\([[47,1,11]]\) quantum QR code[1]

Description

A \([[47,1,11]]\) self-dual doubly-even CSS code constructed from a binary quadratic-residue code. It is one of the shortest known qubit stabilizer codes of distance eleven that realizes the full Clifford group transversally [2].

More generally, shortening the \([48,24,12]\) extended QR code on \(j\) bits yields a self-orthogonal doubly even code whose CSS code is, for \(j=1,\dots,15\), the \([[47,1,11]]\), \([[46,2,10]]\), \([[45,3,9]]\), \([[44,4,8]]\), \([[43,5,7]]\), \([[42,6,6]]\), \([[41,7,6]]\), \([[40,8,5]]\), \([[39,9,5]]\), \([[38,10,5]]\), \([[37,11,4]]\), \([[36,12,4]]\), \([[35,13,4]]\), \([[34,14,3]]\), and \([[33,15,3]]\) code, respectively. Since \(PSL(2,47)\) is 3-transitive, the shortened code is independent of which bits are chosen for \(j\leq 3\); for larger \(j\) it depends on the choice, e.g., \(j=6\) also yields a \([[42,6,7]]\) code.