← back to the board
[[170,32,14]] d ≤
n
170
k
32
d
14
kd²/n
36.894
w
10
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[14, 46, 50, 51, 54, 70, 78, 83, 108, 110, 128, 135, 140, 150]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[8, 40, 42, 48, 58, 61, 62, 68, 96, 125, 131, 149, 151, 163]
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 10 · H_Z 10
qubit degrees H_X 5 · H_Z 5
trapping sets H_X (1,5)×170 (2,6)×85 (3,7)×340 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 170 (2,6): 85 (2,8): 3655 (3,7): 340 (3,9): 12495 (3,11): 100895 (3,13): 8840
trapping sets H_Z (1,5)×170 (2,6)×85 (3,7)×340 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 170 (2,6): 85 (2,8): 3655 (3,7): 340 (3,9): 12495 (3,11): 100895 (3,13): 8840

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Free-action lifted product over Z_85 with the published A and B supports from arXiv:2608.08996v1.
model human (claimed, not verified)
builds on arXiv:2608.08996
date 2026-08-28
notes Reconstructed from the paper Supplementary Information; 20,000 pure-Python RIS trials per side at seed 20260828 found no lighter logical. Checked against the board before submission: exact_duplicate_of null, wl_equivalent_of null, dominated_by [] — not equivalent to any existing entry.
family lifted product (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

[[170,32,14]] — free-action lifted product over Z_85, reconstructed from arXiv:2608.08996v1

Direction & hypothesis

Cell: unrestricted x any weight. The verifier derives this from the matrices rather than from a self-declared label: there is no layout, so the locality class is unrestricted, and the max check weight is 10, which is past weight-8, so the only cell this entry joins is unrestricted x any weight.

The opening was structural rather than statistical. Against the 1440 entries the board held when this was staged, this is the only one with n <= 170, k >= 32 and d >= 14 at the same time. The next entries reaching that corner are codes/200-40-14.json (n=200, k=40, d=14, w=15) and codes/254-42-14.json (n=254, k=42, d=14, w=10). At n=170 the board already held codes/170-52-3.json (k=52, d=3), codes/170-34-8.json (k=34, d=8), codes/170-16-10.json (k=16, d=10), codes/170-8-20.json (k=8, d=20) and codes/170-2-13.json (k=2, d=13) — every point on the k-d plane at n=170 except the k >= 32, d >= 14 corner.

That corner could be filled by reconstruction instead of search: arXiv:2608.08996v1 publishes a free-action lifted product over Z_85 with exactly [[170,32,14]], and its Supplementary Information gives both supports.

What was searched

No parameter sweep — this is a reconstruction, not a discovery. The two supports over Z_85 were taken from the paper, the matrices rebuilt from them, structurally verified, and then put through the same board screen a searched candidate faces: an entry is only worth submitting if it is non-dominated on (n, k, d, w) in a cell it joins. This one came back non-dominated.

The counting is consistent throughout: |A| = |B| = 5, so every check has weight 10, n = 2 * 85 = 170, and the ranks of H_X and H_Z leave k = 32.

Evidence trail

  • Structure: verify/qldpc_verify.py recomputes n=170 and k=32 from the ranks

    • of the two matrices (85 X checks and 85 Z checks), max check weight 10, CSS commutation, and the locality and weight classes the track grid uses.

  • verify/validate_candidate.py returned passed: true on this candidate:

    • structural checks ok, refutation refuted: false (no lighter logical in 8000 RIS trials, seed 1161346672), exact_duplicate_of: null, wl_equivalent_of: null, board_advancing: true, dominated_by: [].

  • Distance: witnesses of weight 14 on both sides, so d <= 14 on each side.

    • This is a witness-backed upper bound, not a certificate. The paper reports d = 14, which agrees with the witnesses; the submission's own provenance records 20,000 pure-Python RIS trials per side at seed 20260828 finding nothing lighter.

  • Where it sits: non-dominated in unrestricted x any weight, the only cell it

    • joins. It dominates no entry and is dominated by none. kd^2/n = 36.894 makes it the highest-scoring entry on the board at n <= 170, ahead of codes/136-34-12.json at 36.000; by that same score it ranks 162 of 1440 overall, against a cell headline bar of 1456.855 held by codes/674-170-76-b.json.

  • What it does not do: it takes no existing point off the frontier. The nearest

    • entries trade an axis instead of losing one — codes/200-40-14.json has more k and more n, codes/254-42-14.json has more k at more n, codes/170-34-8.json has more k but only d=8 at w=9. The contribution is a new trade-off point in an empty corner, not a takeover.

Dead ends

  • Nothing was searched and failed on this path; the swept searches behind other

    • staged candidates are reported with those entries.

  • The honest limit is scope: no board entry at n <= 170 reaches k >= 32 and

    • d >= 14 together, so no entry could be dominated by this one. The frontier gains a point rather than losing one, which is what a reconstruction of a published code can offer.

Tools

Python 3, the repository's lifted-product construction helpers, the ./qldpc submission builder, and the trusted verification stack (verify/qldpc_verify.py, verify/validate_candidate.py). provenance.model is human: the matrices are a reconstruction of published data and no search produced them. Compute: seconds, no solver time.

Reproduction

Free-action 2BGA (lifted product) over Z_85 with A = {0, 3, 4, 10, 67} and B = {0, 5, 29, 31, 37}, from the Supplementary Information of arXiv:2608.08996v1.

Let C_S be the 85x85 circulant over GF(2) whose first row is the indicator of S, with indices read mod 85. The submitted matrices are

H_X = [ C_{-A} | C_{-B} ] H_Z = [ C_{B} | C_{A} ]

which is what the checked-in file contains: the first 85 columns of H_X carry support (0, 18, 75, 81, 82) = -A, the last 85 carry (0, 48, 54, 56, 80) = -B, and H_Z carries B then A in that order. Each row of checks.X and checks.Z is the 10-element support of one check. Rebuild the two matrices from those supports, then re-run verify/validate_candidate.py before trusting the distance.

Parity checks

X-checks 85 (max weight 10) · Z-checks 85 (max weight 10)
H_X (85 checks, sparse supports)
[0, 18, 75, 81, 82, 85, 133, 139, 141, 165] [1, 19, 76, 82, 83, 86, 134, 140, 142, 166] [2, 20, 77, 83, 84, 87, 135, 141, 143, 167] [0, 3, 21, 78, 84, 88, 136, 142, 144, 168] [0, 1, 4, 22, 79, 89, 137, 143, 145, 169] [1, 2, 5, 23, 80, 85, 90, 138, 144, 146] [2, 3, 6, 24, 81, 86, 91, 139, 145, 147] [3, 4, 7, 25, 82, 87, 92, 140, 146, 148] [4, 5, 8, 26, 83, 88, 93, 141, 147, 149] [5, 6, 9, 27, 84, 89, 94, 142, 148, 150] [0, 6, 7, 10, 28, 90, 95, 143, 149, 151] [1, 7, 8, 11, 29, 91, 96, 144, 150, 152] [2, 8, 9, 12, 30, 92, 97, 145, 151, 153] [3, 9, 10, 13, 31, 93, 98, 146, 152, 154] [4, 10, 11, 14, 32, 94, 99, 147, 153, 155] [5, 11, 12, 15, 33, 95, 100, 148, 154, 156] [6, 12, 13, 16, 34, 96, 101, 149, 155, 157] [7, 13, 14, 17, 35, 97, 102, 150, 156, 158] [8, 14, 15, 18, 36, 98, 103, 151, 157, 159] [9, 15, 16, 19, 37, 99, 104, 152, 158, 160] [10, 16, 17, 20, 38, 100, 105, 153, 159, 161] [11, 17, 18, 21, 39, 101, 106, 154, 160, 162] [12, 18, 19, 22, 40, 102, 107, 155, 161, 163] [13, 19, 20, 23, 41, 103, 108, 156, 162, 164] [14, 20, 21, 24, 42, 104, 109, 157, 163, 165] [15, 21, 22, 25, 43, 105, 110, 158, 164, 166] [16, 22, 23, 26, 44, 106, 111, 159, 165, 167] [17, 23, 24, 27, 45, 107, 112, 160, 166, 168] [18, 24, 25, 28, 46, 108, 113, 161, 167, 169] [19, 25, 26, 29, 47, 85, 109, 114, 162, 168] [20, 26, 27, 30, 48, 86, 110, 115, 163, 169] [21, 27, 28, 31, 49, 85, 87, 111, 116, 164] [22, 28, 29, 32, 50, 86, 88, 112, 117, 165] [23, 29, 30, 33, 51, 87, 89, 113, 118, 166] [24, 30, 31, 34, 52, 88, 90, 114, 119, 167] [25, 31, 32, 35, 53, 89, 91, 115, 120, 168] [26, 32, 33, 36, 54, 90, 92, 116, 121, 169] [27, 33, 34, 37, 55, 85, 91, 93, 117, 122] [28, 34, 35, 38, 56, 86, 92, 94, 118, 123] [29, 35, 36, 39, 57, 87, 93, 95, 119, 124] [30, 36, 37, 40, 58, 88, 94, 96, 120, 125] [31, 37, 38, 41, 59, 89, 95, 97, 121, 126] [32, 38, 39, 42, 60, 90, 96, 98, 122, 127] [33, 39, 40, 43, 61, 91, 97, 99, 123, 128] [34, 40, 41, 44, 62, 92, 98, 100, 124, 129] [35, 41, 42, 45, 63, 93, 99, 101, 125, 130] [36, 42, 43, 46, 64, 94, 100, 102, 126, 131] [37, 43, 44, 47, 65, 95, 101, 103, 127, 132] [38, 44, 45, 48, 66, 96, 102, 104, 128, 133] [39, 45, 46, 49, 67, 97, 103, 105, 129, 134] [40, 46, 47, 50, 68, 98, 104, 106, 130, 135] [41, 47, 48, 51, 69, 99, 105, 107, 131, 136] [42, 48, 49, 52, 70, 100, 106, 108, 132, 137] [43, 49, 50, 53, 71, 101, 107, 109, 133, 138] [44, 50, 51, 54, 72, 102, 108, 110, 134, 139] [45, 51, 52, 55, 73, 103, 109, 111, 135, 140] [46, 52, 53, 56, 74, 104, 110, 112, 136, 141] [47, 53, 54, 57, 75, 105, 111, 113, 137, 142] [48, 54, 55, 58, 76, 106, 112, 114, 138, 143] [49, 55, 56, 59, 77, 107, 113, 115, 139, 144] [50, 56, 57, 60, 78, 108, 114, 116, 140, 145] [51, 57, 58, 61, 79, 109, 115, 117, 141, 146] [52, 58, 59, 62, 80, 110, 116, 118, 142, 147] [53, 59, 60, 63, 81, 111, 117, 119, 143, 148] [54, 60, 61, 64, 82, 112, 118, 120, 144, 149] [55, 61, 62, 65, 83, 113, 119, 121, 145, 150] [56, 62, 63, 66, 84, 114, 120, 122, 146, 151] [0, 57, 63, 64, 67, 115, 121, 123, 147, 152] [1, 58, 64, 65, 68, 116, 122, 124, 148, 153] [2, 59, 65, 66, 69, 117, 123, 125, 149, 154] [3, 60, 66, 67, 70, 118, 124, 126, 150, 155] [4, 61, 67, 68, 71, 119, 125, 127, 151, 156] [5, 62, 68, 69, 72, 120, 126, 128, 152, 157] [6, 63, 69, 70, 73, 121, 127, 129, 153, 158] [7, 64, 70, 71, 74, 122, 128, 130, 154, 159] [8, 65, 71, 72, 75, 123, 129, 131, 155, 160] [9, 66, 72, 73, 76, 124, 130, 132, 156, 161] [10, 67, 73, 74, 77, 125, 131, 133, 157, 162] [11, 68, 74, 75, 78, 126, 132, 134, 158, 163] [12, 69, 75, 76, 79, 127, 133, 135, 159, 164] [13, 70, 76, 77, 80, 128, 134, 136, 160, 165] [14, 71, 77, 78, 81, 129, 135, 137, 161, 166] [15, 72, 78, 79, 82, 130, 136, 138, 162, 167] [16, 73, 79, 80, 83, 131, 137, 139, 163, 168] [17, 74, 80, 81, 84, 132, 138, 140, 164, 169]
H_Z (85 checks, sparse supports)
[0, 5, 29, 31, 37, 85, 88, 89, 95, 152] [1, 6, 30, 32, 38, 86, 89, 90, 96, 153] [2, 7, 31, 33, 39, 87, 90, 91, 97, 154] [3, 8, 32, 34, 40, 88, 91, 92, 98, 155] [4, 9, 33, 35, 41, 89, 92, 93, 99, 156] [5, 10, 34, 36, 42, 90, 93, 94, 100, 157] [6, 11, 35, 37, 43, 91, 94, 95, 101, 158] [7, 12, 36, 38, 44, 92, 95, 96, 102, 159] [8, 13, 37, 39, 45, 93, 96, 97, 103, 160] [9, 14, 38, 40, 46, 94, 97, 98, 104, 161] [10, 15, 39, 41, 47, 95, 98, 99, 105, 162] [11, 16, 40, 42, 48, 96, 99, 100, 106, 163] [12, 17, 41, 43, 49, 97, 100, 101, 107, 164] [13, 18, 42, 44, 50, 98, 101, 102, 108, 165] [14, 19, 43, 45, 51, 99, 102, 103, 109, 166] [15, 20, 44, 46, 52, 100, 103, 104, 110, 167] [16, 21, 45, 47, 53, 101, 104, 105, 111, 168] [17, 22, 46, 48, 54, 102, 105, 106, 112, 169] [18, 23, 47, 49, 55, 85, 103, 106, 107, 113] [19, 24, 48, 50, 56, 86, 104, 107, 108, 114] [20, 25, 49, 51, 57, 87, 105, 108, 109, 115] [21, 26, 50, 52, 58, 88, 106, 109, 110, 116] [22, 27, 51, 53, 59, 89, 107, 110, 111, 117] [23, 28, 52, 54, 60, 90, 108, 111, 112, 118] [24, 29, 53, 55, 61, 91, 109, 112, 113, 119] [25, 30, 54, 56, 62, 92, 110, 113, 114, 120] [26, 31, 55, 57, 63, 93, 111, 114, 115, 121] [27, 32, 56, 58, 64, 94, 112, 115, 116, 122] [28, 33, 57, 59, 65, 95, 113, 116, 117, 123] [29, 34, 58, 60, 66, 96, 114, 117, 118, 124] [30, 35, 59, 61, 67, 97, 115, 118, 119, 125] [31, 36, 60, 62, 68, 98, 116, 119, 120, 126] [32, 37, 61, 63, 69, 99, 117, 120, 121, 127] [33, 38, 62, 64, 70, 100, 118, 121, 122, 128] [34, 39, 63, 65, 71, 101, 119, 122, 123, 129] [35, 40, 64, 66, 72, 102, 120, 123, 124, 130] [36, 41, 65, 67, 73, 103, 121, 124, 125, 131] [37, 42, 66, 68, 74, 104, 122, 125, 126, 132] [38, 43, 67, 69, 75, 105, 123, 126, 127, 133] [39, 44, 68, 70, 76, 106, 124, 127, 128, 134] [40, 45, 69, 71, 77, 107, 125, 128, 129, 135] [41, 46, 70, 72, 78, 108, 126, 129, 130, 136] [42, 47, 71, 73, 79, 109, 127, 130, 131, 137] [43, 48, 72, 74, 80, 110, 128, 131, 132, 138] [44, 49, 73, 75, 81, 111, 129, 132, 133, 139] [45, 50, 74, 76, 82, 112, 130, 133, 134, 140] [46, 51, 75, 77, 83, 113, 131, 134, 135, 141] [47, 52, 76, 78, 84, 114, 132, 135, 136, 142] [0, 48, 53, 77, 79, 115, 133, 136, 137, 143] [1, 49, 54, 78, 80, 116, 134, 137, 138, 144] [2, 50, 55, 79, 81, 117, 135, 138, 139, 145] [3, 51, 56, 80, 82, 118, 136, 139, 140, 146] [4, 52, 57, 81, 83, 119, 137, 140, 141, 147] [5, 53, 58, 82, 84, 120, 138, 141, 142, 148] [0, 6, 54, 59, 83, 121, 139, 142, 143, 149] [1, 7, 55, 60, 84, 122, 140, 143, 144, 150] [0, 2, 8, 56, 61, 123, 141, 144, 145, 151] [1, 3, 9, 57, 62, 124, 142, 145, 146, 152] [2, 4, 10, 58, 63, 125, 143, 146, 147, 153] [3, 5, 11, 59, 64, 126, 144, 147, 148, 154] [4, 6, 12, 60, 65, 127, 145, 148, 149, 155] [5, 7, 13, 61, 66, 128, 146, 149, 150, 156] [6, 8, 14, 62, 67, 129, 147, 150, 151, 157] [7, 9, 15, 63, 68, 130, 148, 151, 152, 158] [8, 10, 16, 64, 69, 131, 149, 152, 153, 159] [9, 11, 17, 65, 70, 132, 150, 153, 154, 160] [10, 12, 18, 66, 71, 133, 151, 154, 155, 161] [11, 13, 19, 67, 72, 134, 152, 155, 156, 162] [12, 14, 20, 68, 73, 135, 153, 156, 157, 163] [13, 15, 21, 69, 74, 136, 154, 157, 158, 164] [14, 16, 22, 70, 75, 137, 155, 158, 159, 165] [15, 17, 23, 71, 76, 138, 156, 159, 160, 166] [16, 18, 24, 72, 77, 139, 157, 160, 161, 167] [17, 19, 25, 73, 78, 140, 158, 161, 162, 168] [18, 20, 26, 74, 79, 141, 159, 162, 163, 169] [19, 21, 27, 75, 80, 85, 142, 160, 163, 164] [20, 22, 28, 76, 81, 86, 143, 161, 164, 165] [21, 23, 29, 77, 82, 87, 144, 162, 165, 166] [22, 24, 30, 78, 83, 88, 145, 163, 166, 167] [23, 25, 31, 79, 84, 89, 146, 164, 167, 168] [0, 24, 26, 32, 80, 90, 147, 165, 168, 169] [1, 25, 27, 33, 81, 85, 91, 148, 166, 169] [2, 26, 28, 34, 82, 85, 86, 92, 149, 167] [3, 27, 29, 35, 83, 86, 87, 93, 150, 168] [4, 28, 30, 36, 84, 87, 88, 94, 151, 169]
Code ID 170-32-14 · download JSON · raw on GitHub