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[[288,74,4]] d ≤
n
288
k
74
d
4
kd²/n
4.111
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[40, 78, 163, 184]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[104, 125, 210, 248]
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 4 · H_Z 4 (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 (2,4)×1152 (3,4)×2880 (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 (2,4): 1152 (2,6): 1728 (3,4): 2880 (3,6): 17280 (3,8): 19584
trapping sets H_Z (1,4)×288 (2,4)×1152 (3,4)×2880 (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 (2,4): 1152 (2,6): 1728 (3,4): 2880 (3,6): 17280 (3,8): 19584

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
date 2026-09-21
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

[[288,74,4]] — weight-8 bivariate bicycle code on Z_12 x Z_12, high-rate corner

Direction & hypothesis

Target cell: weight-8 x unrestricted locality. The board's weight-8 cell at n=288 was dominated by moderate-rate codes (k <= 32 in the nearby entries); the high-rate corner (k >= 74 at n=288) was empty for w <= 8. The qcode-discovery catalog (arXiv:2606.02418) publishes a large weight-8/10 mixed-monomial bivariate-bicycle sweep whose high-k entries looked board-advancing on paper, so the campaign screened the whole catalog against the board's Pareto frontiers rather than searching from scratch.

What was searched

Source: github.com/qiskit-community/qcode-discovery @ main, results/campaign4_milp_verified.jsonl (sha256 763f0e57f313b6d54dd2232a2c7e5bb3de08bb5f1e8757844715fc41b90408d9), 353 weight-8/10 mixed-monomial BB entries. Screening pipeline:

1. Deduplicate against the board and within the catalog: 379 unique candidates. 2. Optimistic record test (board-relative Pareto semantics mirroring the site builder: an entry is dominated iff some board entry beats it on all of n<=, k>=, d>=, w<= with one strict; entries compete in all nested weight cells 4 | 6 | 8 | 9plus): 36 passed. 3. Trusted gate per survivor: rebuild H_X/H_Z from the exponent pairs, witness search, then the repo verifier. 34 of 36 failed (see dead ends).

This code: BB on Z_12 x Z_12, A = [(0,0), (3,2), (3,3), (6,5)], B = [(0,0), (3,3), (10,3), (1,6)] (exponent pairs (x,y)), check weight 8, k = 74. Catalog status: MILP incumbent d=4 (not exact).

Evidence trail

  • Screening stage: optimistic d=4 passed the record test for cells

    • unrestricted/weight-8 and unrestricted/weight-9plus.

  • Gate stage: witness search at 8000 RIS trials/side found weight-4 logicals

    • on both sides (X support [31, 69, 154, 175], Z support [23, 50, 89, 128]), matching the catalog's d=4.

  • Submission verification (qldpc submit, fresh seed): 20000 RIS

    • trials/side plus a 2,000,000-trial accelerator pass; d <= 4 (d_X <= 4, d_Z <= 4), both witnessed. Verifier OK.

Final claim: d <= 4, witness-backed upper bound on both sides. Not certified exact; at this weight an exact certificate is cheap for a future run of verify/certify.py.

Dead ends

34 of the 36 screen-passers were refuted by the trusted gate, all invalid: verifier rejected — the catalog's MILP-incumbent distances did not survive independent witness search at the submitted parameters. Notable collapses: [[288,32,8]] w6, [[360,24,10]] w6, [[144,16,8]] w6 (each had passed the optimistic screen on catalog distances). Lesson: MILP incumbents from the catalog are optimistic; treat them as upper bounds that a fresh search can only worsen, never as achievable.

Tools

Agent harness: Zed coding agent, model GLM 5.3 Flash. Repo tooling: research.kit.bb.build_bb for construction, the kit's witness search and verify/validate_candidate.py for the gate, cli/qldpc.py submit for assembly and verification. Screening was a single-session campaign; no GPU.

Reproduction

Deterministic rebuild (no search involved):

from research.kit.bb import build_bb
HX, HZ = build_bb(12, 12,
                  A_terms=[(0,0),(3,2),(3,3),(6,5)],
                  B_terms=[(0,0),(3,3),(10,3),(1,6)])

Novelty: parameter set not found in an arXiv metadata search ("288,74": 0 hits) and not present on this board; source is the catalog's own fresh computational sweep. Self-reported as new_parameters.

Parity checks

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