section 11.2). Then comparing the codes listed in Table 24.4 (C1 = (7,4)) and Table 24.5 = C1 = (7,3)), we observe that the second is included in the first, namely the property C2 C C1, i.e., C2 is a subgroup of C1. According to the previous section, such a condition (along with the condition that C2 must have the same error-correction capability t as C1) makes the dual code C2 fully eligible to construct a CSS(C1, C2) quantum code of dimension 2k1-k2 = 24-3 = 2. The construction of CSS(C1, C2) is made by applying the definition in Eq. (24.36), which, given X E C1, requires one to determine all the |C2| = 8 binary codewords + y forming the cosets X +C2, and to sum up the |x + y) quantum states thus defined. Figure 24.6 provides the full list of cosets of C2 in C1. The top row lists the codewords of C2, while the left column lists the codewords of C1 (called X = 0, 1, 2, whose elements that are in C2 have been highlighted with a dark background. From the table information, it is readily observed that all the common elements X E C1, C2 belong to the same coset 0 + C2, while all elements x' E C1, x' & C2 belong to the same coset 2 + C2 (which could as well be called 15 + C2, for instance). Therefore, taking any codeword pair X, x' with such properties is sufficient to generate and define the two codewords of CSS(C1, C2), which we shall call |0), |1). For instance, we can use X = 0000000 (x E C1, C2) and x' = 1111111(x" E C1, x' C2). We, thus, obtain: |0) = |0000000 |0000000> + |0111001) + |1011010) + |1100011) (24.53) |1101100) + |10101