Affine permutation matrices · Quantum 10, 2205 (2026)
Kasai codes
Sparse qubit CSS codes built from affine permutation matrices, with commutation constraints tailored to the checks that define the code.
Construction idea
Affine permutation matrices (APMs) represent permutations obtained by multiplication and translation modulo the block size. Their algebra lets us choose which pairs of permutation blocks commute while retaining regular check-matrix degrees and controlling short cycles.
In this construction, CSS orthogonality is imposed on the active check rows. The unused rows of the larger parent matrices are designed so that they do not automatically become low-weight logical operators. This addresses a specific source of distance loss when check rows are removed to increase the coding rate.
Paper and representative example
Breaking the Orthogonality Barrier in Quantum LDPC Codes, Quantum 10, 2205 (2026), develops the construction and evaluates joint sum-product BP (SPA) with post-processing. The arXiv version provides the full paper.
The paper's girth-8, (3,12)-regular example has 9,216 physical qubits, 4,612 logical qubits, and a reported distance upper bound of 48. This is an upper bound, not a claim that the minimum distance equals 48. The catalog records instance-specific parameters and distance evaluations.
Individual codes and distance evaluation
- Kasai Code Catalog, including the distance upper-bound table
- P=216 candidate catalog
- P=2688 candidate catalog
- Definitions of the distance upper bounds
- How to read APM parameter files
Software and presentations
- APM-CSS construction demo
- Joint SPA + post-processing decoder demo
- Post-processing algorithms
- Slides and speaker scripts
- Software and code data · Research FAQ
For other construction principles, see PP codes and other QLDPC constructions.