Pair-partition construction · Okada–Kasai

PP codes

From commuting sparse checks to distance verification and complete logical bases.

Constructing commuting checks

Pair-partition (PP) codes use circulant permutation matrices (CPMs) as building blocks. Pair partitions match terms in each X/Z check product so that they cancel over the binary field. Linear equations on the circulant shifts enforce this cancellation; pairing graphs help screen a design before searching for its shifts.

This separates the combinatorial design from the numerical choice of shifts and gives each code a compact, reproducible description.

Verifying distance

The construction paper gives a complete search for nontrivial logical operators through a chosen weight. Completing the search without a witness proves the corresponding lower bound; finding a nontrivial logical operator gives an upper bound. Cyclic symmetry reduces the support enumeration, while matrix preprocessing and stabilizer-membership tests still depend on the code size. See Section V-A of Pair-Partition Constructions for CPM-Based Quantum LDPC Codes.

From PP checks to complete logical bases

A further development of PP codes constructs physical X and Z representatives for every encoded qubit. Starting from binary CPM–PP checks, we assign coefficients from the four-element field, construct and normalize logical representatives, and expand the matrices into binary form. This produces new qubit CSS codes from the PP starting point.

A complete canonical basis supplies an independent X/Z pair for each logical qubit, with the required commutation relations between pairs. The construction preserves sparse representatives under stated algebraic conditions. Its cost must be assessed alongside the checks: in the paper's worked example, maximum binary check weight increases from 8 to 10. Basis weights and completeness do not by themselves establish the code's minimum distance.

Papers and reproducible data

Pair-Partition Constructions for CPM-Based Quantum LDPC Codes — PP checks and distance verification.
PP-code construction data, CPM instances, and verification records · construction repository
Low-Weight Canonical Logical Bases from Pair-Partition Codes — the quaternary PP extension and complete logical bases.
Logical-basis matrices and verification software