\([[18,4,4]]\) 6.6.6 color code[1,2]

Description

Self-dual pure CSS code that is the 6.6.6 (honeycomb) color code on the torus \(\mathbb{Z}^2/3\mathbb{Z}^2\). The nine hexagonal faces each host a weight-six \(X\)-type and a weight-six \(Z\)-type stabilizer generator, and the 18 qubits sit on the vertices of the tiling, i.e., on the faces of the underlying triangular lattice.

The faces of the honeycomb tiling of a torus \(\mathbb{Z}^2/L\) are three-colorable only when \(L\) lies in a particular index-three sublattice of \(\mathbb{Z}^2\). Among all such tori, this is the smallest one whose 6.6.6 color code has distance four; the smaller admissible tori give the \([[6,4,2]]\) code on three hexagons and a \([[12,4,2]]\) code on six hexagons, both of distance two.

The code is simultaneously the bivariate bicycle code with \(\ell=m=3\), \(A=x+1+y^2\) and \(B=y+1+x^2\) [3], i.e., an Abelian 2BGA code over \(\mathbb{Z}_3\times\mathbb{Z}_3\). It is equivalently the \(\ell=m=3\) member of the bivariate bicycle family \(A=1+x+xy\), \(B=1+y+xy\), which yields 6.6.6 color codes whenever \(\ell\) and \(m\) are multiples of three [4; Exam. 1].

In the bivariate-bicycle qubit ordering \(L_{3a+b},R_{3a+b}\), where the group element \(x^a y^b\) labels the left and right blocks, the stabilizer tableau of the nine \(X\)-type faces is \begin{align} \begin{smallmatrix} X & I & X & X & I & I & I & I & I & X & X & I & I & I & I & X & I & I \\ I & I & I & I & X & X & I & I & X & I & I & X & X & I & X & I & I & I \\ I & X & I & I & I & I & X & X & I & I & I & I & I & X & I & I & X & X \\ X & X & I & I & X & I & I & I & I & I & X & X & I & I & I & I & X & I \\ I & I & I & X & I & X & X & I & I & X & I & I & X & X & I & I & I & I \\ I & I & X & I & I & I & I & X & X & I & I & I & I & I & X & X & I & X \\ I & X & X & I & I & X & I & I & I & X & I & X & I & I & I & I & I & X \\ I & I & I & X & X & I & I & X & I & I & X & I & I & X & X & I & I & I \\ X & I & I & I & I & I & X & I & X & I & I & I & X & I & I & X & X & I \end{smallmatrix}~, \tag*{(1)}\end{align} listed in color classes of three: rows 1–3, rows 4–6, and rows 7–9 each partition all 18 qubits. The \(Z\)-type generators are supported on the same nine strings because \(C_X=C_Z\).

The classical code \(C=C_X=C_Z\) is a self-orthogonal \([18,7]\) code with weight enumerator \(1+18x^6+45x^8+45x^{10}+18x^{12}+x^{18}\). It is even but not doubly even, so this is not a divisible code.