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arXiv:2610.13880

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

Gideon Lee

quant-phSubmitted Oct 9, 2026• Announced Oct 12, 2026• 0 citations• Revised Oct 12, 2026 (v2)
v1 · Oct 9, 2026v2 · Oct 12, 2026
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Abstract

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.

Comments
59 pages: 18 main text + 20 appendices + 8 references + 13 code tables and supplementary figures after the references. 17 figures, 11 tables; v2: corrected literature credits in Tables 3 and 9 Code database: https://github.com/gideonlsx/bicycle-race

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

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Computing the minimum distance of a quantum LDPC code is easy in one direction and expensive in the other: exhibiting a light logical operator is a search, while proving that no lighter one exists is what actually decides the parameter. This note proves an orbit reduction for the second half. If a group of code automorphisms acts on the qubits with r orbits, the minimum distance is the smallest of r restricted problems, one per orbit representative, so the number of solver calls is the number of orbits and not one. We give the a priori condition under which restricting to a single qubit is legitimate, namely transitivity on all qubits, and a counterexample showing what happens without it: on a hypergraph product code with n = 27 and d = 3, the widely used tool dist-m4ri prints a pair of equal bounds, which its own documentation calls a confirmed exact distance, and that value is 6. The reduction is coupled to a proof producing pipeline in which every nonexistence claim is re-verified by an independent checker, drat-trim or, for the largest closure, cake_lpr, whose soundness is proved inside CakeML. This closes with replayable certificates the distances of the five smaller bivariate bicycle codes of Bravyi et al., up to and including the code with n = 288 and d = 18, whose exactness was published with no replayable artifact and is established here at both ends. For the two larger codes it yields the first lower bounds a third party can re-check from the deposited files: d at least 18 for the code with n = 360, against a published upper bound of 24 and against d at least 11 for the best previously re-executable method, and a first lower bound of any kind for the code with n = 756. A separate and much cheaper ingredient, a sweep over logical operators supported on unions of cosets of a subgroup, tightens two upper bounds printed in a recent table of two block group algebra codes. The code printed there as [[140, 8, at most 16]] carries a logical operator of weight 14; the code printed as [[144, 10, at most 16]] has distance exactly 12, established at both ends and so closed four below the published bound. Both witnesses are re-checked outside any solver by two rank computations over the field with two elements, so neither of these two upper bounds rests on a solver at all. Both entries are printed with the inequality sign in that reference, which states no method for the tables they sit in; nothing it proves is contradicted here, and no error of its authors is corrected. What is improved are two bounds they offered as bounds. Finally we show that the two instance form of the reduction is a consequence of commutativity and not of the two block structure. On two block group algebra codes over non-abelian groups, taken from the published catalogue, the translation subgroup has two orbits for one code and twelve for another, so the number of solver calls has to be computed from the group before any search rather than assumed. On three such codes we certify the minimum distance. The argument that makes one orientation enough in the abelian case does not apply there, and we check that it does not, so both orientations are certified separately rather than assumed equal; on these three codes the two turn out to coincide, which is a measured fact about each code and not a consequence of that argument. The published values for these codes come from randomised search, and a certificate of nonexistence a reader can replay is, to our knowledge, new for non-abelian codes of this family. What this work does not claim is stated explicitly in the note: the restriction itself is not our idea, it is an option of dist-m4ri present since 2020; a machine checkable chain covering a symmetry breaking step is not a first; our proofs are checked by an independent checker and not by the kernel of a proof assistant; and the encoding step is not formally verified. A public question asking whether the orbit reduction is already in the literature was posted on MathOverflow on 28 August 2026 and carried no answer at the time of deposit. Every table of numbers in the note is generated from the deposited artifacts by a script that refuses to emit a row it cannot source. 27 pages. Verification scripts and certificate digests included as separate archives. MSC 2020: 81P73, 94B05, 68Q25, 20C05.

•Aug 29, 2026•0 citations
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 10810^8108 two-block codes with up to 320320320 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≤200n \le 200n≤200 to reach kd2/n=12kd^2/n=12kd2/n=12, no weight-six two-block code with n≤300n \le 300n≤300 beats [[288,16,16]][[288,16,16]][[288,16,16]], and the known [[64,18,8]][[64,18,8]][[64,18,8]] is the best two-block code by kd2/nkd^2/nkd2/n across weights five to nine with n≤94n \le 94n≤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 nnn and non-abelian groups at large nnn, with the crossover appearing to move out as the check weight grows; and high quality codes become 100100100 to 100010001000 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]][[102,20,10]][[102,20,10]] and [[120,24,10]][[120,24,10]][[120,24,10]] at check weight nine, which we simulate in noisy syndrome-extraction circuits, and [[186,20,18]][[186,20,18]][[186,20,18]], a weight-nine code with every check in a 5×55\times55×5 window. At check weights seven to nine, our best codes exceed the best published kd2/nkd^2/nkd2/n with exact distance at the same check weight in most length bands up to 320320320 qubits. All codes are available in a public database.
Quantum Bicycle LDPC Codes with High kd2/nkd^2/nkd2/n from Divisor-Driven Search
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[[42,12,4]]_2[[42,12,4]]2​ and [[62,12,4]]2[[62,12,4]]_2[[62,12,4]]2​ and produces a family of codes with competitive figure of merit kd2/nkd^2/nkd2/n, including [[66,20,7]]2[[66,20,7]]_2[[66,20,7]]2​ with kd2/n=14.85kd^2/n=14.85kd2/n=14.85, above the bivariate bicycle code [[144,12,12]]2[[144,12,12]]_2[[144,12,12]]2​ (kd2/n=12kd^2/n=12kd2/n=12) at less than half the block length, together with [[46,2,8]]2[[46,2,8]]_2[[46,2,8]]2​, [[66,2,9]]2[[66,2,9]]_2[[66,2,9]]2​, [[66,4,8]]2[[66,4,8]]_2[[66,4,8]]2​, [[66,6,8]]2[[66,6,8]]_2[[66,6,8]]2​ and, at n=90n=90n=90, [[90,16,6]]2[[90,16,6]]_2[[90,16,6]]2​, [[90,18,6]]2[[90,18,6]]_2[[90,18,6]]2​, [[90,20,6]]2[[90,20,6]]_2[[90,20,6]]2​. An exhaustive census at n=48n=48n=48 delineates the boundary of this picture: we exhibit a [[48,10,6]]2[[48,10,6]]_2[[48,10,6]]2​ code from a minimal 484848-element group (the Aydin--Tamo--Barg realization uses 727272 elements), and prove that distance 555 forces a stabilizer-rank loss, which excludes [[48,10,5]]2[[48,10,5]]_2[[48,10,5]]2​ from the weight-888 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.
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-666 checks, exemplified by [[144,12,12]][[144,12,12]][[144,12,12]] with kd2/n=12kd^2/n=12kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-888 checks (weight-444 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=2dim⁡R/(A,B)k=2\dim R/(A,B)k=2dimR/(A,B) (forcing even kkk), a 4ℓm4\ell m4ℓm-element symmetry group on generator pairs, an X/ZX/ZX/Z distance equality dX=dZd_X=d_ZdX​=dZ​, 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=144n=144n=144 the pipeline returns a census of 535353 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]][[144,6,d\ge 15]][[144,6,d≥15]] exceeds the benchmark distance 121212 (certified d≥15d\ge 15d≥15), [[144,10,12]][[144,10,12]][[144,10,12]] reaches it with weight-888 checks, and [[144,16,10]][[144,16,10]][[144,16,10]] encodes a third more logical qubits at kd2/n=11.11kd^2/n=11.11kd2/n=11.11 (7.4%7.4\%7.4% below benchmark) while decoding no worse. At n=72n=72n=72, [[72,14,8]][[72,14,8]][[72,14,8]] attains kd2/n=12.44kd^2/n=12.44kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-888 codes at ≈0.1%\approx 0.1\%≈0.1% pseudo-threshold versus ≈0.4%\approx 0.4\%≈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.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group GGG and two subsets of GGG whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]][[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]][[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]][[ 85, 8, 9 ]][[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>nkd > nkd>n. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%0.2\%0.2%, approximately matching that of the [[144,12,12]][[ 144, 12, 12 ]][[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
Code surgery measures a logical operator of a quantum LDPC code with O(d)O(d)O(d) rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to 131313 where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of A4A_4A4​, S4S_4S4​ and SL(2,3) with gain three for a best logical, up to [[336,26,12]][[336,26,12]][[336,26,12]]. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
We study finite-length qudit quantum low-density parity-check (LDPC) codes from translation-invariant CSS constructions on two-dimensional tori with twisted boundary conditions. Recent qubit work [PRX Quantum 6, 020357 (2025)] showed that, within the bivariate-bicycle viewpoint, twisting generalized toric patterns can significantly improve the finite-size distance of qubit codes. Building on this insight, we extend the construction and the search to qudit codes over finite fields. Using algebraic methods based on Laurent-polynomial ideals and Gröbner-basis computations over Fq\mathbb{F}_qFq​, we compute the number of logical qudits on any twisted torus and identify compact codes with favorable rate--distance tradeoffs. Overall, for the finite sizes explored, twisted-torus qudit constructions typically achieve larger estimated distances than their untwisted counterparts and, at the same number of physical and logical qudits, larger estimated distances than previously reported twisted qubit instances. The best new codes are tabulated.
Computing the minimum distance of a quantum LDPC code is easy in one direction and expensive in the other: exhibiting a light logical operator is a search, while proving that no lighter one exists is what actually decides the parameter. This note proves an orbit reduction for the second half. If a group of code automorphisms acts on the qubits with r orbits, the minimum distance is the smallest of r restricted problems, one per orbit representative, so the number of solver calls is the number of orbits and not one. We give the a priori condition under which restricting to a single qubit is legitimate, namely transitivity on all qubits, and a counterexample showing what happens without it: on a hypergraph product code with n = 27 and d = 3, the widely used tool dist-m4ri prints a pair of equal bounds, which its own documentation calls a confirmed exact distance, and that value is 6. The reduction is coupled to a proof producing pipeline in which every nonexistence claim is re-verified by an independent checker, drat-trim or, for the largest closure, cake_lpr, whose soundness is proved inside CakeML. This closes with replayable certificates the distances of the five smaller bivariate bicycle codes of Bravyi et al., up to and including the code with n = 288 and d = 18, whose exactness was published with no replayable artifact and is established here at both ends. For the two larger codes it yields the first lower bounds a third party can re-check from the deposited files: d at least 18 for the code with n = 360, against a published upper bound of 24 and against d at least 11 for the best previously re-executable method, and a first lower bound of any kind for the code with n = 756. A separate and much cheaper ingredient, a sweep over logical operators supported on unions of cosets of a subgroup, tightens two upper bounds printed in a recent table of two block group algebra codes. The code printed there as [[140, 8, at most 16]] carries a logical operator of weight 14; the code printed as [[144, 10, at most 16]] has distance exactly 12, established at both ends and so closed four below the published bound. Both witnesses are re-checked outside any solver by two rank computations over the field with two elements, so neither of these two upper bounds rests on a solver at all. Both entries are printed with the inequality sign in that reference, which states no method for the tables they sit in; nothing it proves is contradicted here, and no error of its authors is corrected. What is improved are two bounds they offered as bounds. Finally we show that the two instance form of the reduction is a consequence of commutativity and not of the two block structure. On two block group algebra codes over non-abelian groups, taken from the published catalogue, the translation subgroup has two orbits for one code and twelve for another, so the number of solver calls has to be computed from the group before any search rather than assumed. On three such codes we certify the minimum distance. The argument that makes one orientation enough in the abelian case does not apply there, and we check that it does not, so both orientations are certified separately rather than assumed equal; on these three codes the two turn out to coincide, which is a measured fact about each code and not a consequence of that argument. The published values for these codes come from randomised search, and a certificate of nonexistence a reader can replay is, to our knowledge, new for non-abelian codes of this family. What this work does not claim is stated explicitly in the note: the restriction itself is not our idea, it is an option of dist-m4ri present since 2020; a machine checkable chain covering a symmetry breaking step is not a first; our proofs are checked by an independent checker and not by the kernel of a proof assistant; and the encoding step is not formally verified. A public question asking whether the orbit reduction is already in the literature was posted on MathOverflow on 28 August 2026 and carried no answer at the time of deposit. Every table of numbers in the note is generated from the deposited artifacts by a script that refuses to emit a row it cannot source. 27 pages. Verification scripts and certificate digests included as separate archives. MSC 2020: 81P73, 94B05, 68Q25, 20C05.