Kasai code[1]

Description

Actively orthogonal CSS code whose lifts are affine permutation matrices \(\mathrm{APM}(a,b,P)\), the permutation matrices of the maps \(x \mapsto ax+b \bmod P\) with \(\gcd(a,P)=1\) [1]. Two such matrices commute exactly when a quadratic congruence modulo \(P\) in their coefficients holds [1]. The commutation table of active orthogonality is therefore a system of congruences solved by the choice of coefficients, which yields regular, high-girth stabilizer generator matrices at encoding rate at or above one half.

Given \(L\), \(J\), and an active set \(\Gamma\), the code is specified by lifts \(\mathcal{F}=\{F_i\}\) and \(\mathcal{G}=\{G_i\}\) drawn from the group of affine permutations on \(P\) letters, with \([F_i,G_j]=0\) for every \((i,j)\in\Gamma\). The stabilizer generator matrices are the first \(J\) block rows of the block-circulant parents built from \(\mathcal{F}\) and \(\mathcal{G}\), giving \(n=LP\) qubits.

Co-designed Kasai codes for reconfigurable qubit arrays additionally require every syndrome-extraction transition permutation to commute with a fixed reference affine permutation [2]. In the qubit ordering along the orbits of the reference permutation, each transition is a cyclic shift within the orbits plus a permutation between them. Each qubit-permutation step of syndrome extraction therefore compiles into a few global cyclic shifts [2]. These codes require girth at least six rather than the girth eight of the original construction. Instances include \([[1152,580,\leq 12]]\), \([[2304,1156,\leq 14]]\), and the movement-compatible \([[2304,1156,\leq 16]]\) and \([[4608,2308,\leq 22]]\) [2].

The original instance is a \((3,12)\)-regular, girth-eight \([[9216,4612,\leq 48]]\) code with \(P=768\) and rate approximately \(0.5004\) [1]. It attains a frame error rate of \(10^{-8}\) on the depolarizing channel at physical error rate \(4\%\) under BP decoding with post-processing [1].