← back to the board
[[288,16,16]] d ≤
n
288
k
16
d
16
kd²/n
14.222
w
6

Share this result

Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[16, 38, 75, 112, 123, 134, 156, 159, 161, 166, 205, 207, 252, 253, 257, 262]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[12, 14, 17, 19, 62, 64, 108, 112, 113, 115, 146, 157, 183, 194, 253, 279]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @willzeng
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (Lin-Pryadko two-block group-algebra) code on the metacyclic C12 : C12 = < x, y | x12=1, y12=1, y x y-1 = x5 > (order 144). H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with support a = [0, 9, 55], b = [0, 26, 37] (element indices; metacyclic element (i,j) = x^i y^j at index i*k+j; dicyclic element a^i b^j at index i*2+j; identity = 0). Found by screen-then-confirm simulated annealing over supports; distance is the lightest logical witnessed in 600000-trial randomized search per the campaign ladder (see REPORT.md).
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-07-14
notes Ladder (trials->d): {'2000': 16, '8000': 16, '60000': 16, '200000': 16}. Surrogate values are upper bounds; confidence upper_bound. Novelty vs literature unverified (order > 100 group, outside Lin-Pryadko exhaustive n<=200 enumeration). Independent adversarial re-check: 3 fresh seeds (910001-910003) x 1M trials/side (gf2_fast), lightest logical 16 each time, nothing lighter. Upper bound only. Lit-check 2026-07-14: no published [[288,16,16]] found; closest published weight-6 neighbor [[224,12,16]] (arXiv:2606.17268); group order 144 is outside the Lin-Pryadko n<=200 exhaustive 2BGA enumeration; novelty vs literature unverified.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[288,16,16]] — annealed 2BGA on the metacyclic group C12⋊C12

Direction & hypothesis

Target: the weight-6 × unrestricted cell (best 13.50, [[288,12,18]]; frontier k-max 12). Hypothesis: nonabelian metacyclic groups of order 100–180 are outside Lin–Pryadko's exhaustive n ≤ 200 enumeration, so novelty is possible, and annealing should out-search blind sampling once tuned.

What was searched

4,700 random screens + ~2,500 anneal evaluations over all nonabelian metacyclic (conjugacy-deduped) and dicyclic groups of order 100–180, support-3 (w6) and support-4 (w8). This code: 2BGA on C12⋊C12 = ⟨x,y | x¹²=y¹²=1, yxy⁻¹=x⁵⟩ (order 144, GAP-independent presentation), a=[0,9,55], b=[0,26,37] (element indices in the kit's enumeration), found by annealing from the screened seed [[288,8,18]].

Method result worth having: the historic ~0.1% anneal acceptance on this family was never a temperature problem — 60–80% of support mutations destroy k and must be auto-rejected before the Metropolis step; among k-viable moves acceptance at T = 1–2 is ~50%. With that fix, T=1.5 with cooling worked.

Evidence trail

Ladder 2k/8k/60k/200k: 16/16/16/16, flat; 600k-trial deep search witnessed weight-16 logicals on both sides. Pre-submission adversarial re-check: 3 fresh seeds × 1M trials/side (gf2_fast), all 16. Claim: **upper bound d ≤ 16**. Gate verdict: board_advancing, dedup clean.

Dead ends

  • [[252,8,26]] (screen) collapsed to 19 by the 8k rung.
  • All n ≥ 330 anneal stars ([[360,18,38]], [[360,12,42]], [[360,12,22]])

    • were left unpackaged: the 1M-trial rung was too expensive on a contended machine, and this family's screen values inflate. Treat those parameters as unconfirmed leads, not results.

  • Dicyclic groups never out-screened metacyclic at equal order.

Tools

Claude Fable 5 agent campaign; annealer in the campaign staging dir (tune/screen/anneal/ladder phases); gf2_fast for deep rungs.

Reproduction

research/kit/group_algebra.py metacyclic builder with the presentation and supports above; construction string in codes/288-16-16.json.

Parity checks

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