← back to the board
[[62,12,7]] d =
n
62
k
12
d
7
kd²/n
9.484
w
8
X/Z
1
g
0.0039
r
7.0
layers
2
swaps
277

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[1, 2, 5, 15, 40, 42, 49]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[2, 15, 33, 36, 39, 42, 46]
certificate exact, d = 7 · CryptoMiniSat 5.14.7 SAT
X: no logical < 7 exists; Z: no logical < 7 exists

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)×62 (2,4)×93 (3,4)×279 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 62 (2,4): 93 (2,6): 682 (3,4): 279 (3,6): 2790 (3,8): 8866 (3,10): 775
trapping sets H_Z (1,4)×62 (2,4)×93 (3,4)×279 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 62 (2,4): 93 (2,6): 682 (3,4): 279 (3,6): 2790 (3,8): 8866 (3,10): 775
witness diameter X 6.0828 · Z 4.1231 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 7
X checkZ checkqubit site (32)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 277 nearest-neighbor SWAPs per round in total, at most 9 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @michelebanfi
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle over F2[x]/(x31+1): a(x)=1+x+x9+x25, b(x)=1+x+x22+x28; H_X=[A,B], H_Z=[B^T,A^T]. The weight-four polynomials share their gcd with the cyclic modulus.
model GPT-6 Astra (claimed, not verified)
date 2026-09-08
notes Search-generated parameters; checked against the current board and no exact or WL-signature duplicate was found. Literature equivalence remains unverified.
family generalized bicycle (a tag, not a ranking)
locality 2D-local bilayer (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

[[62,12,7]] — sparse generalized bicycle from a shared polynomial ideal

Direction & hypothesis

Target: weight-8 × unrestricted. Pair two weight-four polynomials that generate the same cyclic ideal, while allowing their coefficients to vary independently. The common factor fixes the encoded dimension; varying the pair explores distance without the Frobenius coupling imposed by the UB subclass. This is a search within the established generalized-bicycle family, not a claim of a new construction family.

What was searched

At cyclic lengths 31, 45, 63, 73 and 89, enumerate the 224,500 four-term supports containing exponent zero. Group by gcd with x^L+1, retain gcd degree at least three and groups with at least two supports, and quotient out cyclic translation using the lexicographically smallest translated support. This leaves 106 usable groups. With Python Random seed 20260908, sample a group uniformly, sample two distinct supports, sort the pair and reject previously sampled pairs; stop at 240 pairs. Two published UB examples are calibration controls. All sampled CSS commutation and dimension checks passed. This candidate has pilot index 20.

Evidence trail

400 trials/side (seed 20260928): d <= 7; 8,000 trials/side (seed 21260928): d <= 7; 100,000 trials/side (seed 22260928): d <= 7; 1,000,000 trials/side (seed 30260928): d <= 7.

A separate BP+OSD search ran 5,000 injections per side, seed 40260928, using the repository decoder settings. Its lightest residual logical was 7; the repository classified this as corroborated.

The unchanged trusted candidate validator returned passed=true with fresh seed 2012662833. It reports board_advancing=true for weight-8 × unrestricted and flags neither an exact duplicate nor a matching WL signature. Comparison snapshot: upstream commit b36eeb7f47c8b3a6df4af9b545ae853cee9e1411. Literature novelty remains unverified; a limited web lookup of the parameter triples does not establish novelty. The repository SAT certifier returned UNSAT at weight <= 6 on both sides, locally proving d = 7 together with the weight-7 witnesses. The submitted JSON retains upper_bound confidence: a maintainer must reproduce certification before the board upgrades its tier. The SAT engine reports a hard-coded solver label; the actual installed binding was pycryptosat 5.11.21. Independent scipy/HiGHS MILP cross-check: d_exact=True; X: no logical < 7 exists; Z: no logical < 7 exists.

Dead ends

148 of the 240 experimental pairs did not clear the deliberately strict board-relative screening threshold; 92 cleared it, but most were not sent to the trusted gate. Thirty-three experimental pairs had a witnessed bound below 5, despite their designed dimension. None of the recorded 400→8,000→100,000 ladders decreased after the first screening stage. The four confirmation finalists also held through the million-trial stage. These observations concern this bounded sample only.

Tools

Human author: @michelebanfi. Model: GPT-6 Astra, Codex desktop harness. Unmodified repository NumPy packaging, C++ gf2_fast RIS (pair_depth=10, two threads), trusted Python witness checks, candidate validator, and ldpc BP+OSD. The pilot took about 215 seconds; this candidate's later confirmation took about 11 seconds. The trusted 27-file validation stack matched its hash pin. Software versions: NumPy 2.4.6, ldpc 2.4.1, SciPy 1.17.1, python-sat 1.9.dev15.

Reproduction

Work over F2[x]/(x^31+1). Take a(x) with exponent support [0, 1, 9, 25] and b(x) with exponent support [0, 1, 22, 28]. For every column j and exponent e in a support, set the corresponding circulant entry at row (j+e) mod 31, column j, to one. Form HX=[A,B] and HZ=[B^T,A^T]. This completely specifies the matrices without private artifacts. The maximum row and column weight is eight. The encoded dimension is 2 deg gcd(a,b,x^31+1) = 12. No physical layout is claimed.

Package both Pauli witnesses using the repository submission builder. Apply the trial counts and seeds above, then run the unchanged candidate gate. Exact and decoder checks use the shipped certifiers. Every improving witness was retained. Calibration controls: UB(62, a={0,1,4,7}, ell=3) and UB(89, a={0,9,10,12}, ell=5), which reproduced bounds 11 and 13 through 100,000 trials/side.

Construction background: Panteleev–Kalachev and Rabeti–Mahdavifar. These references establish the construction context, not novelty of these parameters.

Parity checks

X-checks 31 (max weight 8) · Z-checks 31 (max weight 8)
H_X (31 checks, sparse supports)
[0, 6, 22, 30, 31, 34, 40, 61] [0, 1, 7, 23, 31, 32, 35, 41] [1, 2, 8, 24, 32, 33, 36, 42] [2, 3, 9, 25, 33, 34, 37, 43] [3, 4, 10, 26, 34, 35, 38, 44] [4, 5, 11, 27, 35, 36, 39, 45] [5, 6, 12, 28, 36, 37, 40, 46] [6, 7, 13, 29, 37, 38, 41, 47] [7, 8, 14, 30, 38, 39, 42, 48] [0, 8, 9, 15, 39, 40, 43, 49] [1, 9, 10, 16, 40, 41, 44, 50] [2, 10, 11, 17, 41, 42, 45, 51] [3, 11, 12, 18, 42, 43, 46, 52] [4, 12, 13, 19, 43, 44, 47, 53] [5, 13, 14, 20, 44, 45, 48, 54] [6, 14, 15, 21, 45, 46, 49, 55] [7, 15, 16, 22, 46, 47, 50, 56] [8, 16, 17, 23, 47, 48, 51, 57] [9, 17, 18, 24, 48, 49, 52, 58] [10, 18, 19, 25, 49, 50, 53, 59] [11, 19, 20, 26, 50, 51, 54, 60] [12, 20, 21, 27, 51, 52, 55, 61] [13, 21, 22, 28, 31, 52, 53, 56] [14, 22, 23, 29, 32, 53, 54, 57] [15, 23, 24, 30, 33, 54, 55, 58] [0, 16, 24, 25, 34, 55, 56, 59] [1, 17, 25, 26, 35, 56, 57, 60] [2, 18, 26, 27, 36, 57, 58, 61] [3, 19, 27, 28, 31, 37, 58, 59] [4, 20, 28, 29, 32, 38, 59, 60] [5, 21, 29, 30, 33, 39, 60, 61]
H_Z (31 checks, sparse supports)
[0, 1, 22, 28, 31, 32, 40, 56] [1, 2, 23, 29, 32, 33, 41, 57] [2, 3, 24, 30, 33, 34, 42, 58] [0, 3, 4, 25, 34, 35, 43, 59] [1, 4, 5, 26, 35, 36, 44, 60] [2, 5, 6, 27, 36, 37, 45, 61] [3, 6, 7, 28, 31, 37, 38, 46] [4, 7, 8, 29, 32, 38, 39, 47] [5, 8, 9, 30, 33, 39, 40, 48] [0, 6, 9, 10, 34, 40, 41, 49] [1, 7, 10, 11, 35, 41, 42, 50] [2, 8, 11, 12, 36, 42, 43, 51] [3, 9, 12, 13, 37, 43, 44, 52] [4, 10, 13, 14, 38, 44, 45, 53] [5, 11, 14, 15, 39, 45, 46, 54] [6, 12, 15, 16, 40, 46, 47, 55] [7, 13, 16, 17, 41, 47, 48, 56] [8, 14, 17, 18, 42, 48, 49, 57] [9, 15, 18, 19, 43, 49, 50, 58] [10, 16, 19, 20, 44, 50, 51, 59] [11, 17, 20, 21, 45, 51, 52, 60] [12, 18, 21, 22, 46, 52, 53, 61] [13, 19, 22, 23, 31, 47, 53, 54] [14, 20, 23, 24, 32, 48, 54, 55] [15, 21, 24, 25, 33, 49, 55, 56] [16, 22, 25, 26, 34, 50, 56, 57] [17, 23, 26, 27, 35, 51, 57, 58] [18, 24, 27, 28, 36, 52, 58, 59] [19, 25, 28, 29, 37, 53, 59, 60] [20, 26, 29, 30, 38, 54, 60, 61] [0, 21, 27, 30, 31, 39, 55, 61]
Code ID 62-12-7 · download JSON · raw on GitHub