CSS codes and classical-code relations

For a binary row h, form $X(h)$ by placing X on each position containing one, and identity elsewhere; define $Z(h)$ similarly. Rows of a matrix $H_X$ specify X-type checks, and rows of $H_Z$ specify Z-type checks. Checks of the same type automatically commute. A mixed pair commutes exactly when its two rows overlap in an even number of positions. Matrix multiplication over the binary field packages all these requirements at once.

In this convention the classical bit patterns obeying Z checks form $\ker H_Z$, while the supports of X stabilizers lie in $\operatorname{row}H_X$. Compatibility means that the latter space is contained in the former. Exchanging X and Z gives the complementary inclusion. This specifies conventions explicitly because textbooks may name the two classical codes differently. Counting independent rows, rather than printed rows, gives the quantum dimension and avoids mistaking a redundant parity for another protected degree of freedom.

Choose the same three-row matrix H for both types, with rows 1111000, 1100110, and 1010101. Each row has weight four and any two rows overlap twice. Thus even self-overlaps and pairwise overlaps give $HH^{\mathsf T}=0$. The three rows are independent, so six independent quantum checks on seven physical qubits leave one logical qubit. The seven columns are precisely the seven distinct nonzero binary triples.

For an error represented by binary vectors x and z, the Z checks report $H_Zx$ and the X checks report $H_Xz$. A single X error is located by its nonzero column in the first syndrome; a single Z is located in the second. A Y error has both components at the same site and triggers both sets. Treating these syndromes separately is algebraically valid, but multiplying their likelihoods requires an additional independence assumption about the physical noise.

The all-seven X and all-seven Z strings commute with every weight-four check and anticommute with each other, so they furnish logical representatives. Smaller representatives can be obtained by multiplying stabilizers. No weight-one or weight-two nonidentity Pauli has zero syndrome: distinct nonzero columns cannot sum to zero. In contrast, columns two, three, and five sum to zero, giving a weight-three Z normalizer element outside the Z stabilizer row space.

Every nonzero row-space word of H has weight four, so that weight-three string is nontrivial on the code. The distance is therefore three, establishing ideal correction of one arbitrary physical-qubit error. This conclusion uses both complementary checks, unlike the three-qubit repetition construction. It still assumes correct initialization, compatible syndrome measurements, and a suitable recovery. Separability of the check matrices helps implementation, but does not automatically make a parity-extraction circuit fault tolerant.