[[144,6,15]] · QEC Challenge

[[144,6,15]] d ≤

144

kd²/n

9.375

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[15, 21, 43, 50, 55, 65, 79, 81, 82, 83, 87, 107, 108, 121, 122]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[0, 10, 14, 15, 19, 45, 46, 49, 53, 74, 77, 86, 93, 121, 143]

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)×144 (2,4)×72 (3,4)×144 (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): 72 (2,6): 1872 (3,4): 144 (3,6): 3264 (3,8): 32400 (3,10): 3240

trapping sets H_Z (1,4)×144 (2,4)×72 (3,4)×144 (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): 72 (2,6): 1872 (3,4): 144 (3,6): 3264 (3,8): 32400 (3,10): 3240

Construction & provenance

authors Liangdong Lu and Ruipan Yang and Guanmin Guo

provenance literature baseline

construction Bivariate bicycle code QC(A,B) on Z_8 x Z_9 with weight-4 generators A = 1 + x⁴ y² + y⁴ + y⁷, B = 1 + x² y⁶ + x⁵ y⁷ + x⁷ y⁸ (weight-8 checks). Reproduction of Table 1 entry 9 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 9, where d>=15 is certified by exhaustive bit-mask verification (their Sections 4.2-4.3, cross-validated against Magma). Check matrices rebuilt independently with this repo's kit; the witness here was found fresh by this repo's RIS surrogate (upper bound d<=15), which together with the paper's certified lower bound pins the true distance at 15 pending server certification. 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

[[144,6,15]] — weight-8 bivariate bicycle code, reproduction of arXiv:2609.06572 Table 1 entry 9

Direction & hypothesis

Target cell: weight-8 × unrestricted. The paper's headline code: a weight-8 BB-type code whose distance strictly exceeds the BB benchmark's 12 at the same length n = 144 — the board's weight-8 cell had no entry at this (n, k) with d >= 15 (all smaller-n codes with d >= 15 carry checks of weight 12-16, so they live outside this cell). Lu, Yang and Guo report it in Table 1, entry 9, with a certified lower bound d >= 15. This entry is a reproduction of that published code, not a new search.

What was searched

No search. Generator polynomials transcribed from the paper's Table 1 and rebuilt with the repo kit: QC(A, B) on Z_8 x Z_9 with

A = 1 + x^4 y^2 + y^4 + y^7
B = 1 + x^2 y^6 + x^5 y^7 + x^7 y^8

(exponents written x^a y^b). The rebuild reproduced the paper's parameters: n = 144, k = 6 (recomputed by the verifier's rank arithmetic), max check weight 8. Notably the paper's subgroup-coset design rule (their Theorem 3.13) explains the high distance: neither generator is divisible by (1+x) or (1+y), so none of the cheap weight-l or weight-m coset logicals exist.

Evidence trail

Source paper: their verifier exhaustively scanned all kernel vectors of

weight <= 14 and found no logical (certified d >= 15); the scan is translation-anchored for completeness (their Theorem 4.5) and cross-validated against Magma (their Remark 4.4).

This rebuild: the kit's RIS surrogate found a weight-15 logical on the X

side and weight-15 on the Z side, so the witness-backed claim is d <= 15. The validation gate (verify/validate_candidate.py) ran an 8000-trial fresh-seed RIS refutation: no lighter logical found; the gate reports the code as board-advancing in the weight-8 x unrestricted cell.

Claim precisely: upper_bound, d <= 15. Source lower bound d >= 15

(exhaustive, published) plus this witness pins the true distance at 15, but the d= tier on this board requires the server certifier, so the entry ships as an upper bound.

Dead ends

Not applicable to a reproduction. As with the paper's other Table 1 entries, most of its weight-8 census is dominated by existing board entries in this cell; this code and [[144,16,10]] (submitted separately, one code per PR) are the two that advance the board.

Tools

Human-authored source paper (Lu, Yang, Guo, arXiv:2609.06572); authorship of the code belongs to them — this entry is seeded as a baseline reproduction. Reproduction harness (by @MathysRennela): 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).

Reproduction

Rebuild (H_X, H_Z) with the kit on the torus Z_8 x Z_9 (n = 2*8*9 = 144):

from bb import build_bb
HX, HZ = build_bb(8, 9,
                  A_terms=[(0,0),(4,2),(0,4),(0,7)],
                  B_terms=[(0,0),(2,6),(5,7),(7,8)])

Then ./qldpc submit <npz> --authors "Liangdong Lu" "Ruipan Yang" "Guanmin Guo" --anonymous --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 9.

Parity checks

X-checks 72 (max weight 8) · Z-checks 72 (max weight 8)

H_X (72 checks, sparse supports)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-6-15 · download JSON · raw on GitHub