The CSS code built from the classical \([7,4,3]\) Hamming code, with all-\(X\) and all-\(Z\) logical operators, packaged with its distance. The Hamming check matrix has as columns the seven nonzero vectors of \(\mathbb {F}_2^3\), so theorem 50 applies with the same three rows on both sides; \(X\) on qubits \(\{ 3,5,6\} \) is the weight-three logical.

For every \(r \ge 3\), the CSS code built from the classical \([2^r-1, 2^r-1-r, 3]\) Hamming code, packaged with its distance; the Steane code (definition 87) is the case \(r = 3\). The columns of the check matrix are the nonzero vectors of \(\mathbb {F}_2^r\), so theorem 50 applies uniformly in \(r\). \(X\) on the first three qubits (columns \(1, 2, 3\), with \(1 \oplus 2 = 3\)) is the weight-three logical.

The nine-qubit Shor code, a repetition code of cat states — historically the first quantum error-correcting code — packaged with its distance. It is degenerate: the two \(X\)-checks cannot tell the three qubits of a block apart, so \(Z_i Z_j\) within a block commutes with every check. It is a stabilizer, though, which is exactly what theorem 51 asks; \(X\) on the first block is the weight-three logical.

The smallest code correcting an arbitrary single-qubit error, and the first non-CSS code in this development: its generators mix \(X\) and \(Z\) on the same qubit, so the CSS machinery does not apply and the distance argument is a direct search over weight-one and weight-two anti-witnesses.

The smallest error-*detecting* code, encoding two qubits with distance two. Its distance is closed by theorem 49.

The parametric generalized-parity family with exactly two stabilizer generators, the all-\(X\) and all-\(Z\) operators. Distance two for every \(m\), again via theorem 49.

The planar \(L \times L\) rotated surface code, an \([[L^2, 1, L]]\) family. It runs on the same homological machinery as the toric code but with boundary, so \(\dim H_1 = 1\) rather than \(2\).

The bit-flip repetition code as a stabilizer code, with \(Z_i Z_{i+1}\) generators, packaged with its distance. It has distance one as a *quantum* code — a single phase error \(Z_0\) commutes with every check but anticommutes with the logical \(X^{\otimes n}\) — which is why it appears here as a degenerate reference point rather than as a usable code.