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[[264,4,26]] d ≤
n
264
k
4
d
26
kd²/n
10.2424
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 26, d_Z ≤ 26 · 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 26 · witness weight 26 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (Perlmutter A100) · found at 3×108 trials · survived 3×108 trials · 2026-10-30
witness operator (support, 26 qubits)
[10, 12, 15, 31, 34, 38, 51, 52, 54, 69, 132, 140, 143, 157, 171, 186, 197, 205, 211, 215, 218, 222, 232, 236, 246, 258]
d_Z 26 · witness weight 26 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (Perlmutter A100) · found at 3×108 trials · survived 3×108 trials · 2026-10-30
witness operator (support, 26 qubits)
[1, 20, 31, 35, 49, 63, 66, 74, 80, 92, 102, 106, 116, 123, 127, 131, 137, 154, 155, 163, 168, 172, 186, 191, 194, 207]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Recipe at other sizes

the stored construction recipe (bb.build_bb, rebuilds this entry at its own parameters) rebuilt at neighboring sizes (l +- 1, m +- 1, and each side halved or doubled, same monomials (reduced modulo the new torus)) and put through the verifier's cheap structural checks only: no distance at any size, no ranking, no tier. A recipe that holds up at other sizes is evidence of a construction rather than a tuned artifact; one that does not is a fact about the recipe, not about this entry.
passes at 7 of 7 sizes · own size [[264,4]] w=8

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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×264 (2,6)×3696 (3,6)×2244 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 264 (2,6): 3696 (3,6): 2244 (3,8): 70884 (3,10): 7392
trapping sets H_Z (1,4)×264 (2,6)×3696 (3,6)×2244 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 264 (2,6): 3696 (3,6): 2244 (3,8): 70884 (3,10): 7392

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle on Z_12 x Z_11 (n = 2lm = 264) with A = xy4 + x6y6 + x10y5 + x5y7 and B = x5y5 + x8y9 + x9y + x6y10, H_X = [A | B], H_Z = [B^T | A^T]; built by research/kit/bb.build_bb(l=12, m=11, A_terms=[[1, 4], [6, 6], [10, 5], [5, 7]], B_terms=[[5, 5], [8, 9], [9, 1], [6, 10]]).
model Claude Claude Fable 5.1 (Claude Code) working for @vprusso (claimed, not verified)
builds on S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low-overhead fault-tolerant quantum memory, Nature 627, 778 (2024), arXiv:2308.07915
date 2026-10-23
notes Unattended loop run on Perlmutter, iteration 358, direction bb-w4-mid, seed 20261375. The loop gate's 8,000-trial search read d = 28; the deeper rungs recorded in notes/ read 26, and the document carries those witnesses.
family bivariate 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

[[264,4,26]] loop bivariate bicycle on Z_12 x Z_11

Direction & hypothesis

Random weight-8 bivariate-bicycle codes between n = 100 and n = 400 fill (n, k) cells where the board's weight-8 entries come from a few structured families; uniform sampling over the torus and the monomials tests whether those families are the best this weight allows.

What was searched

Direction bb-w4-mid of the loop samples bivariate-bicycle codes on Z_l x Z_m with l in [6, 16], m in [4, 12], four monomials per polynomial (check weight 8), and n in [100, 400]. Each iteration draws fresh candidates, screens them at 400 and then 20,000 RIS trials (gf2_fast), keeps the twelve best by kd^2/n that no board entry or earlier staged document dominates, and gates them with research/kit/submit.make_submission at 8,000 NumPy trials per side plus verify/validate_candidate.py. This code is iteration 358 (seed 20261375). By 29 October the direction had run 801 iterations over 106,817 candidates and staged 474; across all five directions the loop staged 1,383 documents in 390.6 CPU hours. The 26 October review kept 49 staged codes that neither the board nor another staged code dominated, and each of the 49 was re-searched on OLCF Lux (10,000,000 gf2_fast trials, two seeds) and on Perlmutter A100s (300,000,000 RIS trials per side, two seeds).

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 20,000 (loop screen, gf2_fast, both sides) | 28 | | 8,000 (loop gate, NumPy, per side) | 28 | | 10,000,000 (Lux gf2_fast, both sides, seed 11) | 26 | | 10,000,000 (Lux gf2_fast, both sides, seed 22) | 26 | | 300,000,000 (GPU RIS, 300M per side, seed 1) | 26 | | 300,000,000 (GPU RIS, 300M per side, seed 2) | 26 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 300,000,000 trials per side. The submitted witness reproduces it: X: d <= 26 (upper_bound), Z: d <= 26 (upper_bound).

Trial counts are per side for the gate and the GPU rungs; the screen and the Lux rungs are gf2_fast budgets shared across both sides (the Lux rung also ran 2,000 NumPy RIS trials per side). Each rung's reading is the lightest logical its own witnesses reach after an independent GF(2) check. The orbit fold (verify/heuristic_distance, 1,000,000 trials per folded code, seeds 101, 202, and 303) found automorphism partitions on 3 of 3 runs and no lighter logical through them. The gate's reading of 28 did not survive the deeper rungs; the document was repackaged at 26 before filing.

Dead ends

None recorded for this candidate.

Tools

Model: Claude Fable 5.1 (Claude Code) working for @vprusso (unattended loop orchestrator on Perlmutter over research/kit/search.screen_adaptive). Repo tooling: research/kit/search.py, research/kit/bb.py, research/kit/submit.py, verify/heuristic_distance.py, verify/ris_gpu.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: Perlmutter CPU for the loop, one OLCF Lux node for the two 10M rungs, and one Perlmutter A100 for the two 300M rungs (about ten minutes per code).

Reproduction

`research/kit/bb.build_bb(l=12, m=11, A_terms=[[1, 4], [6, 6], [10, 5], [5, 7]], B_terms=[[5, 5], [8, 9], [9, 1], [6, 10]])` returns (H_X, H_Z); the repro manifest replays that call through research/kit/rebuild.py and compares fingerprints. The distance rungs are verify/heuristic_distance.estimate with --fast-trials 10000000 (seeds 11 and 22) and `verify/ris_gpu.py --trials 300000000` (seeds 1 and 2) on the staged document.

Parity checks

X-checks 132 (max weight 8) · Z-checks 132 (max weight 8)
H_X (132 checks, sparse supports)
[15, 62, 72, 115, 192, 208, 229, 232] [16, 63, 73, 116, 193, 198, 230, 233] [17, 64, 74, 117, 194, 199, 220, 234] [18, 65, 75, 118, 195, 200, 221, 235] [19, 55, 76, 119, 196, 201, 222, 236] [20, 56, 66, 120, 197, 202, 223, 237] [21, 57, 67, 110, 187, 203, 224, 238] [11, 58, 68, 111, 188, 204, 225, 239] [12, 59, 69, 112, 189, 205, 226, 240] [13, 60, 70, 113, 190, 206, 227, 241] [14, 61, 71, 114, 191, 207, 228, 231] [26, 73, 83, 126, 203, 219, 240, 243] [27, 74, 84, 127, 204, 209, 241, 244] [28, 75, 85, 128, 205, 210, 231, 245] [29, 76, 86, 129, 206, 211, 232, 246] [30, 66, 87, 130, 207, 212, 233, 247] [31, 67, 77, 131, 208, 213, 234, 248] [32, 68, 78, 121, 198, 214, 235, 249] [22, 69, 79, 122, 199, 215, 236, 250] [23, 70, 80, 123, 200, 216, 237, 251] [24, 71, 81, 124, 201, 217, 238, 252] [25, 72, 82, 125, 202, 218, 239, 242] [5, 37, 84, 94, 214, 230, 251, 254] [6, 38, 85, 95, 215, 220, 252, 255] [7, 39, 86, 96, 216, 221, 242, 256] [8, 40, 87, 97, 217, 222, 243, 257] [9, 41, 77, 98, 218, 223, 244, 258] [10, 42, 78, 88, 219, 224, 245, 259] [0, 43, 79, 89, 209, 225, 246, 260] [1, 33, 80, 90, 210, 226, 247, 261] [2, 34, 81, 91, 211, 227, 248, 262] [3, 35, 82, 92, 212, 228, 249, 263] [4, 36, 83, 93, 213, 229, 250, 253] [16, 48, 95, 105, 133, 225, 241, 262] [17, 49, 96, 106, 134, 226, 231, 263] [18, 50, 97, 107, 135, 227, 232, 253] [19, 51, 98, 108, 136, 228, 233, 254] [20, 52, 88, 109, 137, 229, 234, 255] [21, 53, 89, 99, 138, 230, 235, 256] [11, 54, 90, 100, 139, 220, 236, 257] [12, 44, 91, 101, 140, 221, 237, 258] [13, 45, 92, 102, 141, 222, 238, 259] [14, 46, 93, 103, 142, 223, 239, 260] [15, 47, 94, 104, 132, 224, 240, 261] [27, 59, 106, 116, 141, 144, 236, 252] [28, 60, 107, 117, 142, 145, 237, 242] [29, 61, 108, 118, 132, 146, 238, 243] [30, 62, 109, 119, 133, 147, 239, 244] [31, 63, 99, 120, 134, 148, 240, 245] [32, 64, 100, 110, 135, 149, 241, 246] [22, 65, 101, 111, 136, 150, 231, 247] [23, 55, 102, 112, 137, 151, 232, 248] [24, 56, 103, 113, 138, 152, 233, 249] [25, 57, 104, 114, 139, 153, 234, 250] [26, 58, 105, 115, 140, 143, 235, 251] [38, 70, 117, 127, 152, 155, 247, 263] [39, 71, 118, 128, 153, 156, 248, 253] [40, 72, 119, 129, 143, 157, 249, 254] [41, 73, 120, 130, 144, 158, 250, 255] [42, 74, 110, 131, 145, 159, 251, 256] [43, 75, 111, 121, 146, 160, 252, 257] [33, 76, 112, 122, 147, 161, 242, 258] [34, 66, 113, 123, 148, 162, 243, 259] [35, 67, 114, 124, 149, 163, 244, 260] [36, 68, 115, 125, 150, 164, 245, 261] [37, 69, 116, 126, 151, 154, 246, 262] [6, 49, 81, 128, 142, 163, 166, 258] [7, 50, 82, 129, 132, 164, 167, 259] [8, 51, 83, 130, 133, 154, 168, 260] [9, 52, 84, 131, 134, 155, 169, 261] [10, 53, 85, 121, 135, 156, 170, 262] [0, 54, 86, 122, 136, 157, 171, 263] [1, 44, 87, 123, 137, 158, 172, 253] [2, 45, 77, 124, 138, 159, 173, 254] [3, 46, 78, 125, 139, 160, 174, 255] [4, 47, 79, 126, 140, 161, 175, 256] [5, 48, 80, 127, 141, 162, 165, 257] [7, 17, 60, 92, 137, 153, 174, 177] [8, 18, 61, 93, 138, 143, 175, 178] [9, 19, 62, 94, 139, 144, 165, 179] [10, 20, 63, 95, 140, 145, 166, 180] [0, 21, 64, 96, 141, 146, 167, 181] [1, 11, 65, 97, 142, 147, 168, 182] [2, 12, 55, 98, 132, 148, 169, 183] [3, 13, 56, 88, 133, 149, 170, 184] [4, 14, 57, 89, 134, 150, 171, 185] [5, 15, 58, 90, 135, 151, 172, 186] [6, 16, 59, 91, 136, 152, 173, 176] [18, 28, 71, 103, 148, 164, 185, 188] [19, 29, 72, 104, 149, 154, 186, 189] [20, 30, 73, 105, 150, 155, 176, 190] [21, 31, 74, 106, 151, 156, 177, 191] [11, 32, 75, 107, 152, 157, 178, 192] [12, 22, 76, 108, 153, 158, 179, 193] [13, 23, 66, 109, 143, 159, 180, 194] [14, 24, 67, 99, 144, 160, 181, 195] [15, 25, 68, 100, 145, 161, 182, 196] [16, 26, 69, 101, 146, 162, 183, 197] [17, 27, 70, 102, 147, 163, 184, 187] [29, 39, 82, 114, 159, 175, 196, 199] [30, 40, 83, 115, 160, 165, 197, 200] [31, 41, 84, 116, 161, 166, 187, 201] [32, 42, 85, 117, 162, 167, 188, 202] [22, 43, 86, 118, 163, 168, 189, 203] [23, 33, 87, 119, 164, 169, 190, 204] [24, 34, 77, 120, 154, 170, 191, 205] [25, 35, 78, 110, 155, 171, 192, 206] [26, 36, 79, 111, 156, 172, 193, 207] [27, 37, 80, 112, 157, 173, 194, 208] [28, 38, 81, 113, 158, 174, 195, 198] [40, 50, 93, 125, 170, 186, 207, 210] [41, 51, 94, 126, 171, 176, 208, 211] [42, 52, 95, 127, 172, 177, 198, 212] [43, 53, 96, 128, 173, 178, 199, 213] [33, 54, 97, 129, 174, 179, 200, 214] [34, 44, 98, 130, 175, 180, 201, 215] [35, 45, 88, 131, 165, 181, 202, 216] [36, 46, 89, 121, 166, 182, 203, 217] [37, 47, 90, 122, 167, 183, 204, 218] [38, 48, 91, 123, 168, 184, 205, 219] [39, 49, 92, 124, 169, 185, 206, 209] [4, 51, 61, 104, 181, 197, 218, 221] [5, 52, 62, 105, 182, 187, 219, 222] [6, 53, 63, 106, 183, 188, 209, 223] [7, 54, 64, 107, 184, 189, 210, 224] [8, 44, 65, 108, 185, 190, 211, 225] [9, 45, 55, 109, 186, 191, 212, 226] [10, 46, 56, 99, 176, 192, 213, 227] [0, 47, 57, 100, 177, 193, 214, 228] [1, 48, 58, 101, 178, 194, 215, 229] [2, 49, 59, 102, 179, 195, 216, 230] [3, 50, 60, 103, 180, 196, 217, 220]
H_Z (132 checks, sparse supports)
[43, 46, 67, 83, 160, 203, 213, 260] [33, 47, 68, 84, 161, 204, 214, 261] [34, 48, 69, 85, 162, 205, 215, 262] [35, 49, 70, 86, 163, 206, 216, 263] [36, 50, 71, 87, 164, 207, 217, 253] [37, 51, 72, 77, 154, 208, 218, 254] [38, 52, 73, 78, 155, 198, 219, 255] [39, 53, 74, 79, 156, 199, 209, 256] [40, 54, 75, 80, 157, 200, 210, 257] [41, 44, 76, 81, 158, 201, 211, 258] [42, 45, 66, 82, 159, 202, 212, 259] [54, 57, 78, 94, 139, 171, 214, 224] [44, 58, 79, 95, 140, 172, 215, 225] [45, 59, 80, 96, 141, 173, 216, 226] [46, 60, 81, 97, 142, 174, 217, 227] [47, 61, 82, 98, 132, 175, 218, 228] [48, 62, 83, 88, 133, 165, 219, 229] [49, 63, 84, 89, 134, 166, 209, 230] [50, 64, 85, 90, 135, 167, 210, 220] [51, 65, 86, 91, 136, 168, 211, 221] [52, 55, 87, 92, 137, 169, 212, 222] [53, 56, 77, 93, 138, 170, 213, 223] [65, 68, 89, 105, 150, 182, 225, 235] [55, 69, 90, 106, 151, 183, 226, 236] [56, 70, 91, 107, 152, 184, 227, 237] [57, 71, 92, 108, 153, 185, 228, 238] [58, 72, 93, 109, 143, 186, 229, 239] [59, 73, 94, 99, 144, 176, 230, 240] [60, 74, 95, 100, 145, 177, 220, 241] [61, 75, 96, 101, 146, 178, 221, 231] [62, 76, 97, 102, 147, 179, 222, 232] [63, 66, 98, 103, 148, 180, 223, 233] [64, 67, 88, 104, 149, 181, 224, 234] [76, 79, 100, 116, 161, 193, 236, 246] [66, 80, 101, 117, 162, 194, 237, 247] [67, 81, 102, 118, 163, 195, 238, 248] [68, 82, 103, 119, 164, 196, 239, 249] [69, 83, 104, 120, 154, 197, 240, 250] [70, 84, 105, 110, 155, 187, 241, 251] [71, 85, 106, 111, 156, 188, 231, 252] [72, 86, 107, 112, 157, 189, 232, 242] [73, 87, 108, 113, 158, 190, 233, 243] [74, 77, 109, 114, 159, 191, 234, 244] [75, 78, 99, 115, 160, 192, 235, 245] [87, 90, 111, 127, 172, 204, 247, 257] [77, 91, 112, 128, 173, 205, 248, 258] [78, 92, 113, 129, 174, 206, 249, 259] [79, 93, 114, 130, 175, 207, 250, 260] [80, 94, 115, 131, 165, 208, 251, 261] [81, 95, 116, 121, 166, 198, 252, 262] [82, 96, 117, 122, 167, 199, 242, 263] [83, 97, 118, 123, 168, 200, 243, 253] [84, 98, 119, 124, 169, 201, 244, 254] [85, 88, 120, 125, 170, 202, 245, 255] [86, 89, 110, 126, 171, 203, 246, 256] [6, 98, 101, 122, 136, 183, 215, 258] [7, 88, 102, 123, 137, 184, 216, 259] [8, 89, 103, 124, 138, 185, 217, 260] [9, 90, 104, 125, 139, 186, 218, 261] [10, 91, 105, 126, 140, 176, 219, 262] [0, 92, 106, 127, 141, 177, 209, 263] [1, 93, 107, 128, 142, 178, 210, 253] [2, 94, 108, 129, 132, 179, 211, 254] [3, 95, 109, 130, 133, 180, 212, 255] [4, 96, 99, 131, 134, 181, 213, 256] [5, 97, 100, 121, 135, 182, 214, 257] [1, 17, 109, 112, 137, 147, 194, 226] [2, 18, 99, 113, 138, 148, 195, 227] [3, 19, 100, 114, 139, 149, 196, 228] [4, 20, 101, 115, 140, 150, 197, 229] [5, 21, 102, 116, 141, 151, 187, 230] [6, 11, 103, 117, 142, 152, 188, 220] [7, 12, 104, 118, 132, 153, 189, 221] [8, 13, 105, 119, 133, 143, 190, 222] [9, 14, 106, 120, 134, 144, 191, 223] [10, 15, 107, 110, 135, 145, 192, 224] [0, 16, 108, 111, 136, 146, 193, 225] [12, 28, 120, 123, 148, 158, 205, 237] [13, 29, 110, 124, 149, 159, 206, 238] [14, 30, 111, 125, 150, 160, 207, 239] [15, 31, 112, 126, 151, 161, 208, 240] [16, 32, 113, 127, 152, 162, 198, 241] [17, 22, 114, 128, 153, 163, 199, 231] [18, 23, 115, 129, 143, 164, 200, 232] [19, 24, 116, 130, 144, 154, 201, 233] [20, 25, 117, 131, 145, 155, 202, 234] [21, 26, 118, 121, 146, 156, 203, 235] [11, 27, 119, 122, 147, 157, 204, 236] [2, 23, 39, 131, 159, 169, 216, 248] [3, 24, 40, 121, 160, 170, 217, 249] [4, 25, 41, 122, 161, 171, 218, 250] [5, 26, 42, 123, 162, 172, 219, 251] [6, 27, 43, 124, 163, 173, 209, 252] [7, 28, 33, 125, 164, 174, 210, 242] [8, 29, 34, 126, 154, 175, 211, 243] [9, 30, 35, 127, 155, 165, 212, 244] [10, 31, 36, 128, 156, 166, 213, 245] [0, 32, 37, 129, 157, 167, 214, 246] [1, 22, 38, 130, 158, 168, 215, 247] [10, 13, 34, 50, 170, 180, 227, 259] [0, 14, 35, 51, 171, 181, 228, 260] [1, 15, 36, 52, 172, 182, 229, 261] [2, 16, 37, 53, 173, 183, 230, 262] [3, 17, 38, 54, 174, 184, 220, 263] [4, 18, 39, 44, 175, 185, 221, 253] [5, 19, 40, 45, 165, 186, 222, 254] [6, 20, 41, 46, 166, 176, 223, 255] [7, 21, 42, 47, 167, 177, 224, 256] [8, 11, 43, 48, 168, 178, 225, 257] [9, 12, 33, 49, 169, 179, 226, 258] [21, 24, 45, 61, 138, 181, 191, 238] [11, 25, 46, 62, 139, 182, 192, 239] [12, 26, 47, 63, 140, 183, 193, 240] [13, 27, 48, 64, 141, 184, 194, 241] [14, 28, 49, 65, 142, 185, 195, 231] [15, 29, 50, 55, 132, 186, 196, 232] [16, 30, 51, 56, 133, 176, 197, 233] [17, 31, 52, 57, 134, 177, 187, 234] [18, 32, 53, 58, 135, 178, 188, 235] [19, 22, 54, 59, 136, 179, 189, 236] [20, 23, 44, 60, 137, 180, 190, 237] [32, 35, 56, 72, 149, 192, 202, 249] [22, 36, 57, 73, 150, 193, 203, 250] [23, 37, 58, 74, 151, 194, 204, 251] [24, 38, 59, 75, 152, 195, 205, 252] [25, 39, 60, 76, 153, 196, 206, 242] [26, 40, 61, 66, 143, 197, 207, 243] [27, 41, 62, 67, 144, 187, 208, 244] [28, 42, 63, 68, 145, 188, 198, 245] [29, 43, 64, 69, 146, 189, 199, 246] [30, 33, 65, 70, 147, 190, 200, 247] [31, 34, 55, 71, 148, 191, 201, 248]
Code ID 264-4-26 · download JSON · raw on GitHub