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\([[18,2,5]]\) BCC code[1]

Description

BCC code on 18 qubits encoding 2 logical qubits with distance 5, found by computer search [1].

The code is defined by \(\mathcal{S}=\{5,11,15,17\}\) in the BCC code framework, with even qubits \((0,2,\ldots,16)\) forming the \(A\)-sublattice and odd qubits \((1,3,\ldots,17)\) forming the \(B\)-sublattice. The subset \(\{5,11,17\}\subset\mathcal{S}\) is closed under adding 6 (mod 18), so the conjugated stabilizer \(UX_0X_6U^\dagger\) has large cancellations, yielding weight-4 stabilizers \(X_{-3}X_0X_3X_6\) and its cyclic shifts by even integers (and likewise with \(X\to Z\)).

A stabilizer tableau for the code is \begin{align} \begin{smallmatrix} X & X & X & I & I & X & I & X & I & I & I & X & I & X & I & X & I & I \\ I & I & X & X & X & I & I & X & I & X & I & I & I & X & I & X & I & X \\ I & X & I & I & X & X & X & I & I & X & I & X & I & I & I & X & I & X \\ I & X & I & X & I & I & X & X & X & I & I & X & I & X & I & I & I & X \\ I & X & I & X & I & X & I & I & X & X & X & I & I & X & I & X & I & I \\ I & I & I & X & I & X & I & X & I & I & X & X & X & I & I & X & I & X \\ I & X & I & I & I & X & I & X & I & X & I & I & X & X & X & I & I & X \\ I & X & I & X & I & I & I & X & I & X & I & X & I & I & X & X & X & I \\ I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I & Z & I & Z & I \\ Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I & Z & I \\ Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I & I & I \\ I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I & Z & I \\ Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I & Z & I \\ Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I & Z & I \\ Z & I & Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z & I & I \\ I & I & Z & I & Z & I & Z & I & I & I & Z & I & Z & I & I & Z & Z & Z \end{smallmatrix}~. \tag*{(1)}\end{align}

Transversal Gates

Transversal Hadamard-SWAP logical gate (inherited from BCC code structure) [1].Logical Hadamard without SWAP: the code is self-dual, so transversal physical Hadamard implements logical \(H^{\otimes 2}\) [1].

Decoding

An ancilla syndrome extraction scheme with 18 shared ancilla qubits (one per data qubit position across two code blocks) detects all single bit-flip errors; a further reduction to 9 entangled ancilla qubits is numerically shown to preserve fault tolerance to distance 5 [1].

Fault Tolerance

An ancilla syndrome extraction scheme with 18 shared ancilla qubits (one per data qubit position across two code blocks) detects all single bit-flip errors; a further reduction to 9 entangled ancilla qubits is numerically shown to preserve fault tolerance to distance 5 [1].

Member of code lists

Primary Hierarchy

Quantum Domain
Galois-qudit Kingdom
Galois-qudit codeQECC Quantum
Galois-qudit USt code
Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
True Galois-qudit stabilizer code
Galois-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Balanced product (BP) codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Lifted-product (LP) code
Two-block group-algebra (2BGA) codesCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized bicycle (GB) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
Bipartite cyclic cluster (BCC) codeCyclic quantum QECC Quantum
Parents
Bipartite cyclic cluster (BCC) code
The \([[18,2,5]]\) BCC code is the smallest BCC code with parameters \([[n,2,5]]\) [1].
Self-dual CSS codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum
The stabilizer group of the \([[18,2,5]]\) BCC code is invariant under transversal Hadamard [1].
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[18,2,5]]\) BCC code

References

[1]
M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865
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Zoo Code ID: stab_18_2_5

Cite as:
“\([[18,2,5]]\) BCC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_18_2_5
BibTeX:
@incollection{eczoo_stab_18_2_5, title={\([[18,2,5]]\) BCC code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_18_2_5} }
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Cite as:

“\([[18,2,5]]\) BCC code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_18_2_5

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/18/stab_18_2_5.yml.