\([[20,8,4]]\) triples code[1]

Description

Self-dual pure CSS code on the 20 three-element subsets of a six-element set. For each of the six points, an \(X\)-type and a \(Z\)-type generator are supported on the ten triples that contain that point.

The underlying classical code is a singly-even self-orthogonal \([20,6,8]\) code with weight enumerator \(1 + 15y^{8} + 32y^{10} + 15y^{12} + y^{20}\). Its codewords are indexed by the subsets \(S\) of the six-element set, with the codeword for \(S\) supported on the triples that meet \(S\) in an odd number of points. The 15 codewords of weight eight correspond to the 15 four-element subsets, and the 15 codewords of weight twelve are their complements.

Its permutation automorphism group is isomorphic to \(\mathbb{Z}_{2}^{5}\rtimes S_{6}\), of order 23040. The group is transitive on the 20 qubits and has rank three, with point-stabilizer orbits of sizes 1, 18, and 1. The group is imprimitive, and its only nontrivial block system is the set of ten pairs of complementary triples.

Rows of the following stabilizer tableau are indexed by the six points, and columns by the 20 triples in lexicographic order [1] \begin{align} \begin{smallmatrix} Z & Z & Z & Z & Z & Z & Z & Z & Z & Z & I & I & I & I & I & I & I & I & I & I \\ Z & Z & Z & Z & I & I & I & I & I & I & Z & Z & Z & Z & Z & Z & I & I & I & I \\ Z & I & I & I & Z & Z & Z & I & I & I & Z & Z & Z & I & I & I & Z & Z & Z & I \\ I & Z & I & I & Z & I & I & Z & Z & I & Z & I & I & Z & Z & I & Z & Z & I & Z \\ I & I & Z & I & I & Z & I & Z & I & Z & I & Z & I & Z & I & Z & Z & I & Z & Z \\ I & I & I & Z & I & I & Z & I & Z & Z & I & I & Z & I & Z & Z & I & Z & Z & Z \end{smallmatrix}~. \tag*{(1)}\end{align}