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[[144,16,10]] d =
n
144
k
16
d
10
kd²/n
11.111
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[7, 8, 20, 37, 38, 50, 67, 68, 118, 125]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[62, 69, 72, 77, 95, 102, 107, 114, 119, 137]
certificate exact, d = 10 · CryptoMiniSat 5.14.7 SAT
X: no logical < 10 exists; Z: no logical < 10 exists

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)×144 (2,4)×144 (3,4)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 144 (2,4): 144 (2,6): 1728 (3,4): 288 (3,6): 4320 (3,8): 28656 (3,10): 2592
trapping sets H_Z (1,4)×144 (2,4)×144 (3,4)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 144 (2,4): 144 (2,6): 1728 (3,4): 288 (3,6): 4320 (3,8): 28656 (3,10): 2592

Construction & provenance

authors Liangdong Lu and Ruipan Yang and Guanmin Guo
provenance literature baseline
construction Bivariate bicycle code QC(A,B) on Z_12 x Z_6 with weight-4 generators A = 1 + x11 + y5 + x9 y5, B = x3 y2 + x8 y2 + x4 y3 + x9 y3 (weight-8 checks). Reproduction of Table 1 entry 1 of arXiv:2609.06572 (Lu, Yang, Guo).
model classical construction (no AI model)
reference arXiv:2609.06572
date 2026-09-06
notes Seeded from the published census of arXiv:2609.06572, Table 1 entry 1, where d=10 is established exactly by exhaustive bit-mask verification cross-validated against Magma (their Remark 4.4). Check matrices rebuilt independently with this repo's kit; the witnesses here were found fresh by this repo's RIS surrogate (upper bound d<=10). Reproduced by @MathysRennela.
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

[[144,16,10]] — weight-8 bivariate bicycle code, reproduction of arXiv:2609.06572 Table 1 entry 1

Direction & hypothesis

Target cell: weight-8 × unrestricted. The board's weight-8 cell at n = 144 was thin: the incumbent at this (n, k) was [[144,16,8]], so a same-size code one point higher in distance was a concrete frontier opening. The source is the published census of Lu, Yang and Guo (arXiv:2609.06572), which develops the structure theory of BB-type codes with weight-4 generator polynomials (weight-8 checks) and reports this code with an exactly verified distance of 10. This entry is a faithful reproduction of their Table 1, entry 1, not a new search: the parameters exist in the literature, and the contribution here is an independently rebuilt, witness-backed board entry.

What was searched

No search. The generator polynomials were transcribed from the paper's Table 1 and rebuilt with the repo kit: QC(A, B) on Z_12 x Z_6 with

  • A = 1 + x^11 + y^5 + x^9 y^5
  • B = x^3 y^2 + x^8 y^2 + x^4 y^3 + x^9 y^3

(exponents written x^a y^b). The rebuild reproduced the paper's parameters exactly before packaging: n = 144, k = 16 (recomputed by the verifier's own rank arithmetic, not trusted from the paper), max check weight 8.

Evidence trail

  • Source paper: the exact distance d = 10 was computed by their bit-mask DFS

    • verifier, cross-validated against Magma's Words enumeration (their Remark 4.4), and their census reproduces the BB benchmark d = 12 as a calibration check.

  • This rebuild: the kit's RIS surrogate found weight-10 logicals on both

    • sides, so the staged claim is a witness-backed upper bound d <= 10. The validation gate (verify/validate_candidate.py) ran an 8000-trial fresh-seed RIS refutation and found no lighter logical; the gate reports the code as board-advancing in the weight-8 x unrestricted cell.

  • Claim precisely: upper_bound, d <= 10. The paper's exhaustive result says

    • the true distance is exactly 10; per board policy that d= tier belongs to the server certifier, not to this submission.

Dead ends

Not applicable to a reproduction. One adjacent finding from evaluating the paper's full Table 1 against the board: most of its weight-8 members ([[72,14,8]], [[144,10,12]], [[144,14,10]], [[144,20,8]], [[144,18,8]], [[144,8,12]]) are dominated by existing weight-8-cell entries here — e.g. [[64,18,8]] and the weight-6 BB benchmark [[144,12,12]] — so only this code and the paper's [[144,6,d>=15]] advance this board.

Tools

Human-authored source paper (Lu, Yang, Guo, arXiv:2609.06572). Reproduction harness: this repository's kit — research/kit/bb.py (build_bb), research/kit/css.py (k recomputation), research/kit/submit.py and the CLI ./qldpc submit (witness search, verification), verify/validate_candidate.py (gate). No AI model produced the code; provenance.model is human.

Reproduction

Rebuild (H_X, H_Z) with the kit on the torus Z_12 x Z_6 (n = 2*12*6 = 144):

from bb import build_bb
HX, HZ = build_bb(12, 6,
                  A_terms=[(0,0),(11,0),(0,5),(9,5)],
                  B_terms=[(3,2),(8,2),(4,3),(9,3)])

Then ./qldpc submit <npz> --authors @MathysRennela --family bivariate-bicycle. The construction string in the JSON carries the same polynomials. Source: arXiv:2609.06572 (https://arxiv.org/abs/2609.06572), Table 1, entry 1.

Parity checks

X-checks 72 (max weight 8) · Z-checks 72 (max weight 8)
H_X (72 checks, sparse supports)
[0, 5, 59, 66, 92, 99, 122, 129] [0, 1, 54, 67, 93, 100, 123, 130] [1, 2, 55, 68, 94, 101, 124, 131] [2, 3, 56, 69, 95, 96, 125, 126] [3, 4, 57, 70, 90, 97, 120, 127] [4, 5, 58, 71, 91, 98, 121, 128] [0, 6, 11, 65, 98, 105, 128, 135] [1, 6, 7, 60, 99, 106, 129, 136] [2, 7, 8, 61, 100, 107, 130, 137] [3, 8, 9, 62, 101, 102, 131, 132] [4, 9, 10, 63, 96, 103, 126, 133] [5, 10, 11, 64, 97, 104, 127, 134] [6, 12, 17, 71, 104, 111, 134, 141] [7, 12, 13, 66, 105, 112, 135, 142] [8, 13, 14, 67, 106, 113, 136, 143] [9, 14, 15, 68, 107, 108, 137, 138] [10, 15, 16, 69, 102, 109, 132, 139] [11, 16, 17, 70, 103, 110, 133, 140] [5, 12, 18, 23, 75, 110, 117, 140] [0, 13, 18, 19, 76, 111, 118, 141] [1, 14, 19, 20, 77, 112, 119, 142] [2, 15, 20, 21, 72, 113, 114, 143] [3, 16, 21, 22, 73, 108, 115, 138] [4, 17, 22, 23, 74, 109, 116, 139] [11, 18, 24, 29, 74, 81, 116, 123] [6, 19, 24, 25, 75, 82, 117, 124] [7, 20, 25, 26, 76, 83, 118, 125] [8, 21, 26, 27, 77, 78, 119, 120] [9, 22, 27, 28, 72, 79, 114, 121] [10, 23, 28, 29, 73, 80, 115, 122] [17, 24, 30, 35, 80, 87, 122, 129] [12, 25, 30, 31, 81, 88, 123, 130] [13, 26, 31, 32, 82, 89, 124, 131] [14, 27, 32, 33, 83, 84, 125, 126] [15, 28, 33, 34, 78, 85, 120, 127] [16, 29, 34, 35, 79, 86, 121, 128] [23, 30, 36, 41, 86, 93, 128, 135] [18, 31, 36, 37, 87, 94, 129, 136] [19, 32, 37, 38, 88, 95, 130, 137] [20, 33, 38, 39, 89, 90, 131, 132] [21, 34, 39, 40, 84, 91, 126, 133] [22, 35, 40, 41, 85, 92, 127, 134] [29, 36, 42, 47, 92, 99, 134, 141] [24, 37, 42, 43, 93, 100, 135, 142] [25, 38, 43, 44, 94, 101, 136, 143] [26, 39, 44, 45, 95, 96, 137, 138] [27, 40, 45, 46, 90, 97, 132, 139] [28, 41, 46, 47, 91, 98, 133, 140] [35, 42, 48, 53, 75, 98, 105, 140] [30, 43, 48, 49, 76, 99, 106, 141] [31, 44, 49, 50, 77, 100, 107, 142] [32, 45, 50, 51, 72, 101, 102, 143] [33, 46, 51, 52, 73, 96, 103, 138] [34, 47, 52, 53, 74, 97, 104, 139] [41, 48, 54, 59, 74, 81, 104, 111] [36, 49, 54, 55, 75, 82, 105, 112] [37, 50, 55, 56, 76, 83, 106, 113] [38, 51, 56, 57, 77, 78, 107, 108] [39, 52, 57, 58, 72, 79, 102, 109] [40, 53, 58, 59, 73, 80, 103, 110] [47, 54, 60, 65, 80, 87, 110, 117] [42, 55, 60, 61, 81, 88, 111, 118] [43, 56, 61, 62, 82, 89, 112, 119] [44, 57, 62, 63, 83, 84, 113, 114] [45, 58, 63, 64, 78, 85, 108, 115] [46, 59, 64, 65, 79, 86, 109, 116] [53, 60, 66, 71, 86, 93, 116, 123] [48, 61, 66, 67, 87, 94, 117, 124] [49, 62, 67, 68, 88, 95, 118, 125] [50, 63, 68, 69, 89, 90, 119, 120] [51, 64, 69, 70, 84, 91, 114, 121] [52, 65, 70, 71, 85, 92, 115, 122]
H_Z (72 checks, sparse supports)
[21, 28, 51, 58, 72, 73, 78, 91] [22, 29, 52, 59, 73, 74, 79, 92] [23, 24, 53, 54, 74, 75, 80, 93] [18, 25, 48, 55, 75, 76, 81, 94] [19, 26, 49, 56, 76, 77, 82, 95] [20, 27, 50, 57, 72, 77, 83, 90] [27, 34, 57, 64, 78, 79, 84, 97] [28, 35, 58, 65, 79, 80, 85, 98] [29, 30, 59, 60, 80, 81, 86, 99] [24, 31, 54, 61, 81, 82, 87, 100] [25, 32, 55, 62, 82, 83, 88, 101] [26, 33, 56, 63, 78, 83, 89, 96] [33, 40, 63, 70, 84, 85, 90, 103] [34, 41, 64, 71, 85, 86, 91, 104] [35, 36, 65, 66, 86, 87, 92, 105] [30, 37, 60, 67, 87, 88, 93, 106] [31, 38, 61, 68, 88, 89, 94, 107] [32, 39, 62, 69, 84, 89, 95, 102] [4, 39, 46, 69, 90, 91, 96, 109] [5, 40, 47, 70, 91, 92, 97, 110] [0, 41, 42, 71, 92, 93, 98, 111] [1, 36, 43, 66, 93, 94, 99, 112] [2, 37, 44, 67, 94, 95, 100, 113] [3, 38, 45, 68, 90, 95, 101, 108] [3, 10, 45, 52, 96, 97, 102, 115] [4, 11, 46, 53, 97, 98, 103, 116] [5, 6, 47, 48, 98, 99, 104, 117] [0, 7, 42, 49, 99, 100, 105, 118] [1, 8, 43, 50, 100, 101, 106, 119] [2, 9, 44, 51, 96, 101, 107, 114] [9, 16, 51, 58, 102, 103, 108, 121] [10, 17, 52, 59, 103, 104, 109, 122] [11, 12, 53, 54, 104, 105, 110, 123] [6, 13, 48, 55, 105, 106, 111, 124] [7, 14, 49, 56, 106, 107, 112, 125] [8, 15, 50, 57, 102, 107, 113, 120] [15, 22, 57, 64, 108, 109, 114, 127] [16, 23, 58, 65, 109, 110, 115, 128] [17, 18, 59, 60, 110, 111, 116, 129] [12, 19, 54, 61, 111, 112, 117, 130] [13, 20, 55, 62, 112, 113, 118, 131] [14, 21, 56, 63, 108, 113, 119, 126] [21, 28, 63, 70, 114, 115, 120, 133] [22, 29, 64, 71, 115, 116, 121, 134] [23, 24, 65, 66, 116, 117, 122, 135] [18, 25, 60, 67, 117, 118, 123, 136] [19, 26, 61, 68, 118, 119, 124, 137] [20, 27, 62, 69, 114, 119, 125, 132] [4, 27, 34, 69, 120, 121, 126, 139] [5, 28, 35, 70, 121, 122, 127, 140] [0, 29, 30, 71, 122, 123, 128, 141] [1, 24, 31, 66, 123, 124, 129, 142] [2, 25, 32, 67, 124, 125, 130, 143] [3, 26, 33, 68, 120, 125, 131, 138] [3, 10, 33, 40, 73, 126, 127, 132] [4, 11, 34, 41, 74, 127, 128, 133] [5, 6, 35, 36, 75, 128, 129, 134] [0, 7, 30, 37, 76, 129, 130, 135] [1, 8, 31, 38, 77, 130, 131, 136] [2, 9, 32, 39, 72, 126, 131, 137] [9, 16, 39, 46, 79, 132, 133, 138] [10, 17, 40, 47, 80, 133, 134, 139] [11, 12, 41, 42, 81, 134, 135, 140] [6, 13, 36, 43, 82, 135, 136, 141] [7, 14, 37, 44, 83, 136, 137, 142] [8, 15, 38, 45, 78, 132, 137, 143] [15, 22, 45, 52, 72, 85, 138, 139] [16, 23, 46, 53, 73, 86, 139, 140] [17, 18, 47, 48, 74, 87, 140, 141] [12, 19, 42, 49, 75, 88, 141, 142] [13, 20, 43, 50, 76, 89, 142, 143] [14, 21, 44, 51, 77, 84, 138, 143]
Code ID 144-16-10 · download JSON · raw on GitHub