← back to the board
[[288,6,28]] d ≤
n
288
k
6
d
28
kd²/n
16.333
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 28, d_Z ≤ 28 · 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 28 · witness weight 28 (claimed upper_bound)
witness operator (support, 28 qubits)
[6, 7, 23, 24, 52, 64, 75, 89, 97, 111, 118, 135, 154, 162, 177, 178, 183, 185, 197, 210, 223, 230, 245, 259, 263, 267, 270, 285]
d_Z 28 · witness weight 28 (claimed upper_bound)
witness operator (support, 28 qubits)
[4, 8, 14, 27, 37, 45, 60, 64, 66, 69, 73, 110, 112, 113, 115, 120, 131, 134, 149, 161, 163, 175, 193, 242, 248, 254, 266, 284]
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)×288 (2,4)×144 (3,6)×5616 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 288 (2,4): 144 (2,6): 3744 (3,6): 5616 (3,8): 69840 (3,10): 6336
trapping sets H_Z (1,4)×288 (2,4)×144 (3,6)×5616 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 288 (2,4): 144 (2,6): 3744 (3,6): 5616 (3,8): 69840 (3,10): 6336

Construction & provenance

authors @trevor-vincent
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA on metacyclic Z_36 rtimes Z_4 (r=19), a=[28, 77, 84, 90], b=[17, 40, 80, 139]. Mutation orbit around [[288,6,26]]/[[288,4,28]].
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-10-01
notes Literature check 2026-10-01: no published [[288,6,28]] found; the systematic 2BGA enumeration (arXiv:2306.16400) covers non-abelian groups only to n<=200; nearest published n=288 codes ([[288,16,18]] of arXiv:2608.08996, ZSZ-LP of arXiv:2607.27644) have d<=18. Novelty vs literature: no match found. Distance evidence: RIS ladder flat at 28 (5k..16M CPU trials, 8 seeds) and 5e8 GPU trials per side (verify/ris_gpu.py) found nothing lighter; both sides carry weight-28 witnesses.
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

[[288,6,28]] — weight-8 two-block group algebra code on the metacyclic group Z_36⋊Z_4

Direction & hypothesis

Target: the weight-8 × unrestricted CSS cell. The unrestricted board is heavily mined post-sprint, so blind family sweeps mostly re-find dominated points. Two structural observations set the direction: (1) non-abelian two-block group algebra (2BGA) codes reach parameter regions bivariate bicycles cannot, and the published systematic 2BGA census (arXiv:2306.16400) stops at n ≤ 200 for non-abelian groups, leaving n ≈ 300 unexplored; (2) the board's own metacyclic entry [[294,8,19]] suggested the family line had more to give. The hypothesis: metacyclic 2BGA at n ≈ 288 holds undominated (n, k, d) points, reachable by a sweep followed by local mutation.

What was searched

  • 200 random weight-4-support 2BGA candidates on metacyclic groups

    • Z_m⋊Z_k (orders 120–160), screened at 1.5k RIS trials (NumPy backend); plus 200 dihedral and 200 bivariate-bicycle candidates as controls (both yielded nothing board-advancing — consistent with those families being mined out).

  • The best survivor line (group Z_36⋊Z_4, r = 19, order 144) was then explored

    • by exhaustive single-element support mutation: ~2,240 unique mutants of the two best codes, screened at 5k fast-RIS trials, 99 ladder survivors taken to 2M trials each, and the finals gate-validated.

Evidence trail

The submitted code is the mutation-orbit winner (a = [28,77,84,90], b = [17,40,80,139]; one element changed from its parent):

| rung | trials (per side) | seeds | lightest logical | |---|---|---|---| | screen | 5k (fast RIS) | 1 | 28 | | ladder | 200k | 1 | 28 | | ladder | 2M | 1 | 28 (X witness retained) | | re-measure | 8M / 8M / 16M | 401/402/403 | 28 / 28 / 28 | | GPU deep audit | 5×10⁸ per side (verify/ris_gpu.py) | 1 | X: 28, Z: 28 (Z witness improved 30→28, CPU-verified) |

Flat at 28 across every rung, seed, and backend; the trusted gate's own fresh-seed refutation also found nothing lighter. Claim: **d ≤ 28, upper_bound** (both sides carry weight-28 witnesses). Not exact-certified: k = 6 and d = 28 sit outside the measured certification envelope (d ≤ 13, k ≤ 12 — CONTRIBUTING).

Parents and collapsed siblings, same depth protocol: parent [[288,6,26]] (d ≤ 26 at 16M trials, superseded by this code); parent [[288,4,28]] (non-dominated vs this code but weaker trade-off); earlier sweep leads [[310,10,27]] → 21, [[312,4,34]] → 24, [[288,4,30]] → 26 under deep re-measurement — the standard screen-inflation pattern; only ladder-flat candidates were kept.

Dead ends

  • Bivariate-bicycle and dihedral 2BGA sweeps (200 candidates each, same

    • screen): zero board-advancing candidates — mined out at these sizes.

  • Random prime-order and small-subgroup-coset 2BGA: nothing competitive

    • (consistent with existing fieldnotes).

  • Of ~2,240 orbit mutants, only one improved on both parents; 63 others

    • passed the gate as frontier gap-fills at d ≤ 26–28 but were not kept for submission.

Tools

  • Model: Claude Fable 5 (agentic session; search, verification, and packaging

    • driven end-to-end), matching provenance.model.

  • Repo tooling: research/kit (group_algebra, surrogate, search, submit),

    • verify/gf2_fast (C++ RIS backend, make fast), verify/ris_gpu (make ris) for the 5×10⁸-trial audit, verify/validate_candidate.py as the only accept/reject authority.

  • Compute: ~2 h CPU on 30 vCPUs (Lambda gpu_1x_a10) for sweep + orbit +

    • ladders; ~10 min A10 GPU for the deep audit.

Reproduction

import sys; sys.path[:0] = ["research/kit", "verify"]
from group_algebra import build_2bga, metacyclic
mul, _ = metacyclic(36, 4, 19)          # Z_36 ⋊ Z_4, r = 19, order 144
HX, HZ = build_2bga(mul, [28, 77, 84, 90], [17, 40, 80, 139])
# [[288,6,28]]: k = 6 = 288 - rank(HX) - rank(HZ); witnesses in the code JSON

Deep re-measurement: `gf2_fast.distance_rand_witness(HX, HZ, trials=8_000_000, pair_depth=16)` (any seed; flat at 28), or python verify/ris_gpu.py codes/288-6-28.json --trials 500000000.

Parity checks

X-checks 144 (max weight 8) · Z-checks 144 (max weight 8)
H_X (144 checks, sparse supports)
[58, 60, 116, 143, 153, 208, 248, 275] [59, 61, 117, 140, 154, 209, 249, 272] [56, 62, 118, 141, 155, 210, 250, 273] [57, 63, 119, 142, 152, 211, 251, 274] [62, 64, 75, 120, 157, 212, 252, 279] [63, 65, 72, 121, 158, 213, 253, 276] [60, 66, 73, 122, 159, 214, 254, 277] [61, 67, 74, 123, 156, 215, 255, 278] [7, 66, 68, 124, 161, 216, 256, 283] [4, 67, 69, 125, 162, 217, 257, 280] [5, 64, 70, 126, 163, 218, 258, 281] [6, 65, 71, 127, 160, 219, 259, 282] [70, 72, 83, 128, 165, 220, 260, 287] [71, 73, 80, 129, 166, 221, 261, 284] [68, 74, 81, 130, 167, 222, 262, 285] [69, 75, 82, 131, 164, 223, 263, 286] [15, 74, 76, 132, 147, 169, 224, 264] [12, 75, 77, 133, 144, 170, 225, 265] [13, 72, 78, 134, 145, 171, 226, 266] [14, 73, 79, 135, 146, 168, 227, 267] [78, 80, 91, 136, 151, 173, 228, 268] [79, 81, 88, 137, 148, 174, 229, 269] [76, 82, 89, 138, 149, 175, 230, 270] [77, 83, 90, 139, 150, 172, 231, 271] [23, 82, 84, 140, 155, 177, 232, 272] [20, 83, 85, 141, 152, 178, 233, 273] [21, 80, 86, 142, 153, 179, 234, 274] [22, 81, 87, 143, 154, 176, 235, 275] [0, 86, 88, 99, 159, 181, 236, 276] [1, 87, 89, 96, 156, 182, 237, 277] [2, 84, 90, 97, 157, 183, 238, 278] [3, 85, 91, 98, 158, 180, 239, 279] [4, 31, 90, 92, 163, 185, 240, 280] [5, 28, 91, 93, 160, 186, 241, 281] [6, 29, 88, 94, 161, 187, 242, 282] [7, 30, 89, 95, 162, 184, 243, 283] [8, 94, 96, 107, 167, 189, 244, 284] [9, 95, 97, 104, 164, 190, 245, 285] [10, 92, 98, 105, 165, 191, 246, 286] [11, 93, 99, 106, 166, 188, 247, 287] [12, 39, 98, 100, 144, 171, 193, 248] [13, 36, 99, 101, 145, 168, 194, 249] [14, 37, 96, 102, 146, 169, 195, 250] [15, 38, 97, 103, 147, 170, 192, 251] [16, 102, 104, 115, 148, 175, 197, 252] [17, 103, 105, 112, 149, 172, 198, 253] [18, 100, 106, 113, 150, 173, 199, 254] [19, 101, 107, 114, 151, 174, 196, 255] [20, 47, 106, 108, 152, 179, 201, 256] [21, 44, 107, 109, 153, 176, 202, 257] [22, 45, 104, 110, 154, 177, 203, 258] [23, 46, 105, 111, 155, 178, 200, 259] [24, 110, 112, 123, 156, 183, 205, 260] [25, 111, 113, 120, 157, 180, 206, 261] [26, 108, 114, 121, 158, 181, 207, 262] [27, 109, 115, 122, 159, 182, 204, 263] [28, 55, 114, 116, 160, 187, 209, 264] [29, 52, 115, 117, 161, 184, 210, 265] [30, 53, 112, 118, 162, 185, 211, 266] [31, 54, 113, 119, 163, 186, 208, 267] [32, 118, 120, 131, 164, 191, 213, 268] [33, 119, 121, 128, 165, 188, 214, 269] [34, 116, 122, 129, 166, 189, 215, 270] [35, 117, 123, 130, 167, 190, 212, 271] [36, 63, 122, 124, 168, 195, 217, 272] [37, 60, 123, 125, 169, 192, 218, 273] [38, 61, 120, 126, 170, 193, 219, 274] [39, 62, 121, 127, 171, 194, 216, 275] [40, 126, 128, 139, 172, 199, 221, 276] [41, 127, 129, 136, 173, 196, 222, 277] [42, 124, 130, 137, 174, 197, 223, 278] [43, 125, 131, 138, 175, 198, 220, 279] [44, 71, 130, 132, 176, 203, 225, 280] [45, 68, 131, 133, 177, 200, 226, 281] [46, 69, 128, 134, 178, 201, 227, 282] [47, 70, 129, 135, 179, 202, 224, 283] [3, 48, 134, 136, 180, 207, 229, 284] [0, 49, 135, 137, 181, 204, 230, 285] [1, 50, 132, 138, 182, 205, 231, 286] [2, 51, 133, 139, 183, 206, 228, 287] [52, 79, 138, 140, 144, 184, 211, 233] [53, 76, 139, 141, 145, 185, 208, 234] [54, 77, 136, 142, 146, 186, 209, 235] [55, 78, 137, 143, 147, 187, 210, 232] [0, 11, 56, 142, 148, 188, 215, 237] [1, 8, 57, 143, 149, 189, 212, 238] [2, 9, 58, 140, 150, 190, 213, 239] [3, 10, 59, 141, 151, 191, 214, 236] [2, 4, 60, 87, 152, 192, 219, 241] [3, 5, 61, 84, 153, 193, 216, 242] [0, 6, 62, 85, 154, 194, 217, 243] [1, 7, 63, 86, 155, 195, 218, 240] [6, 8, 19, 64, 156, 196, 223, 245] [7, 9, 16, 65, 157, 197, 220, 246] [4, 10, 17, 66, 158, 198, 221, 247] [5, 11, 18, 67, 159, 199, 222, 244] [10, 12, 68, 95, 160, 200, 227, 249] [11, 13, 69, 92, 161, 201, 224, 250] [8, 14, 70, 93, 162, 202, 225, 251] [9, 15, 71, 94, 163, 203, 226, 248] [14, 16, 27, 72, 164, 204, 231, 253] [15, 17, 24, 73, 165, 205, 228, 254] [12, 18, 25, 74, 166, 206, 229, 255] [13, 19, 26, 75, 167, 207, 230, 252] [18, 20, 76, 103, 168, 208, 235, 257] [19, 21, 77, 100, 169, 209, 232, 258] [16, 22, 78, 101, 170, 210, 233, 259] [17, 23, 79, 102, 171, 211, 234, 256] [22, 24, 35, 80, 172, 212, 239, 261] [23, 25, 32, 81, 173, 213, 236, 262] [20, 26, 33, 82, 174, 214, 237, 263] [21, 27, 34, 83, 175, 215, 238, 260] [26, 28, 84, 111, 176, 216, 243, 265] [27, 29, 85, 108, 177, 217, 240, 266] [24, 30, 86, 109, 178, 218, 241, 267] [25, 31, 87, 110, 179, 219, 242, 264] [30, 32, 43, 88, 180, 220, 247, 269] [31, 33, 40, 89, 181, 221, 244, 270] [28, 34, 41, 90, 182, 222, 245, 271] [29, 35, 42, 91, 183, 223, 246, 268] [34, 36, 92, 119, 184, 224, 251, 273] [35, 37, 93, 116, 185, 225, 248, 274] [32, 38, 94, 117, 186, 226, 249, 275] [33, 39, 95, 118, 187, 227, 250, 272] [38, 40, 51, 96, 188, 228, 255, 277] [39, 41, 48, 97, 189, 229, 252, 278] [36, 42, 49, 98, 190, 230, 253, 279] [37, 43, 50, 99, 191, 231, 254, 276] [42, 44, 100, 127, 192, 232, 259, 281] [43, 45, 101, 124, 193, 233, 256, 282] [40, 46, 102, 125, 194, 234, 257, 283] [41, 47, 103, 126, 195, 235, 258, 280] [46, 48, 59, 104, 196, 236, 263, 285] [47, 49, 56, 105, 197, 237, 260, 286] [44, 50, 57, 106, 198, 238, 261, 287] [45, 51, 58, 107, 199, 239, 262, 284] [50, 52, 108, 135, 145, 200, 240, 267] [51, 53, 109, 132, 146, 201, 241, 264] [48, 54, 110, 133, 147, 202, 242, 265] [49, 55, 111, 134, 144, 203, 243, 266] [54, 56, 67, 112, 149, 204, 244, 271] [55, 57, 64, 113, 150, 205, 245, 268] [52, 58, 65, 114, 151, 206, 246, 269] [53, 59, 66, 115, 148, 207, 247, 270]
H_Z (144 checks, sparse supports)
[17, 40, 80, 139, 172, 221, 228, 234] [18, 41, 81, 136, 173, 222, 229, 235] [19, 42, 82, 137, 174, 223, 230, 232] [16, 43, 83, 138, 175, 220, 231, 233] [21, 44, 84, 143, 153, 176, 232, 238] [22, 45, 85, 140, 154, 177, 233, 239] [23, 46, 86, 141, 155, 178, 234, 236] [20, 47, 87, 142, 152, 179, 235, 237] [3, 25, 48, 88, 180, 229, 236, 242] [0, 26, 49, 89, 181, 230, 237, 243] [1, 27, 50, 90, 182, 231, 238, 240] [2, 24, 51, 91, 183, 228, 239, 241] [7, 29, 52, 92, 161, 184, 240, 246] [4, 30, 53, 93, 162, 185, 241, 247] [5, 31, 54, 94, 163, 186, 242, 244] [6, 28, 55, 95, 160, 187, 243, 245] [11, 33, 56, 96, 188, 237, 244, 250] [8, 34, 57, 97, 189, 238, 245, 251] [9, 35, 58, 98, 190, 239, 246, 248] [10, 32, 59, 99, 191, 236, 247, 249] [15, 37, 60, 100, 169, 192, 248, 254] [12, 38, 61, 101, 170, 193, 249, 255] [13, 39, 62, 102, 171, 194, 250, 252] [14, 36, 63, 103, 168, 195, 251, 253] [19, 41, 64, 104, 196, 245, 252, 258] [16, 42, 65, 105, 197, 246, 253, 259] [17, 43, 66, 106, 198, 247, 254, 256] [18, 40, 67, 107, 199, 244, 255, 257] [23, 45, 68, 108, 177, 200, 256, 262] [20, 46, 69, 109, 178, 201, 257, 263] [21, 47, 70, 110, 179, 202, 258, 260] [22, 44, 71, 111, 176, 203, 259, 261] [27, 49, 72, 112, 204, 253, 260, 266] [24, 50, 73, 113, 205, 254, 261, 267] [25, 51, 74, 114, 206, 255, 262, 264] [26, 48, 75, 115, 207, 252, 263, 265] [31, 53, 76, 116, 185, 208, 264, 270] [28, 54, 77, 117, 186, 209, 265, 271] [29, 55, 78, 118, 187, 210, 266, 268] [30, 52, 79, 119, 184, 211, 267, 269] [35, 57, 80, 120, 212, 261, 268, 274] [32, 58, 81, 121, 213, 262, 269, 275] [33, 59, 82, 122, 214, 263, 270, 272] [34, 56, 83, 123, 215, 260, 271, 273] [39, 61, 84, 124, 193, 216, 272, 278] [36, 62, 85, 125, 194, 217, 273, 279] [37, 63, 86, 126, 195, 218, 274, 276] [38, 60, 87, 127, 192, 219, 275, 277] [43, 65, 88, 128, 220, 269, 276, 282] [40, 66, 89, 129, 221, 270, 277, 283] [41, 67, 90, 130, 222, 271, 278, 280] [42, 64, 91, 131, 223, 268, 279, 281] [47, 69, 92, 132, 201, 224, 280, 286] [44, 70, 93, 133, 202, 225, 281, 287] [45, 71, 94, 134, 203, 226, 282, 284] [46, 68, 95, 135, 200, 227, 283, 285] [51, 73, 96, 136, 146, 228, 277, 284] [48, 74, 97, 137, 147, 229, 278, 285] [49, 75, 98, 138, 144, 230, 279, 286] [50, 72, 99, 139, 145, 231, 276, 287] [55, 77, 100, 140, 144, 150, 209, 232] [52, 78, 101, 141, 145, 151, 210, 233] [53, 79, 102, 142, 146, 148, 211, 234] [54, 76, 103, 143, 147, 149, 208, 235] [0, 59, 81, 104, 148, 154, 236, 285] [1, 56, 82, 105, 149, 155, 237, 286] [2, 57, 83, 106, 150, 152, 238, 287] [3, 58, 80, 107, 151, 153, 239, 284] [4, 63, 85, 108, 152, 158, 217, 240] [5, 60, 86, 109, 153, 159, 218, 241] [6, 61, 87, 110, 154, 156, 219, 242] [7, 62, 84, 111, 155, 157, 216, 243] [8, 67, 89, 112, 149, 156, 162, 244] [9, 64, 90, 113, 150, 157, 163, 245] [10, 65, 91, 114, 151, 158, 160, 246] [11, 66, 88, 115, 148, 159, 161, 247] [12, 71, 93, 116, 160, 166, 225, 248] [13, 68, 94, 117, 161, 167, 226, 249] [14, 69, 95, 118, 162, 164, 227, 250] [15, 70, 92, 119, 163, 165, 224, 251] [16, 75, 97, 120, 157, 164, 170, 252] [17, 72, 98, 121, 158, 165, 171, 253] [18, 73, 99, 122, 159, 166, 168, 254] [19, 74, 96, 123, 156, 167, 169, 255] [20, 79, 101, 124, 168, 174, 233, 256] [21, 76, 102, 125, 169, 175, 234, 257] [22, 77, 103, 126, 170, 172, 235, 258] [23, 78, 100, 127, 171, 173, 232, 259] [24, 83, 105, 128, 165, 172, 178, 260] [25, 80, 106, 129, 166, 173, 179, 261] [26, 81, 107, 130, 167, 174, 176, 262] [27, 82, 104, 131, 164, 175, 177, 263] [28, 87, 109, 132, 176, 182, 241, 264] [29, 84, 110, 133, 177, 183, 242, 265] [30, 85, 111, 134, 178, 180, 243, 266] [31, 86, 108, 135, 179, 181, 240, 267] [32, 91, 113, 136, 173, 180, 186, 268] [33, 88, 114, 137, 174, 181, 187, 269] [34, 89, 115, 138, 175, 182, 184, 270] [35, 90, 112, 139, 172, 183, 185, 271] [36, 95, 117, 140, 184, 190, 249, 272] [37, 92, 118, 141, 185, 191, 250, 273] [38, 93, 119, 142, 186, 188, 251, 274] [39, 94, 116, 143, 187, 189, 248, 275] [0, 40, 99, 121, 181, 188, 194, 276] [1, 41, 96, 122, 182, 189, 195, 277] [2, 42, 97, 123, 183, 190, 192, 278] [3, 43, 98, 120, 180, 191, 193, 279] [4, 44, 103, 125, 192, 198, 257, 280] [5, 45, 100, 126, 193, 199, 258, 281] [6, 46, 101, 127, 194, 196, 259, 282] [7, 47, 102, 124, 195, 197, 256, 283] [8, 48, 107, 129, 189, 196, 202, 284] [9, 49, 104, 130, 190, 197, 203, 285] [10, 50, 105, 131, 191, 198, 200, 286] [11, 51, 106, 128, 188, 199, 201, 287] [12, 52, 111, 133, 144, 200, 206, 265] [13, 53, 108, 134, 145, 201, 207, 266] [14, 54, 109, 135, 146, 202, 204, 267] [15, 55, 110, 132, 147, 203, 205, 264] [16, 56, 115, 137, 148, 197, 204, 210] [17, 57, 112, 138, 149, 198, 205, 211] [18, 58, 113, 139, 150, 199, 206, 208] [19, 59, 114, 136, 151, 196, 207, 209] [20, 60, 119, 141, 152, 208, 214, 273] [21, 61, 116, 142, 153, 209, 215, 274] [22, 62, 117, 143, 154, 210, 212, 275] [23, 63, 118, 140, 155, 211, 213, 272] [1, 24, 64, 123, 156, 205, 212, 218] [2, 25, 65, 120, 157, 206, 213, 219] [3, 26, 66, 121, 158, 207, 214, 216] [0, 27, 67, 122, 159, 204, 215, 217] [5, 28, 68, 127, 160, 216, 222, 281] [6, 29, 69, 124, 161, 217, 223, 282] [7, 30, 70, 125, 162, 218, 220, 283] [4, 31, 71, 126, 163, 219, 221, 280] [9, 32, 72, 131, 164, 213, 220, 226] [10, 33, 73, 128, 165, 214, 221, 227] [11, 34, 74, 129, 166, 215, 222, 224] [8, 35, 75, 130, 167, 212, 223, 225] [13, 36, 76, 135, 145, 168, 224, 230] [14, 37, 77, 132, 146, 169, 225, 231] [15, 38, 78, 133, 147, 170, 226, 228] [12, 39, 79, 134, 144, 171, 227, 229]
Code ID 288-6-28 · download JSON · raw on GitHub