\([[1152,580,\leq 12]]\) co-designed Kasai code[1]

Description

Kasai code with \(J=3\), \(L=12\), and lift size \(P=96\) whose affine permutations are chosen so that every transition permutation of syndrome extraction commutes with a fixed reference affine permutation having three orbits of length \(32\) [1]. In the qubit ordering along those orbits, each transition is a cyclic shift within the orbits plus a permutation between them, which yields a three-by-thirty-two layout with a syndrome-extraction schedule of constant qubit-movement cost [1].

The code has \(n=LP=1152\) qubits and girth at least six, and non-commutativity is imposed on the index pairs \((0,3)\) and \((1,2)\) [1]. The twelve affine permutations \(f_i(x)=a_ix+b_i\) and \(g_i(x)=c_ix+d_i\) defining the \(G\)-lift are tabulated in Ref. [1].

The code is permutation equivalent to the GALA code \(\mathrm{GALA}_{12,3}(S_3\times\mathbb{Z}_{32})\), and its \([[2304,1156]]\) sibling to \(\mathrm{GALA}_{12,3}(S_3\times\mathbb{Z}_2\times\mathbb{Z}_{32})\) [2]. The non-Abelian factor \(S_3\) and Abelian factor \(\mathbb{Z}_{32}\) of that presentation reproduce the reference-permutation orbit structure and recover the same movement schedule from the group data [2].