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[[240,20,19]] d ≤
n
240
k
20
d
19
kd²/n
30.083
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)
witness operator (support, 19 qubits)
[15, 20, 32, 53, 61, 63, 79, 83, 104, 107, 140, 145, 152, 178, 180, 199, 228, 235, 236]
d_Z 19 · witness weight 19 (claimed upper_bound)
witness operator (support, 19 qubits)
[10, 23, 48, 49, 51, 57, 64, 73, 80, 81, 101, 105, 107, 143, 163, 169, 200, 220, 226]
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)×240 (2,6)×3360 (3,6)×2160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 240 (2,6): 3360 (3,6): 2160 (3,8): 64080 (3,10): 6720
trapping sets H_Z (1,4)×240 (2,6)×3360 (3,6)×2160 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 240 (2,6): 3360 (3,6): 2160 (3,8): 64080 (3,10): 6720

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_12 x Z_10: A = 1 + x3y9 + x9y6 + x10y5, B = 1 + x2y + x3y8 + x10; H_X = [A | B], H_Z = [B^T | A^T]
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-10-17
notes Not equivalent to 240-20-18.json despite the shared Weisfeiler-Leman signature: at a matched budget (research/audits/leader_audit.py pair, 10000000 gf2_fast trials per seed, seeds 4242,5353, pair depth 8) the peer reads d <= 18 and this code d <= 19; equivalent codes have equal distance, so a weight-18 logical would exist here too. The pair audit's decision was 'credible: both claims held at matched depth'.
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

[[240,20,19]] — weight-8 bivariate bicycle code on Z_12 x Z_10

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 >= 9 and n from 221 to 240 ran through 224-24-16, 232-62-12, 240-10-22, 240-18-20, 240-22-18, 240-24-17, 240-28-16. The hypothesis was that random 4 + 4-term bivariate bicycle (BB) codes on tori with 2lm in 240..340, prefiltered for k >= 16, still reach points there that the board does not hold.

[[240,20,19]] does: no board entry has n <= 240, k >= 20, d >= 19 and check weight <= 8. Against [[240,20,18]] (same n, k and weight) the gain is in d only; it also beats other entries, so the gate does not classify the overall gain as d-only. The pair was re-measured at matched depth (see Evidence trail).

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 240 <= 2lm <= 340; the torus is drawn uniformly from that list and the other monomials uniformly from the torus.

This sweep (seed 623) drew 500,010 codes in a parallel generate-and-rank prefilter; 2,265 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; 15 advanced the board frontier at screen depth and 15 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 <= 19, not an exact distance.

| stage | trials | lightest logical | |---|---|---| | screen (gf2_fast) | 400 | 19 | | confirm | 20,000 | 19 | | confirm | 200,000 | 19 | | confirm | 2,000,000 | 19 | | deep ladder, 3 fresh seeds (68057, 75976, 83895), gf2_fast | 20,000,000 each | 19, 19, 19 |

Matched-depth pair audit (research/audits/leader_audit.py pair, 10,000,000 gf2_fast trials per seed, seeds 4242, 5353, pair depth 8): this code reads d <= 19, 240-20-18 reads d <= 18; decision "credible: both claims held at matched depth".

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 14 of 15 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 8.3 CPU-hours on this code and its sweep.

Reproduction

Torus Z_12 x Z_10 (x of order 12, y of order 10), terms (i, j) = x^i y^j:

A = 1 + x^3 y^9 + x^9 y^6 + x^10 y^5, B = 1 + x^2 y + x^3 y^8 + x^10.

import sys; sys.path[:0] = ["research/kit", "verify"]
from bb import build_bb
from css import compute_k
HX, HZ = build_bb(12, 10, [(0, 0), (3, 9), (9, 6), (10, 5)],
                  [(0, 0), (2, 1), (3, 8), (10, 0)])   # n = 240
assert compute_k(HX, HZ) == 20

Ladder: surrogate.distance_rand_witness(HX, HZ, trials=20_000_000, seed=<s>, backend="fast", pair_depth=8) with s in {68057, 75976, 83895} returns logicals of weight 19, 19, 19.

Parity checks

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