Generalized Bicycle codes

Generalized Bicycle codes are univariate quasi-cyclic two-block CSS codes. Both circulants come from the same quotient ring and therefore commute, giving a compact algebraic representation tied to cyclic shifts, classical cyclic codes, and syndrome relations.

Choose $a(x),b(x)$ in $R=\mathbb F_q[x]/(x^\ell-1)$. For the one-step cyclic-shift permutation , let and , so With , . The field, , polynomials, and shift orientation are identity data.

Under number-theoretic conditions GB codes can reach asymptotic rate $1/4$, and sparse families with square-root distance lower bounds exist. Unrestricted polynomial weight permits linear-distance constructions, but that is not a fixed-check-weight qLDPC linear-distance theorem. Nontrivial weight-four GB codes are especially close to surface codes.

Cyclic shifts, swapping $a,b$, taking reciprocals, and multiplying by invertible polynomials generate code equivalences and physical permutations. Their induced logical Clifford action must be computed on a logical basis; equivalence operations are not automatically executable fault-tolerant gates.

BP-OSD is a representative finite-length baseline. A specific 6-limited GB family reports an approximately 15% threshold under depolarizing code-capacity noise. The number is tied to that family, BP-OSD configuration, and word-error criterion, not arbitrary GB or circuit-level noise.

Cyclic syndrome codes provide candidate measurement redundancy, but single-shot or two-shot performance requires a dedicated algorithm and noisy-syndrome proof or simulation. Static polynomials do not determine ancilla schedules, hook errors, or circuit thresholds.

An instance stores (q,ℓ,a(x),b(x)), coefficient ordering, shift orientation, a normalized polynomial-equivalence class, expanded matrices, and logical operators. QECDB currently has no exact generalized bicycle filter, so broader 2BGA or bivariate-bicycle lists are not substitutes.

For lookup, verify $\ell$ and the polynomials, normalize by allowed unit multiplication, swapping, reciprocity, and exponent automorphisms, and then compare matrix hashes. [[n,k,d]] or the word ‘Bicycle’ alone cannot distinguish GB, BB, original Bicycle, and other 2BGA codes.