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Stronger bound for impure distance-three stabilizer codes

An impure [[n,k,3]][[n,k,3]] stabilizer code is a quantum stabilizer code encoding kk logical qubits into nn physical qubits, with minimum distance 33 and with nontrivial stabilizer elements of weight less than the minimum distance. Stronger impure-code bound. Any impure [[n,k,3]][[n,k,3]] stabilizer code satisfies

log⁡2 ⁣(3(n−2)+1)≤(n−1)−k.\log_{2}\!\left(3\left(n-2\right)+1\right)\leq \left(n-1\right)-k.

The bound is motivated by the gap between the authors' new bound and the best known impure codes, together with the intuition that the best distance-three impure code should have exactly one weight-two stabilizer element. Choosing a good syndrome-measurement code for a specific stabilizer code with stabilizer elements of varied weights remains one of the biggest open questions in designing quantum data-syndrome codes.

Sources & referencesView supporting material

Primary source

Andrew Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, arXiv:2302.01527 (2023).

quantum-coding-theoryquantum-error-correctionstabilizer-codes

Progress summary

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The proposed stronger bound remains an unproved conjecture, with no reported counterexample or verification.

A 2023 paper proposed the stronger inequality for every impure [[n,k,3]][[n,k,3]] stabilizer code. It was motivated by the best known impure [[22,15,3]][[22,15,3]] code and the conjecture that an optimal code has exactly one weight-22 stabilizer element.

Known results

  • A 2009 paper gave necessary-and-sufficient existence conditions for general qubit [[n,k,3]][[n,k,3]] stabilizer codes, with r=n−kr=n-k satisfying r≥⌈log⁡2(3n+1)⌉+ϵnr\geq\lceil\log_2(3n+1)\rceil+\epsilon_n.
  • The 2023 paper proved the weaker impure-code bound log⁡2(4(n−k+1))≤n−k\log_2(4(n-k+1))\leq n-k.

2023 conjecture; no reported resolution by August 2026

The stronger inequality remains stated as a conjecture. The retrieved literature reports no proof, counterexample, verification, withdrawal, or retraction concerning this formulation.

Current status (as of August 2026): The stronger impure-code bound is unsettled; only the weaker bound and broader distance-three classification results are established in the retrieved sources.

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