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[[180,94,4]] d ≤
n
180
k
94
d
4
kd²/n
8.356
w
12
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[95, 110, 128, 149]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[11, 97, 128, 154]
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 12 · H_Z 12
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×180 (2,2)×235 (3,3)×5840 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 180 (2,2): 235 (2,4): 2500 (3,3): 5840 (3,5): 47125 (3,7): 5885
trapping sets H_Z (1,3)×180 (2,2)×235 (3,3)×5840 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 180 (2,2): 235 (2,4): 2500 (3,3): 5840 (3,5): 47125 (3,7): 5885

Circuit tier

syndrome-extraction memory circuits committed under circuits/180-94-4/ · canonical noise recipe, 4 rounds, stim 1.16.0
d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[21421, 21422]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[11688, 11754, 13329]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Cornucopia code (Lu, Li & Deng, arXiv:2608.02773; block-convolutional CSS construction with the check matrices given in the Methods of arXiv:2610.19073): q=5, n=36q=180 data qubits in twelve 3x5 tiles L0..L5,R0..R5 plus nine 3x5 X-check and nine 3x5 Z-check tiles; HX=[A_(j-i)|B_(j-i)], HZ=[B-1_(i-j)|A-1_(i-j)] with the coordinate permutations A_p,B_p:(r,c)|->(rho(p,r),c+s[p]) on Z_3 x Z_5, row maps rho_A(0)=r+1, rho_A(1)=2r+2, rho_B(2)=r+1, rho_B(3)=2r+2 (identity otherwise), and column shifts sA=[0, 2, 2, 1, 1, 4], sB=[1, 1, 4, 2, 0, 0]. These shifts were found by a random shift search; the paper publishes instances only at q=7,16,25,29,49,64,79. Weight-12 checks, k=18q+4=94.
model MiMo-V2.6-Flash (claimed, not verified)
builds on arXiv:2608.02773, arXiv:2610.19073
date 2026-10-15
family other (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

[[180,94,4]] - Cornucopia code at q=5, a new member of the published family

Direction & hypothesis

Target: the weight-9plus x unrestricted cell on the CSS board (checks of weight 12, no 2D-local layout). The Cornucopia architecture paper (arXiv:2610.19073) closes by saying its construction "carries over to other codes of the family", and the family (Lu, Li & Deng, arXiv:2608.02773) is n = 36q, k = 18q + 4 with weight-12 checks - but the paper publishes instances only at q = 7, 16, 25, 29, 49, 64, 79, of which only q=7 and q=16 fit this board's n <= 700 cap and both are already board entries. So the unpublished block sizes under the cap, q = 4..19, were the opening: every one of them is a rate-1/2 point no CSS board entry can dominate (an entry dominating [[180,94,d]] would need n <= 180 and k >= 94, i.e. rate at least 0.522 at that size, and the board has none).

What was searched

The family's only free parameters are its 12 column shifts: check matrices HX = [A_(j-i) | B_(j-i)], HZ = [B^-1_(i-j) | A^-1_(i-j)] over Z_3 x Z_q coordinate permutations (row maps r+1 for A0 and B2, 2r+2 for A1 and B3, identity otherwise). CSS commutation holds for *every* shift choice: the only noncommuting row-map pairs, (A0,B3) and (A1,B2), enter block-row differences i'-i = 3 (mod 6), which never occur for i,i' in {0,1,2}; so the search is over shifts alone, with k recomputed exactly on every draw.

At q=5: 1200 uniformly random shift pairs, screened at 100 RIS trials each (accelerated backend); the top 24 re-measured at 4000; the top 4 re-measured at 100000; then the packaged candidate re-verified by submit.make_submission (8000 numpy trials/side) and by the submission CLI (800 python RIS + 300000 accelerated trials/side). Stage-1 reads at q=5: 2 -> 705, 4 -> 495. Seed 17 throughout the sweep, 0 for the CLI witness search. One design fact from the sweep: a read is the lightest logical *found*, so it is always >= the true d - stage-1 ranking cannot discard a better code, only fail to see how good it is.

Evidence trail

Ladder for this candidate (fast-RIS read -> numpy witness -> CLI):

  • stage reads at 100, 4000, 100000 fast-RIS trials: 4, 4, 4; packaging numpy witness search: 8000 trials/side -> d <= 4.
  • final claim: witness-backed upper bound d <= 4, not exact (the

    • exact tier needs the server MILP; k = 94 puts this beyond its envelope).

  • k = 94 = 18*5+4 recomputed by the verifier; CSS commutation checked;

    • max check weight 12.

  • Gate (verify/validate_candidate.py): passed: true, refuted: false

    • (its own refutation found no lighter logical), board_advancing: true, d_only_gain: false, dominated_by: [].

  • The 1200 draws at this q never read above 4; deeper stages confirmed

    • 4. kd^2/n = 8.36.

Dead ends

  • ~0.0% of random shift sets drop rank (k < 18q+4): the two

    • redundancies per check type are not always the only ones. Rejected on an exact GF(2) rank, not a heuristic. At q=5: 0 of 1200 draws.

  • At q=7 and q=16 the best random-shift reads (6 and 8) only *equal* the

    • published-instance board entries [[252,130,6]] and [[576,292,8]], so nothing was staged there: a tie is not a submission.

  • Across the whole campaign (q=4..19, 19,200 draws) no candidate anywhere read

    • above 8 at any depth: uniform random shifts do not reach d=9/10 under n <= 700; the published family first reaches d=10 at q=25 (n=900, over cap). A designed search (hill-climbing on the plateau of 8-reads) is the untried rung, a wider uniform sweep is not.

Tools

Model: MiMo-V2.6-Flash (matches provenance.model), driving an OpenCode agent harness. Repo tooling: research/kit/css.py (exact k, CSS check), research/kit/surrogate.py (RIS surrogate, fast backend verify/gf2_fast), research/kit/submit.py packaging, verify/validate_candidate.py (the trusted gate), ./qldpc submit (witness search, verifier, circuit tier). Compute: ~9 CPU-hours for the whole 16-q campaign; this q's share is the 1200+24*4000+4*100000 screened trials above.

Reproduction

Rebuild (H_X, H_Z) exactly: q = 5, n = 36q, data qubit (b,r,c) (index (3b+r)q+c, b = 0..5 tiles L, b = 6..11 tiles R, r in Z_3, c in Z_q); HX[(i,r,c), (j, A_(j-i)(r,c))] = 1 and on the R tile with B; HZ[(i,r,c), (j, B^-1_(i-j)(r,c))] = 1 and on the R tile with A; each A_p, B_p : (r,c) |-> (rho(p,r), c + s[p]) mod q with rho_A(0)=r+1, rho_A(1)=2r+2, rho_B(2)=r+1, rho_B(3)=2r+2, identity otherwise, and shifts sA = [0, 2, 2, 1, 1, 4], sB = [1, 1, 4, 2, 0, 0]. Weight-12 checks, k = 18q + 4. Verify: uv run python verify/qldpc_verify.py codes/180-94-4.json.

Parity checks

X-checks 45 (max weight 12) · Z-checks 45 (max weight 12)
H_X (45 checks, sparse supports)
[5, 27, 32, 46, 61, 79, 91, 106, 129, 147, 150, 165] [6, 28, 33, 47, 62, 75, 92, 107, 125, 148, 151, 166] [7, 29, 34, 48, 63, 76, 93, 108, 126, 149, 152, 167] [8, 25, 30, 49, 64, 77, 94, 109, 127, 145, 153, 168] [9, 26, 31, 45, 60, 78, 90, 105, 128, 146, 154, 169] [10, 22, 37, 51, 66, 84, 96, 111, 134, 142, 155, 170] [11, 23, 38, 52, 67, 80, 97, 112, 130, 143, 156, 171] [12, 24, 39, 53, 68, 81, 98, 113, 131, 144, 157, 172] [13, 20, 35, 54, 69, 82, 99, 114, 132, 140, 158, 173] [14, 21, 36, 50, 65, 83, 95, 110, 133, 141, 159, 174] [0, 17, 42, 56, 71, 89, 101, 116, 124, 137, 160, 175] [1, 18, 43, 57, 72, 85, 102, 117, 120, 138, 161, 176] [2, 19, 44, 58, 73, 86, 103, 118, 121, 139, 162, 177] [3, 15, 40, 59, 74, 87, 104, 119, 122, 135, 163, 178] [4, 16, 41, 55, 70, 88, 100, 115, 123, 136, 164, 179] [4, 20, 42, 47, 61, 76, 90, 106, 121, 144, 162, 165] [0, 21, 43, 48, 62, 77, 91, 107, 122, 140, 163, 166] [1, 22, 44, 49, 63, 78, 92, 108, 123, 141, 164, 167] [2, 23, 40, 45, 64, 79, 93, 109, 124, 142, 160, 168] [3, 24, 41, 46, 60, 75, 94, 105, 120, 143, 161, 169] [9, 25, 37, 52, 66, 81, 95, 111, 126, 149, 157, 170] [5, 26, 38, 53, 67, 82, 96, 112, 127, 145, 158, 171] [6, 27, 39, 54, 68, 83, 97, 113, 128, 146, 159, 172] [7, 28, 35, 50, 69, 84, 98, 114, 129, 147, 155, 173] [8, 29, 36, 51, 65, 80, 99, 110, 125, 148, 156, 174] [14, 15, 32, 57, 71, 86, 100, 116, 131, 139, 152, 175] [10, 16, 33, 58, 72, 87, 101, 117, 132, 135, 153, 176] [11, 17, 34, 59, 73, 88, 102, 118, 133, 136, 154, 177] [12, 18, 30, 55, 74, 89, 103, 119, 134, 137, 150, 178] [13, 19, 31, 56, 70, 85, 104, 115, 130, 138, 151, 179] [1, 19, 35, 57, 62, 76, 90, 105, 121, 136, 159, 177] [2, 15, 36, 58, 63, 77, 91, 106, 122, 137, 155, 178] [3, 16, 37, 59, 64, 78, 92, 107, 123, 138, 156, 179] [4, 17, 38, 55, 60, 79, 93, 108, 124, 139, 157, 175] [0, 18, 39, 56, 61, 75, 94, 109, 120, 135, 158, 176] [6, 24, 40, 52, 67, 81, 95, 110, 126, 141, 164, 172] [7, 20, 41, 53, 68, 82, 96, 111, 127, 142, 160, 173] [8, 21, 42, 54, 69, 83, 97, 112, 128, 143, 161, 174] [9, 22, 43, 50, 65, 84, 98, 113, 129, 144, 162, 170] [5, 23, 44, 51, 66, 80, 99, 114, 125, 140, 163, 171] [11, 29, 30, 47, 72, 86, 100, 115, 131, 146, 154, 167] [12, 25, 31, 48, 73, 87, 101, 116, 132, 147, 150, 168] [13, 26, 32, 49, 74, 88, 102, 117, 133, 148, 151, 169] [14, 27, 33, 45, 70, 89, 103, 118, 134, 149, 152, 165] [10, 28, 34, 46, 71, 85, 104, 119, 130, 145, 153, 166]
H_Z (45 checks, sparse supports)
[4, 15, 30, 58, 71, 79, 100, 106, 124, 139, 153, 178] [0, 16, 31, 59, 72, 75, 101, 107, 120, 135, 154, 179] [1, 17, 32, 55, 73, 76, 102, 108, 121, 136, 150, 175] [2, 18, 33, 56, 74, 77, 103, 109, 122, 137, 151, 176] [3, 19, 34, 57, 70, 78, 104, 105, 123, 138, 152, 177] [9, 20, 35, 53, 61, 84, 90, 111, 129, 144, 158, 173] [5, 21, 36, 54, 62, 80, 91, 112, 125, 140, 159, 174] [6, 22, 37, 50, 63, 81, 92, 113, 126, 141, 155, 170] [7, 23, 38, 51, 64, 82, 93, 114, 127, 142, 156, 171] [8, 24, 39, 52, 60, 83, 94, 110, 128, 143, 157, 172] [14, 25, 40, 48, 66, 89, 95, 116, 134, 149, 163, 168] [10, 26, 41, 49, 67, 85, 96, 117, 130, 145, 164, 169] [11, 27, 42, 45, 68, 86, 97, 118, 131, 146, 160, 165] [12, 28, 43, 46, 69, 87, 98, 119, 132, 147, 161, 166] [13, 29, 44, 47, 65, 88, 99, 115, 133, 148, 162, 167] [4, 19, 30, 45, 73, 86, 103, 115, 121, 139, 154, 168] [0, 15, 31, 46, 74, 87, 104, 116, 122, 135, 150, 169] [1, 16, 32, 47, 70, 88, 100, 117, 123, 136, 151, 165] [2, 17, 33, 48, 71, 89, 101, 118, 124, 137, 152, 166] [3, 18, 34, 49, 72, 85, 102, 119, 120, 138, 153, 167] [9, 24, 35, 50, 68, 76, 98, 105, 126, 144, 159, 173] [5, 20, 36, 51, 69, 77, 99, 106, 127, 140, 155, 174] [6, 21, 37, 52, 65, 78, 95, 107, 128, 141, 156, 170] [7, 22, 38, 53, 66, 79, 96, 108, 129, 142, 157, 171] [8, 23, 39, 54, 67, 75, 97, 109, 125, 143, 158, 172] [14, 29, 40, 55, 63, 81, 93, 110, 131, 149, 164, 178] [10, 25, 41, 56, 64, 82, 94, 111, 132, 145, 160, 179] [11, 26, 42, 57, 60, 83, 90, 112, 133, 146, 161, 175] [12, 27, 43, 58, 61, 84, 91, 113, 134, 147, 162, 176] [13, 28, 44, 59, 62, 80, 92, 114, 130, 148, 163, 177] [11, 19, 34, 45, 60, 88, 93, 118, 130, 136, 154, 169] [12, 15, 30, 46, 61, 89, 94, 119, 131, 137, 150, 165] [13, 16, 31, 47, 62, 85, 90, 115, 132, 138, 151, 166] [14, 17, 32, 48, 63, 86, 91, 116, 133, 139, 152, 167] [10, 18, 33, 49, 64, 87, 92, 117, 134, 135, 153, 168] [1, 24, 39, 50, 65, 83, 98, 113, 120, 141, 159, 174] [2, 20, 35, 51, 66, 84, 99, 114, 121, 142, 155, 170] [3, 21, 36, 52, 67, 80, 95, 110, 122, 143, 156, 171] [4, 22, 37, 53, 68, 81, 96, 111, 123, 144, 157, 172] [0, 23, 38, 54, 69, 82, 97, 112, 124, 140, 158, 173] [6, 29, 44, 55, 70, 78, 103, 108, 125, 146, 164, 179] [7, 25, 40, 56, 71, 79, 104, 109, 126, 147, 160, 175] [8, 26, 41, 57, 72, 75, 100, 105, 127, 148, 161, 176] [9, 27, 42, 58, 73, 76, 101, 106, 128, 149, 162, 177] [5, 28, 43, 59, 74, 77, 102, 107, 129, 145, 163, 178]
Code ID 180-94-4 · download JSON · raw on GitHub