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[[144,10,16]] d ≤
n
144
k
10
d
16
kd²/n
17.778
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[10, 16, 43, 52, 53, 58, 64, 66, 73, 104, 110, 117, 128, 132, 133, 141]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[4, 28, 41, 48, 64, 68, 73, 80, 84, 94, 95, 105, 124, 129, 134, 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)×144 (3,4)×216 (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): 216 (3,6): 4656 (3,8): 28656 (3,10): 2376
trapping sets H_Z (1,4)×144 (2,4)×144 (3,4)×216 (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): 216 (3,6): 4656 (3,8): 28656 (3,10): 2376

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty new parameter set claimed by submitter
construction Bivariate bicycle code QC(A,B) on Z_12 x Z_6 with weight-4 generators A = x y2 + x2 + x4 y + x5 y2, B = x5 + x5 y5 + x6 y4 + x9 y5 (weight-8 checks). Found by a weight-8 gap sweep over the (k,d) gaps of the weight-8 cell frontier and grids not searched by arXiv:2609.06572.
model Omen Alpha 1.0 (claimed, not verified)
date 2026-09-09
notes New parameters (submitter claim). Distance: RIS upper bound d<=16 (20k trials), corroborated by BP+OSD at 200k trials/side (weight-16 logicals, nothing lighter); exact certification not run.
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

[[144,10,16]] — weight-8 bivariate bicycle code on Z_12 x Z_6, from the weight-8 gap sweep

Direction & hypothesis

Target cell: weight-8 × unrestricted. The sweep followed the construction family of arXiv:2609.06572 (weight-4 generator polynomials, weight-8 checks) but targeted the (k,d) gaps of the board's weight-8-cell Pareto frontier and grids/weight profiles that paper did not search. This candidate aims at the k = 10 slot: the board's frontier point there was [[144,10,12]], so any non-dominated claim needs d >= 13.

What was searched

13,200 constant-term-normalized random pairs (normalization complete per arXiv:2609.06572 Cor. 3.9) across 13 grid/weight configurations (~59.5k samples after degenerate rejection), screened at 300 RIS trials (400 records), 169 shortlisted records deep-screened at 20,000 RIS trials. This code was one of 51 non-dominated survivors. The sweep was a single throwaway script feeding the kit's screen; its record lists are local working output, not committed — the method described here plus the polynomials below is the full recipe. Seeds: numpy default_rng(20260909), screen seeds 11 and 23.

Evidence trail

Confirmation ladder for this code (per side unless noted):

  • 300-trial RIS screen: upper bound d <= 16.
  • 20,000-trial RIS (deep stage): d <= 16 held.
  • 200,000-trial BP+OSD syndrome-decoder search (independent mechanism):

    • lightest logical found is weight 16 on both X and Z, nothing lighter.

  • Validation gate (verify/validate_candidate.py): passed; fresh-seed RIS

    • refutation found no lighter logical; no exact or WL-equivalent board entry.

Claim precisely: witness-backed upper bound, d <= 16. Two independent mechanisms agree at 16; exact certification was not run, so the d= tier is not claimed.

Dead ends

  • The sibling co-leader at [[144,10,16]] on the Z_36 x Z_2 grid

    • (A = x^8 + x^16 y + x^31 y + x^32 y, B = 1 + x^5 y + x^20 + x^27) passed the same confirmation and is held locally as a backup; only one of the two equal-parameter codes is submitted.

  • The five [[144,14,12]] survivors of the same sweep are dominated by this

    • campaign's [[144,16,12]] (higher k, equal d) and were not submitted.

  • The n=72 targets ([[72,10,>=11]], [[72,12,>=9]]) produced no survivors

    • within this budget and stay open.

Tools

Model: Omen Alpha 1.0 (opencode agent). Repo tooling: research/kit/bb.py (build_bb), research/kit/css.py (k), research/kit/search.py (screen, gf2_fast backend), research/kit/distance.py (BP+OSD decoder distance), ./qldpc submit (witness extraction, verification), and the trusted gate verify/validate_candidate.py. Single machine; screening ~40 min plus ~3 min decoder confirmation per code.

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=[(1,2),(2,0),(4,1),(5,2)],
                  B_terms=[(5,0),(5,5),(6,4),(9,5)])

Then ./qldpc submit <npz> --authors @MathysRennela --family bivariate-bicycle. The construction string in the JSON carries the same polynomials.

Parity checks

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