arXiv:2608.09115

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Liangdong Lu, Guanmin Guo, Yang Liu, Ruipan Yang

cs.ITquant-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations• Revised Sep 1, 2026 (v3)

Abstract

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

Comments: v2: Added references on decoding comparison

Related papers

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

quant-pharXiv:2609.06572Sep 8, 20260 citations

Bicycle Race: small two-block codes on the Pareto frontier

Gideon Lee

quant-pharXiv:2610.13880Oct 12, 20260 citations

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

quant-pharXiv:2609.39395Oct 1, 20260 citations

Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

Sheida Rabeti, Hessam Mahdavifar

arXiv:2605.14173May 13, 20260 citations

Breaking the bicycle frame: Coset-based quantum LDPC codes

Arda Aydin, Itzhak Tamo, Alexander Barg

quant-pharXiv:2606.17268Jun 17, 20260 citations

Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

quant-pharXiv:2603.17703Mar 19, 20260 citations

arXiv:2608.09115

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven SearchQuantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Liangdong Lu, Guanmin Guo, Yang Liu, Ruipan Yang

cs.ITquant-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations• Revised Sep 1, 2026 (v3)

arXiv HTML PDF

Abstract

\F_2[x]/(x^{l}-1)

: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over

\F_4

, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes

[[42,12,4]]_2

and

[[62,12,4]]_2

and produces a family of codes with competitive figure of merit

kd^2/n

, including

[[66,20,7]]_2

with

kd^2/n=14.85

, above the bivariate bicycle code

[[144,12,12]]_2

(

kd^2/n=12

) at less than half the block length, together with

[[46,2,8]]_2

[[66,2,9]]_2

[[66,4,8]]_2

[[66,6,8]]_2

and, at

n=90

[[90,16,6]]_2

[[90,18,6]]_2

[[90,20,6]]_2

. An exhaustive census at

n=48

delineates the boundary of this picture: we exhibit a

[[48,10,6]]_2

code from a minimal

48

-element group (the Aydin--Tamo--Barg realization uses

72

elements), and prove that distance

5

forces a stabilizer-rank loss, which excludes

[[48,10,5]]_2

from the weight-

8

symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

Comments: v2: Added references on decoding comparison

Related papers

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight- $6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$ . We develop the algebraic structure theory of BB-type codes with weight- $8$ checks (weight- $4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$ ), a $4\ell m$ -element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$ , and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$ ), $[[144,10,12]]$ reaches it with weight- $8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ( $7.4\%$ below benchmark) while decoding no worse. At $n=72$ , $[[72,14,8]]$ attains $kd^2/n=12.44$ ---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight- $8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

quant-pharXiv:2609.06572Sep 8, 20260 citations

Bicycle Race: small two-block codes on the Pareto frontier

Gideon Lee

Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about $10^8$ two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with $n \le 200$ to reach $kd^2/n=12$ , no weight-six two-block code with $n \le 300$ beats $[[288,16,16]]$ , and the known $[[64,18,8]]$ is the best two-block code by $kd^2/n$ across weights five to nine with $n \le 94$ . The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small $n$ and non-abelian groups at large $n$ , with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$ , a weight-nine code with every check in a $5\times5$ window. At check weights seven to nine, our best codes exceed the best published $kd^2/n$ with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about $10^8$ two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with $n \le 200$ to reach $kd^2/n=12$, no weight-six two-block code with $n \le 300$ beats $[[288,16,16]]$, and the known $[[64,18,8]]$ is the best two-block code by $kd^2/n$ across weights five to nine with $n \le 94$. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small $n$ and non-abelian groups at large $n$, with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$, a weight-nine code with every check in a $5\times5$ window. At check weights seven to nine, our best codes exceed the best published $kd^2/n$ with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.

quant-pharXiv:2610.13880Oct 12, 20260 citations

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$ -logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$ . Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$ .Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.

quant-pharXiv:2609.39395Oct 1, 20260 citations

Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

Sheida Rabeti, Hessam Mahdavifar

We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$ ) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

arXiv:2605.14173May 13, 20260 citations

Breaking the bicycle frame: Coset-based quantum LDPC codes

Arda Aydin, Itzhak Tamo, Alexander Barg

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$ , $[[96,8,10]]$ , and $[[224,12,16]]$ , as well as weight-8 codes with parameters $[[84,16,8]]$ , $[[112,16,10]]$ , $[[128,16,12]]$ , and $[[168,16,15]]$ . Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$ , including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

quant-pharXiv:2606.17268Jun 17, 20260 citations

Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$ . In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$ , exceeding the best multivariate bicycle code of Voss et al.\ ( $7.9\%$ , $kd^2/n = 2.67$ ). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$ . On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$ . We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ( $kd^2/n = 6.48$ ) and $[[128,20,8]]$ ( $kd^2/n = 10.0$ ). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $kd^2/n = 2.67$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ($kd^2/n = 6.48$) and $[[128,20,8]]$ ($kd^2/n = 10.0$). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

quant-pharXiv:2603.17703Mar 19, 20260 citations

Quantum Bicycle LDPC Codes with High

kd^2/n

from Divisor-Driven Search

\F_2[x]/(x^{l}-1)

: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over

\F_4

[[42,12,4]]_2

and

[[62,12,4]]_2

and produces a family of codes with competitive figure of merit

kd^2/n

, including

[[66,20,7]]_2

with

kd^2/n=14.85

, above the bivariate bicycle code

[[144,12,12]]_2

(

kd^2/n=12

) at less than half the block length, together with

[[46,2,8]]_2

[[66,2,9]]_2

[[66,4,8]]_2

[[66,6,8]]_2

and, at

n=90

[[90,16,6]]_2

[[90,18,6]]_2

[[90,20,6]]_2

. An exhaustive census at

n=48

delineates the boundary of this picture: we exhibit a

[[48,10,6]]_2

code from a minimal

48

-element group (the Aydin--Tamo--Barg realization uses

72

elements), and prove that distance

5

forces a stabilizer-rank loss, which excludes

[[48,10,5]]_2

from the weight-

8

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-

6

checks, exemplified by

[[144,12,12]]

with

kd^2/n=12

. We develop the algebraic structure theory of BB-type codes with weight-

8

checks (weight-

4

generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula

k=2\dim R/(A,B)

(forcing even

k

), a

4\ell m

-element symmetry group on generator pairs, an

X/Z

distance equality

d_X=d_Z

, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At

n=144

the pipeline returns a census of

53

codes whose strongest members surpass the BB benchmark:

[[144,6,d\ge 15]]

exceeds the benchmark distance

12

(certified

d\ge 15

[[144,10,12]]

reaches it with weight-

8

checks, and

[[144,16,10]]

encodes a third more logical qubits at

kd^2/n=11.11

(

7.4\%

below benchmark) while decoding no worse. At

n=72

[[72,14,8]]

attains

kd^2/n=12.44

---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-

8

codes at

\approx 0.1\%

pseudo-threshold versus

\approx 0.4\%

for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

10^8

two-block codes with up to

320

qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with

n \le 200

to reach

kd^2/n=12

, no weight-six two-block code with

n \le 300

beats

[[288,16,16]]

, and the known

[[64,18,8]]

is the best two-block code by

kd^2/n

across weights five to nine with

n \le 94

. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small

n

and non-abelian groups at large

n

, with the crossover appearing to move out as the check weight grows; and high quality codes become

100

1000

times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are

[[102,20,10]]

and

[[120,24,10]]

at check weight nine, which we simulate in noisy syndrome-extraction circuits, and

[[186,20,18]]

, a weight-nine code with every check in a

5\times5

window. At check weights seven to nine, our best codes exceed the best published

kd^2/n

with exact distance at the same check weight in most length bands up to

320

qubits. All codes are available in a public database.

Z

-logical quotient

K/S

over

\mathbb F_2[x,y]/(x^\ell-1,y^m-1)

and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity

d_Z=\min(d_{\mathcal A},d_{\mathcal C})

. Using the Frobenius structure of the finite group algebra, we prove

r_{\mathcal A}=r_{\mathcal C}=k/2

for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances

4,6,10,10,12,18

of the six standard BB codes of lengths

18

288

and enumerates every minimum-weight logical operator. The resulting census shows that

[[108,8,10]]

is the only one of the six whose distance is attained in a single component, with

d_{\mathcal C}=10

and

d_{\mathcal A}=12

10^3

) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

[[48,8,6]]

[[96,8,10]]

, and

[[224,12,16]]

, as well as weight-8 codes with parameters

[[84,16,8]]

[[112,16,10]]

[[128,16,12]]

, and

[[168,16,15]]

. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth

w+2

, including initialization and measurement steps, for any code with a maximum stabilizer weight of

w

from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of

\approx0.65\%

for the weight-6 family and

\approx0.35\%

for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

[[140,6,14]]

code with

kd^2/n = 8.40

. In the code-capacity setting, the

[[140,6,14]]

code achieves a pseudothreshold of

8.0\%

and

kd^2/n = 8.40

, exceeding the best multivariate bicycle code of Voss et al.\ (

7.9\%

kd^2/n = 2.67

). With circuit-level depolarizing noise, pseudothresholds reach

0.59\%

for

[[140,6,14]]

and

0.53\%

for

[[84,6,10]]

. On the SI1000 superconducting noise model, the

[[140,6,14]]

code achieves a per-round per-observable rate of

5.6 \times 10^{-5}

p = 0.20\%

. We additionally present two self-dual codes with weight-8 stabilizers:

[[54,14,5]]

(

kd^2/n = 6.48

) and

[[128,20,8]]

(

kd^2/n = 10.0

). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.