Maximal Cube Root (MCR) code

Maximal Cube Root (MCR) code[1]

Description

Qubit GB code whose defining polynomials guarantee a large automorphism group and many fold-transversal CX gates. The construction ties both properties to a nontrivial cube root of unity in a polynomial quotient ring [1].

Let \(R_{\ell}=\mathbb{F}_2[x]/\langle x^{\ell}-1\rangle\), let \(f\) divide \(x^{\ell}-1\), and set \(\widehat f=(x^{\ell}-1)/f\). An MCR code is specified by \((\ell,f,p,q)\), where \(\gcd(p,q,x^{\ell}-1)=1\), through \begin{align} H_X=\left(\operatorname{circ}(pf)\middle|\operatorname{circ}(qf)\right) \quad\text{and}\quad H_Z=\left(\operatorname{circ}(qf)^T\middle|\operatorname{circ}(pf)^T\right). \tag*{(1)}\end{align} The circulant size \(\ell\) is odd, and every irreducible factor of \(\widehat f\) over \(\mathbb{F}_2\) has even degree. The polynomial \(q\) is invertible in \(S=R_{\ell}/\langle\widehat f\rangle\), and the ratio \(r=pq^{-1}\) obeys \begin{align} r^2+r+1=0\quad\text{in }S. \tag*{(2)}\end{align} These conditions imply that the code has length \(n=2\ell\) and dimension \(k=2\deg f\) [1].

Protection

Examples encoding two logical qubits have parameters \([[18,2,5]]\), \([[22,2,6]]\), \([[30,2,7]]\), \([[50,2,9]]\), \([[54,2,10]]\), \([[58,2,11]]\), and \([[66,2,13]]\). Their respective minimum stabilizer-generator weights are \(8,8,8,12,16,12,12\) [1].

Higher-dimensional examples include \([[30,6,5]]\), \([[66,6,8]]\), \([[78,6,9]]\), \([[90,10,10]]\), \([[102,6,11]]\), \([[102,18,d\leq 12]]\), and \([[110,10,10]]\) codes [1].

Transversal and Permutation-Based Gates

A multiplier satisfying \(r(x^j)=r\) or \(r(x^j)=r^{-1}\) simultaneously yields a qubit-permutation automorphism and two fold-transversal CX gates. The physical pairing depends on which equality holds [1].The \([[18,2,5]]\), \([[22,2,6]]\), \([[54,2,10]]\), and \([[66,2,13]]\) examples generate the full two-qubit Clifford group using automorphisms and fold-transversal gates [1].The \([[102,18,d\leq 12]]\) example has 76 listed logical Clifford actions from automorphisms and fold-transversal gates [1].

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

Parents

Qubit CSS code

Generalized bicycle (GB) codeCSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

MCR codes are binary GB codes whose transfer ratio is a nontrivial cube root of unity in a quotient of \(R_{\ell}\) [1].

Maximal Cube Root (MCR) code

Children

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

The \([[18,2,5]]\) BCC code is permutation-equivalent to the MCR instance with \(\ell=9\), \(f=x+1\), \(p=1\), and \(q=x^7+x^4+x^3+x\) [1,2].

\([[22,2,6]]\) shortened Golay code

The \([[22,2,6]]\) shortened Golay code is permutation-equivalent to the MCR code with \(\ell=11\), \(f=x+1\), \(p=1\), and \(q=x^9+x^5+x^4+x^3+x\) [1].

References

[1]: A. Davenport, J. Blue, and I. Chuang, “Generalized Bicycle Codes as Cyclic Submodules and their Automorphism Structure”, (2026) arXiv:2606.05044
[2]: M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865

Page edit log

Victor V. Albert (2026-08-28) — most recent

Cite as:

“Maximal Cube Root (MCR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/maximal_cube_root, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/generalized_bicycle/maximal_cube_root.yml.