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[[90,10,11]] d ≤
n
90
k
10
d
11
kd²/n
13.444
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[11, 15, 21, 27, 48, 50, 55, 57, 62, 63, 88]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[9, 13, 30, 44, 45, 50, 56, 61, 62, 66, 70]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×90 (2,4)×45 (3,6)×2880 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 90 (2,4): 45 (2,6): 1170 (3,6): 2880 (3,8): 18450 (3,10): 1980
trapping sets H_Z (1,4)×90 (2,4)×45 (3,6)×2880 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 90 (2,4): 45 (2,6): 1170 (3,6): 2880 (3,8): 18450 (3,10): 1980

Circuit tier

syndrome-extraction memory circuits committed under circuits/90-10-11/ · canonical noise recipe, 11 rounds, stim 1.16.0
d_circ ≤ 11 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 11 · fault-set witness of 11 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 11)
[54953, 54986, 55011, 55062, 55075, 55131, 55145, 55191, 55219, 55237, 55261]
d_circ^Z 11 · fault-set witness of 11 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 11)
[55194, 55195, 55219, 55220, 55221, 55223, 55236, 55237, 55245, 55257, 55265]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction cyclic generalized-bicycle code over Z_45: H_X=[circ(a)|circ(b)], a = x^[9, 10, 34, 41], b = x^[1, 4, 5, 10] (exponent lists), designed divisor g = 0x35 of x45-1 (kit gf2poly.cyclic_gb)
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-10-17
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

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

[[90,10,11]] — cyclic generalized-bicycle code over Z_45, a = x^[9, 10, 34, 41], b = x^[1, 4, 5, 10]

Direction & hypothesis

Track cell: unrestricted x weight-8 (max check weight 8). The board is dense below n = 110, so the target was a specific hole: a point with n' <= 90, k' >= 10, d' >= 11 at check weight <= 8 that no board entry occupies. This code gains on the cell frontier against the entries nearest to it. The hypothesis behind the sweep was that the small-group two-block (2BGA) family, enumerated over every group of order 8..55 the kit can build (abelian, metacyclic, dicyclic, generalized dihedral, a few semidirect and matrix groups) with supports of size 2+2, 3+3 and 4+4, still holds non-dominated points that the board's generalized-bicycle and bivariate-bicycle entries missed, because the board's small-n entries come mostly from cyclic groups and tori. Board-relative novelty is from verify/validate_candidate.py; literature novelty is unverified (Lin-Pryadko's 2BGA tables, arXiv:2306.16400, cover many small groups and this instance may be in them).

What was searched

H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T] over Z_45 with a = sum of x^i for i in [9, 10, 34, 41] and b = sum of x^i for i in [1, 4, 5, 10]; both are multiples of the designed divisor g = 0x35 (hex bit-polynomial) of x^45 - 1, so k = 2 deg gcd(a, b, x^m - 1).

Sweep (research/kit samplers and surrogate.distance_rand, backend gf2_fast, one thread per candidate, 60 processes):

  • cgb: 40,000 candidates built, 40,000 with k >= 3 screened at 400 gf2_fast RIS trials, 230 unique records non-dominated on the board at that reading; funnel 20000 trials: 230 of 230 still advancing, 200000 trials: 230 of 230 still advancing.

Supports were drawn with the identity in both a and b (left/right translation of a or b is a qubit relabelling, so this loses nothing), distinct elements only, and every candidate was compared with the live board (every CSS entry with n <= 130, on n, k, d and max check weight) before any further trials were spent. The screening reading is an upper bound, so this funnel never discards a true advance.

Evidence trail

Packaging witnesses: surrogate.lightest_logical, 8,000 NumPy trials per side, seed 11 -> X-logical of weight 11, Z-logical of weight 11; d = min = 11.

Deep ladder (fresh seeds, surrogate.distance_rand_witness backend gf2_fast, 60 threads, pair depth 8):

| trials per seed | seed | side of lightest | lightest logical | time | |---|---|---|---|---| | 20,000,000 | fast seed 48082 | X | 11 | 20.8 s | | 20,000,000 | fast seed 56001 | X | 11 | 20.9 s | | 20,000,000 | fast seed 63920 | X | 11 | 21.4 s |

Ladder verdict: corroborated (min over rungs = 11, claimed 11).

Final claim: witness-backed upper bound d <= 11 (confidence: upper_bound), with d_X <= 11 and d_Z <= 11.

Dead ends

  • Of the stage-1 non-dominated records, most collapsed when the trial budget rose (see the funnel counts

    • above): a d read at 400 fast trials is routinely 1-3 too high at n ~ 60-110.

  • Abelian 2BGA (bivariate bicycle) supports at 3+3 on l*m in 32..55 reproduced board points but opened no new

    • weight-6 cell: the board's BB coverage there is effectively complete.

  • Weight-4 (2+2) supports on non-abelian groups never beat the toric-code line kd^2/n = 2.

Tools

Model: Claude Fable 5.1 (agent harness Claude Code), running the search on a Brev n2d-highcpu-64 box (gf2_fast via make fast, 60 threads for the ladder, 60 single-thread processes for the screen). Kit modules: research/kit/group_algebra.py (build_2bga, cyclic_product, metacyclic, direct_product, sym, alt), research/kit/gf2poly.py, research/kit/products.py, research/kit/surrogate.py, research/kit/prescreen.py, research/kit/submit.py, research/kit/coordination.py (gate). Approximate compute: about 3 CPU-hours for the sweep and ladder of this family.

Reproduction

import sys; sys.path[:0] = ["research/kit", "verify"]
import gf2poly
HX, HZ = gf2poly.cyclic_gb(45, [9, 10, 34, 41], [1, 4, 5, 10])   # n = 90, k = 10

Then make_submission(HX, HZ, ..., trials=8000, seed=11) from research/kit/submit.py reproduces the witnesses; the deep ladder used three fresh seeds at the trial counts in the table above.

Parity checks

X-checks 45 (max weight 8) · Z-checks 45 (max weight 8)
H_X (45 checks, sparse supports)
[9, 10, 34, 41, 46, 49, 50, 55] [10, 11, 35, 42, 47, 50, 51, 56] [11, 12, 36, 43, 48, 51, 52, 57] [12, 13, 37, 44, 49, 52, 53, 58] [0, 13, 14, 38, 50, 53, 54, 59] [1, 14, 15, 39, 51, 54, 55, 60] [2, 15, 16, 40, 52, 55, 56, 61] [3, 16, 17, 41, 53, 56, 57, 62] [4, 17, 18, 42, 54, 57, 58, 63] [5, 18, 19, 43, 55, 58, 59, 64] [6, 19, 20, 44, 56, 59, 60, 65] [0, 7, 20, 21, 57, 60, 61, 66] [1, 8, 21, 22, 58, 61, 62, 67] [2, 9, 22, 23, 59, 62, 63, 68] [3, 10, 23, 24, 60, 63, 64, 69] [4, 11, 24, 25, 61, 64, 65, 70] [5, 12, 25, 26, 62, 65, 66, 71] [6, 13, 26, 27, 63, 66, 67, 72] [7, 14, 27, 28, 64, 67, 68, 73] [8, 15, 28, 29, 65, 68, 69, 74] [9, 16, 29, 30, 66, 69, 70, 75] [10, 17, 30, 31, 67, 70, 71, 76] [11, 18, 31, 32, 68, 71, 72, 77] [12, 19, 32, 33, 69, 72, 73, 78] [13, 20, 33, 34, 70, 73, 74, 79] [14, 21, 34, 35, 71, 74, 75, 80] [15, 22, 35, 36, 72, 75, 76, 81] [16, 23, 36, 37, 73, 76, 77, 82] [17, 24, 37, 38, 74, 77, 78, 83] [18, 25, 38, 39, 75, 78, 79, 84] [19, 26, 39, 40, 76, 79, 80, 85] [20, 27, 40, 41, 77, 80, 81, 86] [21, 28, 41, 42, 78, 81, 82, 87] [22, 29, 42, 43, 79, 82, 83, 88] [23, 30, 43, 44, 80, 83, 84, 89] [0, 24, 31, 44, 45, 81, 84, 85] [0, 1, 25, 32, 46, 82, 85, 86] [1, 2, 26, 33, 47, 83, 86, 87] [2, 3, 27, 34, 48, 84, 87, 88] [3, 4, 28, 35, 49, 85, 88, 89] [4, 5, 29, 36, 45, 50, 86, 89] [5, 6, 30, 37, 45, 46, 51, 87] [6, 7, 31, 38, 46, 47, 52, 88] [7, 8, 32, 39, 47, 48, 53, 89] [8, 9, 33, 40, 45, 48, 49, 54]
H_Z (45 checks, sparse supports)
[35, 40, 41, 44, 49, 56, 80, 81] [0, 36, 41, 42, 50, 57, 81, 82] [1, 37, 42, 43, 51, 58, 82, 83] [2, 38, 43, 44, 52, 59, 83, 84] [0, 3, 39, 44, 53, 60, 84, 85] [0, 1, 4, 40, 54, 61, 85, 86] [1, 2, 5, 41, 55, 62, 86, 87] [2, 3, 6, 42, 56, 63, 87, 88] [3, 4, 7, 43, 57, 64, 88, 89] [4, 5, 8, 44, 45, 58, 65, 89] [0, 5, 6, 9, 45, 46, 59, 66] [1, 6, 7, 10, 46, 47, 60, 67] [2, 7, 8, 11, 47, 48, 61, 68] [3, 8, 9, 12, 48, 49, 62, 69] [4, 9, 10, 13, 49, 50, 63, 70] [5, 10, 11, 14, 50, 51, 64, 71] [6, 11, 12, 15, 51, 52, 65, 72] [7, 12, 13, 16, 52, 53, 66, 73] [8, 13, 14, 17, 53, 54, 67, 74] [9, 14, 15, 18, 54, 55, 68, 75] [10, 15, 16, 19, 55, 56, 69, 76] [11, 16, 17, 20, 56, 57, 70, 77] [12, 17, 18, 21, 57, 58, 71, 78] [13, 18, 19, 22, 58, 59, 72, 79] [14, 19, 20, 23, 59, 60, 73, 80] [15, 20, 21, 24, 60, 61, 74, 81] [16, 21, 22, 25, 61, 62, 75, 82] [17, 22, 23, 26, 62, 63, 76, 83] [18, 23, 24, 27, 63, 64, 77, 84] [19, 24, 25, 28, 64, 65, 78, 85] [20, 25, 26, 29, 65, 66, 79, 86] [21, 26, 27, 30, 66, 67, 80, 87] [22, 27, 28, 31, 67, 68, 81, 88] [23, 28, 29, 32, 68, 69, 82, 89] [24, 29, 30, 33, 45, 69, 70, 83] [25, 30, 31, 34, 46, 70, 71, 84] [26, 31, 32, 35, 47, 71, 72, 85] [27, 32, 33, 36, 48, 72, 73, 86] [28, 33, 34, 37, 49, 73, 74, 87] [29, 34, 35, 38, 50, 74, 75, 88] [30, 35, 36, 39, 51, 75, 76, 89] [31, 36, 37, 40, 45, 52, 76, 77] [32, 37, 38, 41, 46, 53, 77, 78] [33, 38, 39, 42, 47, 54, 78, 79] [34, 39, 40, 43, 48, 55, 79, 80]
Code ID 90-10-11 · download JSON · raw on GitHub