← back to the board
[[224,18,16]] d ≤
n
224
k
18
d
16
kd²/n
20.571
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[2, 15, 25, 31, 35, 66, 114, 136, 137, 140, 165, 197, 200, 210, 211, 214]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[5, 8, 9, 35, 47, 50, 51, 57, 60, 92, 114, 118, 120, 134, 147, 195]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×112 (2,4)×336 (3,3)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 112 (1,4): 112 (2,4): 336 (2,5): 1344 (2,6): 672 (3,3): 112 (3,5): 1008 (3,6): 11872 (3,7): 17584 (3,8): 6048 (3,9): 2016 (3,10): 448
trapping sets H_Z (1,3)×112 (2,4)×336 (3,3)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 112 (1,4): 112 (2,4): 336 (2,5): 1344 (2,6): 672 (3,3): 112 (3,5): 1008 (3,6): 11872 (3,7): 17584 (3,8): 6048 (3,9): 2016 (3,10): 448

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle on Z_28 x Z_4: A = 1 + xy3 + x17y2, B = 1 + x10y + x18y2 + x23; 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 224-18-15.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 <= 15 and this code d <= 16; equivalent codes have equal distance, so a weight-15 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

[[224,18,16]] — weight-7 bivariate bicycle code on Z_28 x Z_4

Direction & hypothesis

Cell weight-8 x unrestricted (maximum check weight 7), CSS board, n between 221 and 400. Before this code, the frontier (board plus this campaign's open PRs) at check weight <= 7, d >= 8 and n from 221 to 224 ran through 224-12-19, 224-18-15, 224-24-10. The hypothesis was that random 3 + 4-term bivariate bicycle (BB) codes on tori with 2lm in 222..300, prefiltered for k >= 12, still reach points there that the board does not hold.

[[224,18,16]] does: no board entry has n <= 224, k >= 18, d >= 16 and check weight <= 7. Against [[224,18,15]] (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). Its maximum check weight is 7 (3 + 4 terms): weight-8 entries may reach these parameters, but no entry of check weight <= 7 does.

What was searched

Random-support BB codes H_X = [A | B], H_Z = [B^T | A^T] with A of 3 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 702) drew 400,020 codes in a parallel generate-and-rank prefilter; 797 had k >= 12 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; 2 advanced the board frontier at screen depth and 2 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 <= 16, not an exact distance.

| stage | trials | lightest logical | |---|---|---| | screen (gf2_fast) | 400 | 16 | | confirm | 20,000 | 16 | | confirm | 200,000 | 16 | | confirm | 2,000,000 | 16 | | deep ladder, 3 fresh seeds (65350, 73269, 81188), gf2_fast | 20,000,000 each | 16, 16, 16 |

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 <= 16, 224-18-15 reads d <= 15; 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 1 of 2 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 4.8 CPU-hours on this code and its sweep.

Reproduction

Torus Z_28 x Z_4 (x of order 28, y of order 4), terms (i, j) = x^i y^j:

A = 1 + x y^3 + x^17 y^2, B = 1 + x^10 y + x^18 y^2 + x^23.

import sys; sys.path[:0] = ["research/kit", "verify"]
from bb import build_bb
from css import compute_k
HX, HZ = build_bb(28, 4, [(0, 0), (1, 3), (17, 2)],
                  [(0, 0), (10, 1), (18, 2), (23, 0)])   # n = 224
assert compute_k(HX, HZ) == 18

Ladder: surrogate.distance_rand_witness(HX, HZ, trials=20_000_000, seed=<s>, backend="fast", pair_depth=8) with s in {65350, 73269, 81188} returns logicals of weight 16, 16, 16.

Parity checks

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