Steane seven-qubit CSS code

The columns of $H$ are the seven distinct nonzero elements of ${\mathbb{F}}_2^3$, so they span ${\mathbb{F}}_2^3$ and $\mathrm{rank}H=3$. The first four columns of $G$ form the identity matrix, so $G$ has full row rank $4$. Direct coordinate dot products give $H{G}^{\mathsf{T}}=0$, so the row space of $G$ lies in $\ker H$. Rank-nullity gives $\dim \ker H=7-3=4$; therefore $\mathrm{row}(G)=C_1$ and $\dim C_1=4$.

The rows of $H$ are independent because $\mathrm{rank}H=3$. Their pairwise dot products are zero: each row has weight four and every two distinct rows overlap in two coordinates. Hence $H{H}^{\mathsf{T}}=0$, so $C_2\subseteq C_1$ and $\dim C_2=3$. The quotient dimension is $k=4-3=1$. Every vector in $C_1=\ker H$ is orthogonal to every row of $H$, so $C_1\subseteq C_2^{\perp }$; both dimensions are $4$ by step 1.1 and [F6], so $C_1=C_2^{\perp }$. Also $H{G}^{\mathsf{T}}=0$ and $C_1=\mathrm{row}(G)$, so $C_2=\mathrm{row}(H)\subseteq C_1^{\perp }$; both dimensions are $3$, hence $C_2=C_1^{\perp }$.

No weight-one vector lies in $C_1$ because every column of $H$ is nonzero. No weight-two vector lies in $C_1$ because its syndrome is the sum of two distinct columns, which is nonzero. The vector $x=1110000$ is in $C_1$ because $h_1+h_2+h_3=0$. Thus $d(C_1)=3$.

Every nonzero word of $C_2$ is a nonzero linear functional evaluated on the seven nonzero vectors of ${\mathbb{F}}_2^3$. Such a functional is $1$ on exactly four of the eight vectors, and is $0$ on the zero vector, so every nonzero word of $C_2$ has weight four. Therefore $x\mathrm{\notin }{C}_2$, $X_L$ and $Z_L$ represent nontrivial logical classes, and their commutator is $(-1{)}^{x\cdot x}=(-1{)}^3=-1$.

The row spaces of $H$ used for the $X$ and $Z$ checks commute because $H{H}^{\mathsf{T}}=0$. Their six labels are independent: the three $X$-type labels are independent among themselves, the three $Z$-type labels are independent among themselves, and the two label halves are separate. Hence $r=6$ and [F4] gives code dimension $2^{7-6}=2$. These checks stabilize the two displayed coset states by [F2].

The columns $h_1,\dots ,h_7$ are nonzero and pairwise distinct. By steps 2.1 and 2.2, $d(C_1)=d(C_2^{\perp })=3$, so [F5] applies with $t=1$. The syndrome formulas in the Example follow by counting the overlaps between $X_j,Z_j,Y_j$ and each ordered check. The three syndrome blocks distinguish the Pauli type, and within each type the nonzero distinct column $h_j$ identifies the site. Thus all $21$ single-site Pauli errors have distinct syndromes and are corrected by [F5].