\([[15,6,3]]\) quantum Hamming subcode

\([[15,6,3]]\) quantum Hamming subcode[1,2]

Description

A pure CSS subcode of the \([[15,7,3]]\) quantum Hamming code obtained by fixing one logical qubit to the \(+1\) eigenspace of its logical \(X\) operator \(X^{\otimes 15}\). Equivalently, \(X^{\otimes 15}\) is adjoined to the stabilizer group [1; Sec. IV.A]. The code can also be obtained from the tesseract color code by removing one qubit.

The \(Z\)-type stabilizers are generated by the \([15,4,8]\) simplex code, while the \(X\)-type stabilizers are generated by the \([15,5,7]\) punctured first-order RM code, i.e., the span of the simplex code and the all-ones vector. A stabilizer tableau for the code, in a qubit ordering for which both classical codes are cyclic, is [2][3; ID 6705228819cca60cf657a8f4] \begin{align} \begin{smallmatrix} Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I & I \\ I & Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I \\ I & I & Z & I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z \\ I & I & I & Z & Z & Z & Z & I & Z & I & Z & Z & I & I & Z \\ X & I & I & I & I & X & I & X & I & I & X & X & I & X & X \\ I & X & I & I & I & X & X & X & X & I & X & I & X & X & I \\ I & I & X & I & I & I & X & X & X & X & I & X & I & X & X \\ I & I & I & X & I & X & I & I & X & X & I & X & X & X & I \\ I & I & I & I & X & I & X & I & I & X & X & I & X & X & X \end{smallmatrix}~. \tag*{(1)}\end{align}

The simultaneous permutation automorphism group of the two classical codes is isomorphic to \(GL(4,2)\), of order 20160, and acts transitively on the qubits [1].

Protection

Since the minimum weight of an \(X\)-type logical operator is 3 while that of a \(Z\)-type logical operator is 4, the code detects \(X\)-type errors on up to 2 qubits and \(Z\)-type errors on up to 3 qubits.

Transversal and Permutation-Based Gates

Qubit permutations combined with transversal CNOT gates realize the full logical linear group \(GL(6,2)\) on one code block and the full \(GL(12,2)\) on pairs of code blocks [1; Sec. IV.A].Transversal \(\sqrt{X}\) preserves the code [2] and implements an order-two logical gate that is a product of Hadamard-conjugated controlled-\(Z\) gates on nine pairs of logical qubits, up to a logical Pauli correction.All logical Clifford gates can be realized as two-fold transversal gates, i.e., by depth-one circuits of two-local code-preserving physical gates [2].

Cousins

\([[15, 7, 3]]\) quantum Hamming code— The \([[15,6,3]]\) code is obtained from the \([[15,7,3]]\) quantum Hamming code by fixing one logical qubit to the \(+1\) eigenspace of its logical \(X\) operator \(X^{\otimes 15}\). Equivalently, \(X^{\otimes 15}\) is adjoined to the stabilizer group [1; Sec. IV.A].
\([[16,6,4]]\) Tesseract color code— The \([[15,6,3]]\) code is obtained from the tesseract color code by removing one qubit, which punctures the underlying first-order RM code on the \(X\) side and shortens it to the simplex code on the \(Z\) side.
\([2^m-1,m,2^{m-1}]\) simplex code— The \(Z\)-type stabilizers of the \([[15,6,3]]\) code are generated by the \([15,4,8]\) simplex code, and its \(X\)-type stabilizers by the simplex code together with the all-ones vector.
\([[15,4,3]]\) quantum Hamming subcode— The \([[15,4,3]]\) code is obtained from the \([[15,6,3]]\) code by fixing two additional logical qubits to the \(+1\) eigenspaces of logical \(X\) operators supported on weight-five codewords of the \([15,7,5]\) BCH code. The two codes have the same \(Z\)-type stabilizer group. Both codes have exactly 35 weight-3 \(X\)-type logical operators. These correspond to the weight-3 codewords of the \([15,11,3]\) Hamming code. Every \(Z\)-type logical operator has even weight at least 4.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeQubit Stabilizer Hamiltonian-based QECC Quantum

Quantum Reed-Muller (RM) codeCSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Quantum Reed-Muller (RM) code

The \([[15,6,3]]\) code is a CSS code constructed from the \([15,5,7]\) punctured first-order RM code and the \([15,4,8]\) shortened first-order RM code.

Qubit BCH codeQubit Stabilizer Hamiltonian-based QECC Quantum

Both classical codes underlying the \([[15,6,3]]\) code are BCH codes: the \([15,5,7]\) punctured first-order RM code is the BCH code of designed distance seven, and the \([15,4,8]\) simplex code is the BCH code of designed distance eight.

Cyclic quantum codeQECC Quantum

The \([[15,6,3]]\) code is cyclic in a qubit ordering in which both underlying classical codes are cyclic.

Small-distance qubit stabilizer codeQubit Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum

\([[15,6,3]]\) quantum Hamming subcode

References

[1]: M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory 534 (2013) arXiv:1302.1035 DOI
[2]: V. V. Albert, “Beyond transversality: structure of Clifford circuits for CSS codes”, (2026) arXiv:2608.05688
[3]: S. Burton, “qecdb.org: Quantum Error Correction Database”, URL

Page edit log

Victor V. Albert (2026-08-27) — most recent
Victor V. Albert (2026-08-26)
Victor V. Albert (2026-08-25)
Victor V. Albert (2026-08-08)

Cite as:

“\([[15,6,3]]\) quantum Hamming subcode”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_15_6_3, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/15/stab_15_6_3.yml.