Quantum code beating the best known [[n,k,d]]

Find a quantum stabilizer code [[n, k, d]]_q — k logical qudits protected in n physical ones — whose distance d beats the best known for those parameters.

Bar to beat. The entry in the quantum section of codetables.de for the chosen [[n, k]]_q, which lists the best known distance and the best known upper bound.

Submission format. The n - k stabilizer generators as Pauli strings, or equivalently the binary symplectic check matrix.

Verifier sketch. Check the generators pairwise commute (their symplectic form vanishes), that they are independent (rank n-k), and then compute the distance: the minimum weight of a Pauli operator commuting with every stabilizer but lying outside the stabilizer group. That last step is a minimum-weight computation on a symplectic code and is exponential in general.

Not written yet because it is the classical minimum-distance problem plus symplectic bookkeeping, so the same budget question applies: an honest verifier exists only for pinned small parameters, and someone has to decide which.

Known. The tables collect constructions (quantum BCH, surface and colour codes, algebraic-geometry codes); recent qLDPC constructions have moved the asymptotic picture but the short-length cells are still contested.

Why. n/k at fixed d is the physical-qubit overhead of a fault-tolerant machine. Improvements at short length translate directly into fewer qubits per logical qubit, the dominant cost in every roadmap.

Ref: codetables.de quantum tables · Wikipedia: Quantum error correction