Twisted \(1\)-group code[1,2]

Description

Block group-representation code realizing particular irreps of particular groups such that a distance of two is automatically guaranteed. Groups which admit irreps with this property are called twisted (unitary) \(1\)-groups and include the binary icosahedral group \(2I\), the \(\Sigma(360\phi)\) subgroup of \(SU(3)\), the family \(\{PSp(2b, 3), b \geq 1\}\), and the alternating groups \(A_{5,6}\). Groups whose irreps are images of the appropriate irreps of twisted \(1\)-groups also yield such properties, e.g., the binary tetrahedral group \(2T\) or qutrit Pauli group \(\Sigma(72\phi)\).

A \(((3,2,2))_3\) code can implement the qutrit Pauli group \(\Sigma(72\phi)\) transversally, a \(((6,3,2))\) code can implement \(A_5\) transversally, a \(((6,2,2))_3\) implements \(2T\) transversally, and a \(((6,5,2))_3\) code implements \(A_6\) transversally.