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[[300,18,24]] d ≤
n
300
k
18
d
24
kd²/n
34.56
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)
[1, 15, 22, 29, 45, 52, 66, 70, 73, 75, 77, 89, 167, 172, 177, 210, 222, 227, 248, 250, 253, 255, 260, 296]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[38, 51, 56, 76, 84, 94, 102, 104, 112, 130, 151, 161, 165, 166, 175, 178, 187, 201, 266, 269, 276, 278, 288, 295]
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)×300 (2,6)×4200 (3,6)×2550 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 300 (2,6): 4200 (3,6): 2550 (3,8): 80550 (3,10): 8400
trapping sets H_Z (1,4)×300 (2,6)×4200 (3,6)×2550 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 300 (2,6): 4200 (3,6): 2550 (3,8): 80550 (3,10): 8400

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_30 x Z_5: A = 1 + x15y4 + x21y2 + x22y2, B = 1 + x3y4 + x5y3 + x7; 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

[[300,18,24]] — weight-8 bivariate bicycle code on Z_30 x Z_5

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 240 to 300 ran through 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, 264-70-12. The hypothesis was that random 4 + 4-term bivariate bicycle (BB) codes on tori with 2lm in 300..400, prefiltered for k >= 18, still reach points there that the board does not hold.

[[300,18,24]] does: no board entry has n <= 300, k >= 18, 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 300 <= 2lm <= 400; the torus is drawn uniformly from that list and the other monomials uniformly from the torus.

This sweep (seed 604) drew 400,020 codes in a parallel generate-and-rank prefilter; 946 had k >= 18 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; 22 advanced the board frontier at screen depth and 21 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; 2 survived that and were staged.

Evidence trail

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

| stage | trials | lightest logical | |---|---|---| | screen (gf2_fast) | 400 | 29 | | confirm | 20,000 | 24 | | confirm | 200,000 | 24 | | confirm | 2,000,000 | 24 | | deep ladder, 3 fresh seeds (64625, 72544, 80463), 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 19 of 21 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 9.5 CPU-hours on this code and its sweep.

Reproduction

Torus Z_30 x Z_5 (x of order 30, y of order 5), terms (i, j) = x^i y^j:

A = 1 + x^15 y^4 + x^21 y^2 + x^22 y^2, B = 1 + x^3 y^4 + x^5 y^3 + x^7.

import sys; sys.path[:0] = ["research/kit", "verify"]
from bb import build_bb
from css import compute_k
HX, HZ = build_bb(30, 5, [(0, 0), (15, 4), (21, 2), (22, 2)],
                  [(0, 0), (3, 4), (5, 3), (7, 0)])   # n = 300
assert compute_k(HX, HZ) == 18

Ladder: surrogate.distance_rand_witness(HX, HZ, trials=10_000_000, seed=<s>, backend="fast", pair_depth=8) with s in {64625, 72544, 80463} returns logicals of weight 24, 24, 24.

Parity checks

X-checks 150 (max weight 8) · Z-checks 150 (max weight 8)
H_X (150 checks, sparse supports)
[0, 79, 107, 112, 150, 169, 178, 185] [1, 75, 108, 113, 151, 165, 179, 186] [2, 76, 109, 114, 152, 166, 175, 187] [3, 77, 105, 110, 153, 167, 176, 188] [4, 78, 106, 111, 154, 168, 177, 189] [5, 84, 112, 117, 155, 174, 183, 190] [6, 80, 113, 118, 156, 170, 184, 191] [7, 81, 114, 119, 157, 171, 180, 192] [8, 82, 110, 115, 158, 172, 181, 193] [9, 83, 111, 116, 159, 173, 182, 194] [10, 89, 117, 122, 160, 179, 188, 195] [11, 85, 118, 123, 161, 175, 189, 196] [12, 86, 119, 124, 162, 176, 185, 197] [13, 87, 115, 120, 163, 177, 186, 198] [14, 88, 116, 121, 164, 178, 187, 199] [15, 94, 122, 127, 165, 184, 193, 200] [16, 90, 123, 128, 166, 180, 194, 201] [17, 91, 124, 129, 167, 181, 190, 202] [18, 92, 120, 125, 168, 182, 191, 203] [19, 93, 121, 126, 169, 183, 192, 204] [20, 99, 127, 132, 170, 189, 198, 205] [21, 95, 128, 133, 171, 185, 199, 206] [22, 96, 129, 134, 172, 186, 195, 207] [23, 97, 125, 130, 173, 187, 196, 208] [24, 98, 126, 131, 174, 188, 197, 209] [25, 104, 132, 137, 175, 194, 203, 210] [26, 100, 133, 138, 176, 190, 204, 211] [27, 101, 134, 139, 177, 191, 200, 212] [28, 102, 130, 135, 178, 192, 201, 213] [29, 103, 131, 136, 179, 193, 202, 214] [30, 109, 137, 142, 180, 199, 208, 215] [31, 105, 138, 143, 181, 195, 209, 216] [32, 106, 139, 144, 182, 196, 205, 217] [33, 107, 135, 140, 183, 197, 206, 218] [34, 108, 136, 141, 184, 198, 207, 219] [35, 114, 142, 147, 185, 204, 213, 220] [36, 110, 143, 148, 186, 200, 214, 221] [37, 111, 144, 149, 187, 201, 210, 222] [38, 112, 140, 145, 188, 202, 211, 223] [39, 113, 141, 146, 189, 203, 212, 224] [2, 40, 119, 147, 190, 209, 218, 225] [3, 41, 115, 148, 191, 205, 219, 226] [4, 42, 116, 149, 192, 206, 215, 227] [0, 43, 117, 145, 193, 207, 216, 228] [1, 44, 118, 146, 194, 208, 217, 229] [2, 7, 45, 124, 195, 214, 223, 230] [3, 8, 46, 120, 196, 210, 224, 231] [4, 9, 47, 121, 197, 211, 220, 232] [0, 5, 48, 122, 198, 212, 221, 233] [1, 6, 49, 123, 199, 213, 222, 234] [7, 12, 50, 129, 200, 219, 228, 235] [8, 13, 51, 125, 201, 215, 229, 236] [9, 14, 52, 126, 202, 216, 225, 237] [5, 10, 53, 127, 203, 217, 226, 238] [6, 11, 54, 128, 204, 218, 227, 239] [12, 17, 55, 134, 205, 224, 233, 240] [13, 18, 56, 130, 206, 220, 234, 241] [14, 19, 57, 131, 207, 221, 230, 242] [10, 15, 58, 132, 208, 222, 231, 243] [11, 16, 59, 133, 209, 223, 232, 244] [17, 22, 60, 139, 210, 229, 238, 245] [18, 23, 61, 135, 211, 225, 239, 246] [19, 24, 62, 136, 212, 226, 235, 247] [15, 20, 63, 137, 213, 227, 236, 248] [16, 21, 64, 138, 214, 228, 237, 249] [22, 27, 65, 144, 215, 234, 243, 250] [23, 28, 66, 140, 216, 230, 244, 251] [24, 29, 67, 141, 217, 231, 240, 252] [20, 25, 68, 142, 218, 232, 241, 253] [21, 26, 69, 143, 219, 233, 242, 254] [27, 32, 70, 149, 220, 239, 248, 255] [28, 33, 71, 145, 221, 235, 249, 256] [29, 34, 72, 146, 222, 236, 245, 257] [25, 30, 73, 147, 223, 237, 246, 258] [26, 31, 74, 148, 224, 238, 247, 259] [4, 32, 37, 75, 225, 244, 253, 260] [0, 33, 38, 76, 226, 240, 254, 261] [1, 34, 39, 77, 227, 241, 250, 262] [2, 30, 35, 78, 228, 242, 251, 263] [3, 31, 36, 79, 229, 243, 252, 264] [9, 37, 42, 80, 230, 249, 258, 265] [5, 38, 43, 81, 231, 245, 259, 266] [6, 39, 44, 82, 232, 246, 255, 267] [7, 35, 40, 83, 233, 247, 256, 268] [8, 36, 41, 84, 234, 248, 257, 269] [14, 42, 47, 85, 235, 254, 263, 270] [10, 43, 48, 86, 236, 250, 264, 271] [11, 44, 49, 87, 237, 251, 260, 272] [12, 40, 45, 88, 238, 252, 261, 273] [13, 41, 46, 89, 239, 253, 262, 274] [19, 47, 52, 90, 240, 259, 268, 275] [15, 48, 53, 91, 241, 255, 269, 276] [16, 49, 54, 92, 242, 256, 265, 277] [17, 45, 50, 93, 243, 257, 266, 278] [18, 46, 51, 94, 244, 258, 267, 279] [24, 52, 57, 95, 245, 264, 273, 280] [20, 53, 58, 96, 246, 260, 274, 281] [21, 54, 59, 97, 247, 261, 270, 282] [22, 50, 55, 98, 248, 262, 271, 283] [23, 51, 56, 99, 249, 263, 272, 284] [29, 57, 62, 100, 250, 269, 278, 285] [25, 58, 63, 101, 251, 265, 279, 286] [26, 59, 64, 102, 252, 266, 275, 287] [27, 55, 60, 103, 253, 267, 276, 288] [28, 56, 61, 104, 254, 268, 277, 289] [34, 62, 67, 105, 255, 274, 283, 290] [30, 63, 68, 106, 256, 270, 284, 291] [31, 64, 69, 107, 257, 271, 280, 292] [32, 60, 65, 108, 258, 272, 281, 293] [33, 61, 66, 109, 259, 273, 282, 294] [39, 67, 72, 110, 260, 279, 288, 295] [35, 68, 73, 111, 261, 275, 289, 296] [36, 69, 74, 112, 262, 276, 285, 297] [37, 65, 70, 113, 263, 277, 286, 298] [38, 66, 71, 114, 264, 278, 287, 299] [44, 72, 77, 115, 150, 265, 284, 293] [40, 73, 78, 116, 151, 266, 280, 294] [41, 74, 79, 117, 152, 267, 281, 290] [42, 70, 75, 118, 153, 268, 282, 291] [43, 71, 76, 119, 154, 269, 283, 292] [49, 77, 82, 120, 155, 270, 289, 298] [45, 78, 83, 121, 156, 271, 285, 299] [46, 79, 84, 122, 157, 272, 286, 295] [47, 75, 80, 123, 158, 273, 287, 296] [48, 76, 81, 124, 159, 274, 288, 297] [54, 82, 87, 125, 153, 160, 275, 294] [50, 83, 88, 126, 154, 161, 276, 290] [51, 84, 89, 127, 150, 162, 277, 291] [52, 80, 85, 128, 151, 163, 278, 292] [53, 81, 86, 129, 152, 164, 279, 293] [59, 87, 92, 130, 158, 165, 280, 299] [55, 88, 93, 131, 159, 166, 281, 295] [56, 89, 94, 132, 155, 167, 282, 296] [57, 85, 90, 133, 156, 168, 283, 297] [58, 86, 91, 134, 157, 169, 284, 298] [64, 92, 97, 135, 154, 163, 170, 285] [60, 93, 98, 136, 150, 164, 171, 286] [61, 94, 99, 137, 151, 160, 172, 287] [62, 90, 95, 138, 152, 161, 173, 288] [63, 91, 96, 139, 153, 162, 174, 289] [69, 97, 102, 140, 159, 168, 175, 290] [65, 98, 103, 141, 155, 169, 176, 291] [66, 99, 104, 142, 156, 165, 177, 292] [67, 95, 100, 143, 157, 166, 178, 293] [68, 96, 101, 144, 158, 167, 179, 294] [74, 102, 107, 145, 164, 173, 180, 295] [70, 103, 108, 146, 160, 174, 181, 296] [71, 104, 109, 147, 161, 170, 182, 297] [72, 100, 105, 148, 162, 171, 183, 298] [73, 101, 106, 149, 163, 172, 184, 299]
H_Z (150 checks, sparse supports)
[0, 115, 127, 136, 150, 193, 198, 226] [1, 116, 128, 137, 151, 194, 199, 227] [2, 117, 129, 138, 152, 190, 195, 228] [3, 118, 125, 139, 153, 191, 196, 229] [4, 119, 126, 135, 154, 192, 197, 225] [5, 120, 132, 141, 155, 198, 203, 231] [6, 121, 133, 142, 156, 199, 204, 232] [7, 122, 134, 143, 157, 195, 200, 233] [8, 123, 130, 144, 158, 196, 201, 234] [9, 124, 131, 140, 159, 197, 202, 230] [10, 125, 137, 146, 160, 203, 208, 236] [11, 126, 138, 147, 161, 204, 209, 237] [12, 127, 139, 148, 162, 200, 205, 238] [13, 128, 135, 149, 163, 201, 206, 239] [14, 129, 136, 145, 164, 202, 207, 235] [1, 15, 130, 142, 165, 208, 213, 241] [2, 16, 131, 143, 166, 209, 214, 242] [3, 17, 132, 144, 167, 205, 210, 243] [4, 18, 133, 140, 168, 206, 211, 244] [0, 19, 134, 141, 169, 207, 212, 240] [6, 20, 135, 147, 170, 213, 218, 246] [7, 21, 136, 148, 171, 214, 219, 247] [8, 22, 137, 149, 172, 210, 215, 248] [9, 23, 138, 145, 173, 211, 216, 249] [5, 24, 139, 146, 174, 212, 217, 245] [2, 11, 25, 140, 175, 218, 223, 251] [3, 12, 26, 141, 176, 219, 224, 252] [4, 13, 27, 142, 177, 215, 220, 253] [0, 14, 28, 143, 178, 216, 221, 254] [1, 10, 29, 144, 179, 217, 222, 250] [7, 16, 30, 145, 180, 223, 228, 256] [8, 17, 31, 146, 181, 224, 229, 257] [9, 18, 32, 147, 182, 220, 225, 258] [5, 19, 33, 148, 183, 221, 226, 259] [6, 15, 34, 149, 184, 222, 227, 255] [0, 12, 21, 35, 185, 228, 233, 261] [1, 13, 22, 36, 186, 229, 234, 262] [2, 14, 23, 37, 187, 225, 230, 263] [3, 10, 24, 38, 188, 226, 231, 264] [4, 11, 20, 39, 189, 227, 232, 260] [5, 17, 26, 40, 190, 233, 238, 266] [6, 18, 27, 41, 191, 234, 239, 267] [7, 19, 28, 42, 192, 230, 235, 268] [8, 15, 29, 43, 193, 231, 236, 269] [9, 16, 25, 44, 194, 232, 237, 265] [10, 22, 31, 45, 195, 238, 243, 271] [11, 23, 32, 46, 196, 239, 244, 272] [12, 24, 33, 47, 197, 235, 240, 273] [13, 20, 34, 48, 198, 236, 241, 274] [14, 21, 30, 49, 199, 237, 242, 270] [15, 27, 36, 50, 200, 243, 248, 276] [16, 28, 37, 51, 201, 244, 249, 277] [17, 29, 38, 52, 202, 240, 245, 278] [18, 25, 39, 53, 203, 241, 246, 279] [19, 26, 35, 54, 204, 242, 247, 275] [20, 32, 41, 55, 205, 248, 253, 281] [21, 33, 42, 56, 206, 249, 254, 282] [22, 34, 43, 57, 207, 245, 250, 283] [23, 30, 44, 58, 208, 246, 251, 284] [24, 31, 40, 59, 209, 247, 252, 280] [25, 37, 46, 60, 210, 253, 258, 286] [26, 38, 47, 61, 211, 254, 259, 287] [27, 39, 48, 62, 212, 250, 255, 288] [28, 35, 49, 63, 213, 251, 256, 289] [29, 36, 45, 64, 214, 252, 257, 285] [30, 42, 51, 65, 215, 258, 263, 291] [31, 43, 52, 66, 216, 259, 264, 292] [32, 44, 53, 67, 217, 255, 260, 293] [33, 40, 54, 68, 218, 256, 261, 294] [34, 41, 50, 69, 219, 257, 262, 290] [35, 47, 56, 70, 220, 263, 268, 296] [36, 48, 57, 71, 221, 264, 269, 297] [37, 49, 58, 72, 222, 260, 265, 298] [38, 45, 59, 73, 223, 261, 266, 299] [39, 46, 55, 74, 224, 262, 267, 295] [40, 52, 61, 75, 151, 225, 268, 273] [41, 53, 62, 76, 152, 226, 269, 274] [42, 54, 63, 77, 153, 227, 265, 270] [43, 50, 64, 78, 154, 228, 266, 271] [44, 51, 60, 79, 150, 229, 267, 272] [45, 57, 66, 80, 156, 230, 273, 278] [46, 58, 67, 81, 157, 231, 274, 279] [47, 59, 68, 82, 158, 232, 270, 275] [48, 55, 69, 83, 159, 233, 271, 276] [49, 56, 65, 84, 155, 234, 272, 277] [50, 62, 71, 85, 161, 235, 278, 283] [51, 63, 72, 86, 162, 236, 279, 284] [52, 64, 73, 87, 163, 237, 275, 280] [53, 60, 74, 88, 164, 238, 276, 281] [54, 61, 70, 89, 160, 239, 277, 282] [55, 67, 76, 90, 166, 240, 283, 288] [56, 68, 77, 91, 167, 241, 284, 289] [57, 69, 78, 92, 168, 242, 280, 285] [58, 65, 79, 93, 169, 243, 281, 286] [59, 66, 75, 94, 165, 244, 282, 287] [60, 72, 81, 95, 171, 245, 288, 293] [61, 73, 82, 96, 172, 246, 289, 294] [62, 74, 83, 97, 173, 247, 285, 290] [63, 70, 84, 98, 174, 248, 286, 291] [64, 71, 80, 99, 170, 249, 287, 292] [65, 77, 86, 100, 176, 250, 293, 298] [66, 78, 87, 101, 177, 251, 294, 299] [67, 79, 88, 102, 178, 252, 290, 295] [68, 75, 89, 103, 179, 253, 291, 296] [69, 76, 85, 104, 175, 254, 292, 297] [70, 82, 91, 105, 153, 181, 255, 298] [71, 83, 92, 106, 154, 182, 256, 299] [72, 84, 93, 107, 150, 183, 257, 295] [73, 80, 94, 108, 151, 184, 258, 296] [74, 81, 90, 109, 152, 180, 259, 297] [75, 87, 96, 110, 153, 158, 186, 260] [76, 88, 97, 111, 154, 159, 187, 261] [77, 89, 98, 112, 150, 155, 188, 262] [78, 85, 99, 113, 151, 156, 189, 263] [79, 86, 95, 114, 152, 157, 185, 264] [80, 92, 101, 115, 158, 163, 191, 265] [81, 93, 102, 116, 159, 164, 192, 266] [82, 94, 103, 117, 155, 160, 193, 267] [83, 90, 104, 118, 156, 161, 194, 268] [84, 91, 100, 119, 157, 162, 190, 269] [85, 97, 106, 120, 163, 168, 196, 270] [86, 98, 107, 121, 164, 169, 197, 271] [87, 99, 108, 122, 160, 165, 198, 272] [88, 95, 109, 123, 161, 166, 199, 273] [89, 96, 105, 124, 162, 167, 195, 274] [90, 102, 111, 125, 168, 173, 201, 275] [91, 103, 112, 126, 169, 174, 202, 276] [92, 104, 113, 127, 165, 170, 203, 277] [93, 100, 114, 128, 166, 171, 204, 278] [94, 101, 110, 129, 167, 172, 200, 279] [95, 107, 116, 130, 173, 178, 206, 280] [96, 108, 117, 131, 174, 179, 207, 281] [97, 109, 118, 132, 170, 175, 208, 282] [98, 105, 119, 133, 171, 176, 209, 283] [99, 106, 115, 134, 172, 177, 205, 284] [100, 112, 121, 135, 178, 183, 211, 285] [101, 113, 122, 136, 179, 184, 212, 286] [102, 114, 123, 137, 175, 180, 213, 287] [103, 110, 124, 138, 176, 181, 214, 288] [104, 111, 120, 139, 177, 182, 210, 289] [105, 117, 126, 140, 183, 188, 216, 290] [106, 118, 127, 141, 184, 189, 217, 291] [107, 119, 128, 142, 180, 185, 218, 292] [108, 115, 129, 143, 181, 186, 219, 293] [109, 116, 125, 144, 182, 187, 215, 294] [110, 122, 131, 145, 188, 193, 221, 295] [111, 123, 132, 146, 189, 194, 222, 296] [112, 124, 133, 147, 185, 190, 223, 297] [113, 120, 134, 148, 186, 191, 224, 298] [114, 121, 130, 149, 187, 192, 220, 299]
Code ID 300-18-24 · download JSON · raw on GitHub