Bivariate bicycle

Let $R=\mathbb{F}_2[x,y]/(x^\ell-1,y^m-1)$ and let $A=a(x,y)$ and $B=b(x,y)$ denote the $\ell m\times\ell m$ binary circulant matrices induced by two bivariate polynomials. A standard CSS presentation is

$H_X=[A\mid B],\qquad H_Z=[B^{\mathsf T}\mid A^{\mathsf T}].$

Because $A$ and $B$ are polynomials in the same commuting cyclic-shift operators, $H_XH_Z^{\mathsf T}=AB+BA=0$ over . The block length is . Choosing three monomials in each polynomial yields the common weight-six BB6 construction.

Under circuit-level noise, BB codes admit a reported pseudo-threshold of about $0.8\%$ with a BP-OSD decoder extended to account for measurement errors. This value depends on the finite instance, extraction circuit, decoder configuration, noise model, and failure criterion. Static code distance $d$ and circuit distance $d_{\mathrm{circ}}$ should therefore be reported separately.

The standard BB6 syndrome-extraction schedule uses seven CNOT layers independently of code length. Circuit-aware BP-OSD extends belief propagation and ordered-statistics decoding to measurement faults. Depth alone does not guarantee distance preservation: for the $[[144,12,12]]$ instance, all $936$ depth-seven schedules obtained by reordering the CNOT layers were reported to satisfy $d_{\mathrm{circ}}\leq10<d$. Other studied approaches include less frequent long-range checks, morphing circuits, BP, BP+OSD, AutDEC, transformer decoders, and matching decoders.

Syndrome extraction for a $[[18,4,4]]$ BB code has been demonstrated on a 32-qubit Kunlun superconducting processor. The same work also reports an $[[18,6,3]]$ variant obtained by removing two checks. These small experimental instances should be distinguished from the $[[144,12,12]]$ Gross-code benchmarks used in many fault-tolerance studies.

BB codes are Abelian 2BGA codes over $\mathbb Z_r\times\mathbb Z_s$ and can be viewed as two-variable multivariate bicycle codes. They are defined on two-dimensional lattices with periodic boundary conditions, while open-boundary variants are also studied. When $r$ and $s$ are coprime, $\mathbb Z_r\times\mathbb Z_s\cong\mathbb Z_{rs}$, relating BB and generalized bicycle constructions. Certain instances are equivalent to families of $6.6.6$ color codes.