Kasai code

Kasai code[1]

Description

Qubit actively orthogonal CSS code whose block-circulant parent matrices are obtained by a \(G\)-lift with affine permutation matrices. Since affine permutations need not commute, the CSS orthogonality condition can be imposed on a chosen subset of block rows while being deliberately broken elsewhere, which allows regular, high-girth stabilizer generator matrices at encoding rate at or above one half.

An affine permutation matrix \(\mathrm{APM}(a,b,P)\) is the permutation matrix of the map \(x \mapsto ax+b \bmod P\) with \(\gcd(a,P)=1\). These form the group of affine permutations on \(P\) letters. 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 this group that commute, \([F_i,G_j]=0\), for every \((i,j)\in\Gamma\), with commutation deliberately broken on at least one pair outside \(\Gamma\). 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-designs for reconfigurable neutral-atom arrays impose the additional requirement that every syndrome-extraction transition permutation commute with a fixed reference affine permutation, so that each atom-rearrangement step compiles into a few global cyclic shifts [2]. That work does not require the girth-eight condition of the original construction, imposing girth at least six instead, and reports much smaller instances such as \([[1152,580,\leq 12]]\), \([[2304,1156,\leq 14]]\), and the movement-compatible \([[2304,1156,\leq 16]]\) and \([[4608,2308,\leq 22]]\) [2].

Protection

The parameters must satisfy \(L \geq 4J\): otherwise active orthogonality forces every block of the parent product to vanish, the latent rows become orthogonal as well, and the construction degenerates to a row-deleted fully orthogonal code [1]. Since the rank of each active matrix is at most \(JP\), the encoding rate is at least \(1-2J/L\), hence at least one half in this regime [3].

Reported distances are latent-based upper bounds obtained from the deleted rows. Certified lower bounds are not established [1]. Girth eight is a structural ceiling rather than merely an achieved value: once \(J \geq 3\) and the required commutation conditions hold, the active Tanner graphs necessarily contain an eight-cycle [1].

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

Decoding

Belief propagation with post-processing, using a library of elementary trapping sets, flip history, and local ordered-statistics decoding [1].

Cousins

Affine-permutation-matrix LDPC (APM-LDPC) code— The blocks of the Kasai parent matrices are affine permutation matrices, the same building blocks used in APM-LDPC codes [1].
Quasi-cyclic QLDPC (QC-QLDPC) code— Kasai codes with trivial multiplier \(a=1\) reduce to the QC-QLDPC codes of Hagiwara and Imai: the lifts are commuting cyclic shifts, so orthogonality holds on all rows and the distance is capped by the row weight when a removed row is independent of the retained rows [1,4].
Group-action lift with active orthogonality (GALA) code— Kasai codes whose reference affine permutation acts freely, or which admit a coprime factorization of the lift size with commuting reductions, are GALA codes [3]. The GALA framework replaces the affine permutations, whose symmetries arise incidentally, with an explicit group product in which the non-Abelian factor governs orthogonality and the Abelian factor governs symmetry.
Pair-partition CPM code— Replacing the circulant permutation matrices by affine permutation matrices extends the pair-partition construction, and a structured subfamily of that extension has the same check matrices as the active rows of Kasai codes. A separate condition on the pair partitions reproduces the complementary latent rows [1,5].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Actively orthogonal CSS codeCSS QLDPC Stabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Actively orthogonal CSS code

Kasai codes are actively orthogonal CSS codes whose lifts are affine permutation matrices [1].

Kasai code

Children

Cornucopia code

Since \(\gcd(q,3)=1\), the Chinese remainder theorem relabels the \(P=3q\) coordinates of each block, identifying \(\mathbb{Z}_3\times\mathbb{Z}_q\) with the ring \(\mathbb{Z}_{3q}\) [6]. Affine maps are preserved under this identification, so each lift entry \((x,y)\mapsto(ax+b,y+s)\) becomes a single affine permutation of \(3q\) letters, whose multiplier reduces to \(a\) modulo \(3\) and to one modulo \(q\) [6].

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

This Kasai code satisfies the reference-affine-permutation commutation condition, which relaxes the girth from eight to six in exchange for qubit-permutation schedules of constant cost [2].

References

[1]: K. Kasai, “Breaking the Orthogonality Barrier in Quantum LDPC Codes”, (2026) arXiv:2601.08824
[2]: C. Zhao, C. Duckering, A. Gu, N. Maskara, and H. Zhou, “Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays”, (2026) arXiv:2604.16209
[3]: W. Yang, C. Duckering, and A. Dua, “Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays”, (2026) arXiv:2608.07431
[4]: M. Hagiwara and H. Imai, “Quantum Quasi-Cyclic LDPC Codes”, 2007 IEEE International Symposium on Information Theory 806 (2007) arXiv:quant-ph/0701020 DOI
[5]: K. Okada and K. Kasai, “Pair-Partition Constructions for CPM-Based Quantum LDPC Codes”, (2026) arXiv:2607.14091
[6]: Z. Lu, W. Li, and D.-L. Deng, “Quantum error correction at ultra-low overhead”, (2026) arXiv:2608.02773

Page edit log

Victor V. Albert (2026-08-25) — most recent
Victor V. Albert (2026-08-22)
Victor V. Albert (2026-08-17)
Victor V. Albert (2026-08-15)

Cite as:

“Kasai code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/kasai, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/active_orthogonality/kasai.yml.