\([[81,3,5]]\) bivariate tricycle code[1]
Alternative Names: \([[81,3,5]]\) BT code.
Description
Bivariate tricycle code that can code switch into a BB code and that admits a depth-two cup-product \(CCZ\) gate [1]. It is defined over \(\mathbb{F}_2[x,y]/(x^3+1,y^9+1)\) by the binomial elements \(a=xy^3+x^2y\), \(b=1+xy^8\), and \(c=x^2y^4+x^2y^6\) [1; Table III].
Dropping \(c\) yields the 2D component code \(LP(a,b)\), the \([[54,2,6]]\) BB code [1]. Since \(|G|=27\) is odd and \(c=(y^3+y^4)a+(xy^6+xy^7)b\) lies in the ideal \((a,b)\), a one-way transversal CNOT couples each BB logical qubit to a distinct tricycle logical qubit [1; Appx. C].
Gates
Depth-two logical \(CCZ\) circuit via the cup product [1].Cousin
- Bivariate bicycle (BB) code— The \([[54,2,6]]\) BB code with \(a=xy^3+x^2y\) and \(b=1+xy^8\) is the 2D component code of the \([[81,3,5]]\) bivariate tricycle code [1].
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Parents
The \([[81,3,5]]\) bivariate tricycle code is the tricycle code over \(\mathbb{Z}_3\times\mathbb{Z}_9\) with binomial elements \(a=xy^3+x^2y\), \(b=1+xy^8\), and \(c=x^2y^4+x^2y^6\) [1; Table III].
Small-distance qubit stabilizer codeQubit Stabilizer Hamiltonian-based Small-distance block quantum QECC Quantum
\([[81,3,5]]\) bivariate tricycle code
References
- [1]
- C. Li, J. Preskill, and Q. Xu, “Transversal dimension jump for product qLDPC codes”, (2026) arXiv:2510.07269
Page edit log
- Victor V. Albert (2026-09-26) — most recent
Cite as:
“\([[81,3,5]]\) bivariate tricycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bt_81_3_5, arXiv:2606.11484