[[276,2,27]] · QEC Challenge

[[276,2,27]] d ≤

276

kd²/n

5.28261

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 27, d_Z ≤ 27 · 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 27 · witness weight 27 (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, 27 qubits)

[19, 22, 33, 40, 51, 70, 88, 94, 101, 106, 109, 116, 119, 134, 143, 147, 151, 167, 192, 193, 195, 229, 238, 245, 246, 263, 266]

d_Z 27 · witness weight 27 (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, 27 qubits)

[0, 39, 54, 56, 90, 101, 106, 107, 109, 120, 124, 129, 135, 138, 160, 161, 170, 183, 208, 209, 233, 238, 245, 246, 251, 268, 273]

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)×276 (2,4)×207 (3,6)×6762 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 276 (2,4): 207 (2,6): 3450 (3,6): 6762 (3,8): 62928 (3,10): 5658

trapping sets H_Z (1,4)×276 (2,4)×207 (3,6)×6762 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 276 (2,4): 207 (2,6): 3450 (3,6): 6762 (3,8): 62928 (3,10): 5658

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Two-block group-algebra code on the dihedral group D_69 (order 138, n = 276): mul, _ = research/kit/group_algebra.dihedral(69); H_X, H_Z = group_algebra.build_2bga(mul, a=[95, 79, 84, 75], b=[60, 122, 54, 113]), with a and b indices into that element list, H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T].

model Claude Fable 5.1 (Claude Code) working for @vprusso (claimed, not verified)

builds on H.-K. Lin and L. P. Pryadko, Quantum two-block group algebra codes, Phys. Rev. A 109, 022407 (2024), arXiv:2306.16400

date 2026-10-24

notes Unattended loop run on Perlmutter, iteration 609, direction dihedral-w4, seed 20261626. The loop gate's 8,000-trial search read d = 30; the deeper rungs recorded in notes/ read 27, and the document carries those 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

[[276,2,27]] loop 2BGA code on the dihedral group D_69

Direction & hypothesis

Two-block group-algebra codes on non-abelian groups are not constrained to the even-distance pattern of the abelian bicycle codes; dihedral groups are the smallest family to test that at weight 8.

What was searched

Direction dihedral-w4 of the loop samples two-block group-algebra codes on dihedral groups D_m with m in [30, 80] and four group elements per block (check weight 8). 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 609 (seed 20261626). By 29 October the direction had run 277 iterations over 33,240 candidates and staged 218; 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) | 27 | | 8,000 (loop gate, NumPy, per side) | 30 | | 10,000,000 (Lux gf2_fast, both sides, seed 11) | 27 | | 10,000,000 (Lux gf2_fast, both sides, seed 22) | 27 | | 300,000,000 (GPU RIS, 300M per side, seed 1) | 27 | | 300,000,000 (GPU RIS, 300M per side, seed 2) | 27 |

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 <= 27 (upper_bound), Z: d <= 27 (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 30 did not survive the deeper rungs; the document was repackaged at 27 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/group_algebra.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

mul, _ = research/kit/group_algebra.dihedral(69) then group_algebra.build_2bga(mul, a=[95, 79, 84, 75], b=[60, 122, 54, 113]) returns (H_X, H_Z) with the element order that function documents; no {constructor, params} recipe exists for this call, so the manifest's construction stage is declared not reproducible and the call is recorded here instead. 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 138 (max weight 8) · Z-checks 138 (max weight 8)

H_X (138 checks, sparse supports)

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

H_Z (138 checks, sparse supports)

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

Code ID 276-2-27 · download JSON · raw on GitHub