[[140,6,10]] · QEC Challenge

[[140,6,10]] d ≤

140

kd²/n

4.286

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 10 · witness weight 10 (claimed exact)

witness operator (support, 10 qubits)

[10, 19, 40, 47, 52, 59, 72, 88, 89, 139]

d_Z 10 · witness weight 10 (claimed exact)

witness operator (support, 10 qubits)

[13, 16, 66, 67, 75, 96, 103, 108, 115, 136]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×70 (2,2)×70 (3,2)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 70 (1,3): 70 (2,2): 70 (2,3): 420 (2,4): 210 (3,2): 70 (3,3): 1330 (3,4): 2310 (3,5): 840 (3,6): 420 (3,7): 70

trapping sets H_Z (1,2)×70 (2,2)×70 (3,2)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 70 (1,3): 70 (2,2): 70 (2,3): 420 (2,4): 210 (3,2): 70 (3,3): 1330 (3,4): 2310 (3,5): 840 (3,6): 420 (3,7): 70

Construction & provenance

authors @cbjuan

provenance submitted through the challenge

novelty novelty not audited

construction Bivariate bicycle code over F_2[x,y]/(x⁵-1, y¹⁴-1), weight-5 mixed-monomial polynomials A = 1 + y¹³ + x⁴ y¹¹, B = 1 + x y⁷, found by an LLM-guided evolutionary search (OpenEvolve, weight-5 mixed-monomial campaign) over CSS BB polynomial pairs. This connected (ell=5, m=14) presentation is BLISS-isomorphic to one of 3 identical connected components of a disconnected parent BB code (ell=15, m=14, k=18, n=420, A = 1 + y¹³ + x¹² y¹¹, B = 1 + x³ y⁷) whose exact distance d=10 was established by MILP in the source repository; the component presentation used here was independently re-verified exact (d_X=d_Z=10, no lighter logical on either side) with this repository's own verify/certify.py (scipy/HiGHS MILP, tlim=600s, 256s wall time).

model GPT 5.6 Sol (claimed, not verified)

builds on an internal weight-5 bivariate-bicycle evolutionary campaign in github.com/qiskit-community/qcode-discovery (branch weight5-campaigns, not part of a public branch at the time of this submission -- this submission's own distance/CSS claims are independently re-derived and re-verified using only this repository's tools and do not rely on that source being externally checkable)

date 2026-09-07

notes Independently corroborated with this repo's own tools before submission: research/kit/surrogate.py distance_rand at 5,000 and 50,000 trials both returned d=10; research/kit/distance.py decoder_distance (BP+OSD, 200,000 injected-error trials) returned d_heuristic=10, verdict=corroborated; research/kit/distance.py exact_distance (scipy/HiGHS MILP) proved d_X=d_Z=10 exact. Discovered by GPT 5.6 Sol (per the source campaign's own per-candidate model-attribution log, alias azure/gpt-5.6-sol); this submission's re-verification and packaging was done by Claude Sonnet 5.

family bivariate bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[140,6,10]] — weight-5 mixed-monomial bivariate bicycle code

Direction & hypothesis

Target: the unrestricted x weight-6 (and looser) cells. This code has maximum check weight 5, one below the weight-6 ceiling, the axis the existing [[140,6,14]] twisted-torus baseline on this board does not beat it on (that entry has weight-6 checks; higher d does not dominate a lower-w code on a different axis). As with the companion [[96,4,10]] submission, the direction here was mining an already-completed weight-5 bivariate-bicycle (BB) evolutionary campaign run in a companion repository for a small, directly-constructible, strongly-evidenced code, rather than running a fresh search in this repository.

What was searched

The source campaign (an internal weight-5 bivariate-bicycle evolutionary campaign in github.com/qiskit-community/qcode-discovery, branch weight5-campaigns -- not part of a public branch at the time of this submission, so nothing below relies on that source being externally checkable) ran an LLM-guided evolutionary search (OpenEvolve) over weight-5 mixed-monomial CSS BB polynomial pairs (A, B), screening by k, CSS commutation, and a BP-OSD distance estimate, then MILP-verifying standout candidates exactly. This code is a direct match: a disconnected parent presentation at (ell=15, m=14) with k=18, n=420 decomposes into 3 identical connected components; one component has an explicit small BB presentation at (ell=5, m=14), A = 1 + y^13 + x^4y^11, B = 1 + xy^7, matched to the parent via BLISS canonical-form isomorphism (evaluation/tanner_equivalence.py in the source repo), which transfers the parent's exact distance to the component.

No fresh search ran in *this* repository; the work here was independent re-verification of an already-found candidate using this repo's own trusted tools, per research/AUTORESEARCH.md.

Evidence trail

Rebuilt (H_X, H_Z) from the polynomials above using this repo's own research/kit/bb.py:build_bb, then re-confirmed every claim independently with this repo's own tools before submitting:

research/kit/css.py:verify_css / compute_k — CSS commutation holds,

k = 6 exactly (GF(2) rank), n = 140, max check weight 5 on both sides.

research/kit/surrogate.py:distance_rand — randomized upper-bound search at

5,000 and again at 50,000 trials, both returning d = 10 (converged).

research/kit/distance.py:decoder_distance — independent BP+OSD mechanism,

200,000 injected-error trials, d_heuristic = 10, verdict = corroborated.

research/kit/distance.py:exact_distance (scipy/HiGHS MILP via

verify/certify.py) — proved exact: no nontrivial logical lighter than weight 10 exists on the X side or the Z side. Wall time 256s at tlim = 600s (k = 6 solves per side, inside the envelope described in CONTRIBUTING.md).

Final claim: exact, d = 10, certified by this repository's own MILP solver, not only by the source repository's internal claim.

Dead ends

Not applicable — this submission mined an existing, already-vetted campaign result rather than running a new search in this repository.

Tools

Discovering model, per the source campaign's own per-candidate model-attribution log: GPT 5.6 Sol (azure/gpt-5.6-sol), matching provenance.model. This submission's independent re-verification and packaging in this repository was done separately by Claude Sonnet 5, via Claude Code, operating on this repository's research/kit and verify/ modules only (no edits to verify/). Compute: a handful of CPU-seconds to minutes per check on a single machine; the MILP certification above was the most expensive step at 256s.

Reproduction

from bb import build_bb
HX, HZ = build_bb(l=5, m=14, A_terms=[(0,0),(0,13),(4,11)], B_terms=[(0,0),(1,7)])

Parity checks

X-checks 70 (max weight 5) · Z-checks 70 (max weight 5)

H_X (70 checks, sparse supports)

[0, 13, 67, 70, 91] [0, 1, 68, 71, 92] [1, 2, 69, 72, 93] [2, 3, 56, 73, 94] [3, 4, 57, 74, 95] [4, 5, 58, 75, 96] [5, 6, 59, 76, 97] [6, 7, 60, 77, 84] [7, 8, 61, 78, 85] [8, 9, 62, 79, 86] [9, 10, 63, 80, 87] [10, 11, 64, 81, 88] [11, 12, 65, 82, 89] [12, 13, 66, 83, 90] [11, 14, 27, 84, 105] [12, 14, 15, 85, 106] [13, 15, 16, 86, 107] [0, 16, 17, 87, 108] [1, 17, 18, 88, 109] [2, 18, 19, 89, 110] [3, 19, 20, 90, 111] [4, 20, 21, 91, 98] [5, 21, 22, 92, 99] [6, 22, 23, 93, 100] [7, 23, 24, 94, 101] [8, 24, 25, 95, 102] [9, 25, 26, 96, 103] [10, 26, 27, 97, 104] [25, 28, 41, 98, 119] [26, 28, 29, 99, 120] [27, 29, 30, 100, 121] [14, 30, 31, 101, 122] [15, 31, 32, 102, 123] [16, 32, 33, 103, 124] [17, 33, 34, 104, 125] [18, 34, 35, 105, 112] [19, 35, 36, 106, 113] [20, 36, 37, 107, 114] [21, 37, 38, 108, 115] [22, 38, 39, 109, 116] [23, 39, 40, 110, 117] [24, 40, 41, 111, 118] [39, 42, 55, 112, 133] [40, 42, 43, 113, 134] [41, 43, 44, 114, 135] [28, 44, 45, 115, 136] [29, 45, 46, 116, 137] [30, 46, 47, 117, 138] [31, 47, 48, 118, 139] [32, 48, 49, 119, 126] [33, 49, 50, 120, 127] [34, 50, 51, 121, 128] [35, 51, 52, 122, 129] [36, 52, 53, 123, 130] [37, 53, 54, 124, 131] [38, 54, 55, 125, 132] [53, 56, 69, 77, 126] [54, 56, 57, 78, 127] [55, 57, 58, 79, 128] [42, 58, 59, 80, 129] [43, 59, 60, 81, 130] [44, 60, 61, 82, 131] [45, 61, 62, 83, 132] [46, 62, 63, 70, 133] [47, 63, 64, 71, 134] [48, 64, 65, 72, 135] [49, 65, 66, 73, 136] [50, 66, 67, 74, 137] [51, 67, 68, 75, 138] [52, 68, 69, 76, 139]

H_Z (70 checks, sparse supports)

[0, 63, 70, 71, 87] [1, 64, 71, 72, 88] [2, 65, 72, 73, 89] [3, 66, 73, 74, 90] [4, 67, 74, 75, 91] [5, 68, 75, 76, 92] [6, 69, 76, 77, 93] [7, 56, 77, 78, 94] [8, 57, 78, 79, 95] [9, 58, 79, 80, 96] [10, 59, 80, 81, 97] [11, 60, 81, 82, 84] [12, 61, 82, 83, 85] [13, 62, 70, 83, 86] [7, 14, 84, 85, 101] [8, 15, 85, 86, 102] [9, 16, 86, 87, 103] [10, 17, 87, 88, 104] [11, 18, 88, 89, 105] [12, 19, 89, 90, 106] [13, 20, 90, 91, 107] [0, 21, 91, 92, 108] [1, 22, 92, 93, 109] [2, 23, 93, 94, 110] [3, 24, 94, 95, 111] [4, 25, 95, 96, 98] [5, 26, 96, 97, 99] [6, 27, 84, 97, 100] [21, 28, 98, 99, 115] [22, 29, 99, 100, 116] [23, 30, 100, 101, 117] [24, 31, 101, 102, 118] [25, 32, 102, 103, 119] [26, 33, 103, 104, 120] [27, 34, 104, 105, 121] [14, 35, 105, 106, 122] [15, 36, 106, 107, 123] [16, 37, 107, 108, 124] [17, 38, 108, 109, 125] [18, 39, 109, 110, 112] [19, 40, 110, 111, 113] [20, 41, 98, 111, 114] [35, 42, 112, 113, 129] [36, 43, 113, 114, 130] [37, 44, 114, 115, 131] [38, 45, 115, 116, 132] [39, 46, 116, 117, 133] [40, 47, 117, 118, 134] [41, 48, 118, 119, 135] [28, 49, 119, 120, 136] [29, 50, 120, 121, 137] [30, 51, 121, 122, 138] [31, 52, 122, 123, 139] [32, 53, 123, 124, 126] [33, 54, 124, 125, 127] [34, 55, 112, 125, 128] [49, 56, 73, 126, 127] [50, 57, 74, 127, 128] [51, 58, 75, 128, 129] [52, 59, 76, 129, 130] [53, 60, 77, 130, 131] [54, 61, 78, 131, 132] [55, 62, 79, 132, 133] [42, 63, 80, 133, 134] [43, 64, 81, 134, 135] [44, 65, 82, 135, 136] [45, 66, 83, 136, 137] [46, 67, 70, 137, 138] [47, 68, 71, 138, 139] [48, 69, 72, 126, 139]

Code ID 140-6-10 · download JSON · raw on GitHub