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[[288,16,24]] d ≤
n
288
k
16
d
24
kd²/n
32.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 24, d_Z ≤ 24 · 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 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[52, 62, 72, 105, 113, 120, 121, 128, 138, 143, 158, 175, 184, 195, 202, 211, 221, 230, 231, 238, 265, 272, 276, 282]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[26, 32, 89, 95, 101, 107, 110, 116, 137, 143, 193, 199, 204, 210, 219, 225, 240, 246, 267, 268, 273, 274, 279, 285]
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 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)×288 (2,6)×4032 (3,6)×1584 (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 (2,6): 4032 (3,6): 1584 (3,8): 79920 (3,10): 8064
trapping sets H_Z (1,4)×288 (2,6)×4032 (3,6)×1584 (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 (2,6): 4032 (3,6): 1584 (3,8): 79920 (3,10): 8064

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_12 x Z_12: A = 1 + y11 + x2y2 + x3, B = 1 + x6y3 + x6y10 + x11y4; H_X = [A | B], H_Z = [B^T | A^T]
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-10-17
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

[[288,16,24]] — weight-8 bivariate bicycle code on Z_12 x Z_12

Direction & hypothesis

Cell weight-8 x unrestricted (maximum check weight 8), CSS board, n between 221 and 400. Before this code, the frontier (board plus this campaign's open PRs) at check weight <= 8, d >= 12 and n from 228 to 288 ran through 232-62-12, 240-10-22, 240-18-20, 240-24-17, 240-28-16, 248-16-22, 248-24-18, 248-66-12, 252-20-21, 252-22-19, 252-26-18, 256-68-12. The hypothesis was that random 4 + 4-term bivariate bicycle (BB) codes on tori with 2lm in 222..300, prefiltered for k >= 16, still reach points there that the board does not hold.

[[288,16,24]] does: no board entry has n <= 288, k >= 16, d >= 24 and check weight <= 8. It is not a d-only gain (no board entry has the same n, k and weight with a lower d).

What was searched

Random-support BB codes H_X = [A | B], H_Z = [B^T | A^T] with A of 4 and B of 4 monomials, both containing 1 (no loss of generality), on every torus Z_l x Z_m with l >= m >= 3 and 222 <= 2lm <= 300; the torus is drawn uniformly from that list and the other monomials uniformly from the torus.

This sweep (seed 631) drew 500,010 codes in a parallel generate-and-rank prefilter; 2,144 had k >= 16 and a connected Tanner graph (research/kit/prescreen.block_structure). Those went through research/kit/search.screen with the gf2_fast backend at 400 trials; 6 advanced the board frontier at screen depth and 5 were taken to 20,000 / 200,000 / 2,000,000 fast trials (research/kit/surrogate.distance_rand_witness, pair depth 8), dropping a candidate as soon as its running bound stopped advancing; 1 survived that and was staged.

Evidence trail

Witness-backed upper bound d <= 24, not an exact distance.

| stage | trials | lightest logical | |---|---|---| | screen (gf2_fast) | 400 | 24 | | confirm | 20,000 | 24 | | confirm | 200,000 | 24 | | confirm | 2,000,000 | 24 | | deep ladder, 3 fresh seeds (67406, 75325, 83244), gf2_fast | 10,000,000 each | 24, 24, 24 |

verify/validate_candidate.py (through research/kit/coordination.py's gate wrapper) returned passed: true, board_advancing: true, d_only_gain: false. Literature novelty unverified; board-relative novelty from verify/validate_candidate.py.

Dead ends

  • In this sweep 4 of 5 screen-depth survivors collapsed below the frontier by 2,000,000 trials: 400-trial readings run several units high at this n.
  • Earlier in this campaign on the same band: cyclic generalized-bicycle codes with a designed divisor at weights (3,3) and (4,4), random-support 2BGA on non-abelian groups at check weight 8, non-abelian lifted products, and weight-6 BB/2BGA at k >= 28 gave nothing that survived 20,000 trials on the frontier.
  • Random 2 + 6 splits at check weight 8 never reached k >= 18 with an advancing distance.

Tools

Model: Claude Opus 5.5 (Claude Code agent, contributor-driven workflow). Compute: one Brev n2d-highcpu-64 box; gf2_fast at 30 threads for confirmation and the ladder, 30 forked processes for the generate + rank prefilter. Kit modules: research/kit/bb.py (build_bb), research/kit/search.py (screen), research/kit/surrogate.py (distance_rand_witness), research/kit/prescreen.py, research/kit/submit.py, research/kit/coordination.py. About 7.0 CPU-hours on this code and its sweep.

Reproduction

Torus Z_12 x Z_12 (x of order 12, y of order 12), terms (i, j) = x^i y^j:

A = 1 + y^11 + x^2 y^2 + x^3, B = 1 + x^6 y^3 + x^6 y^10 + x^11 y^4.

import sys; sys.path[:0] = ["research/kit", "verify"]
from bb import build_bb
from css import compute_k
HX, HZ = build_bb(12, 12, [(0, 0), (0, 11), (2, 2), (3, 0)],
                  [(0, 0), (6, 3), (6, 10), (11, 4)])   # n = 288
assert compute_k(HX, HZ) == 16

Ladder: surrogate.distance_rand_witness(HX, HZ, trials=10_000_000, seed=<s>, backend="fast", pair_depth=8) with s in {67406, 75325, 83244} returns logicals of weight 24, 24, 24.

Parity checks

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