[[144,8,15]] · QEC Challenge

[[144,8,15]] d ≤

144

kd²/n

12.5

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)

[3, 11, 17, 38, 57, 61, 85, 102, 107, 116, 122, 123, 125, 128, 138]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[8, 18, 20, 26, 39, 41, 43, 44, 61, 84, 85, 89, 90, 106, 129]

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): 3888 (3,8): 30672 (3,10): 3168

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): 3888 (3,8): 30672 (3,10): 3168

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_18 x Z_4 with weight-4 generators A = 1 + x y² + x⁹ y³ + x¹⁴ y, B = x⁸ y² + x y + x¹² + x¹² y¹ (weight-8 checks). Found by a random weight-8 gap sweep over grids not searched by arXiv:2609.06572, targeting the [[144,8,>=13]] gap of the weight-8 cell frontier.

model Omen Alpha 1.0 (claimed, not verified)

date 2026-09-09

notes New parameters (submitter claim): arXiv:2609.06572 swept only the (12,6) and (8,9) grids; this is from the unsearched (18,4) grid. Distance: RIS upper bound d<=15 (20k trials), corroborated by BP+OSD at 200k trials/side (weight-15 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

[[144,8,15]] — weight-8 bivariate bicycle code on Z_18 x Z_4, from the weight-8 gap sweep

Direction & hypothesis

Target cell: weight-8 × unrestricted. Following the construction family of arXiv:2609.06572 (BB-type codes with weight-4 generator polynomials, i.e. weight-8 checks), the sweep targeted the frontier gaps that paper left open: grids it never searched ((18,4), (24,3), (36,2) at n=144), asymmetric generator weights (3+5, 2+6 — still weight-8 checks), and specific (k,d) gaps computed against the board's weight-8-cell Pareto frontier, in particular [[144,8,>=13]] between [[144,6,15]] and [[144,10,12]].

What was searched

13,200 constant-term-normalized random pairs (the normalization is 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 kept), then 169 shortlisted records deep-screened at 20,000 RIS trials. The sweep was a single throwaway script (random pair sampler feeding the kit's screen); its full record lists are local working output and are not committed — the method is fully described here and the winner is rebuilt from the polynomials below. 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 <= 15 (stage 1).
20,000-trial RIS (stage 2): d <= 15 held.
200,000-trial BP+OSD syndrome-decoder search (independent mechanism):

lightest logical found is weight 15 on both X and Z, nothing lighter.

Validation gate (verify/validate_candidate.py): passed, 8,000-trial

fresh-seed RIS refutation found no lighter logical; dedup found no exact or WL-equivalent board entry; computed board-advancing in the weight-8 x unrestricted cell.

Claim precisely: witness-backed upper bound d <= 15. Two independent search mechanisms agree at 15 with nothing lighter; exact certification (MILP) was not run, so the d= tier is not claimed.

Dead ends

The four n=72 configurations (6,6), (9,4), (12,3), (18,2) produced no

deep-screen survivors within this budget; the n=72 targets ([[72,10,>=11]], [[72,12,>=9]]) stay open.

The 2+6 asymmetric-weight configs produced no non-dominated records.
Of 117 non-dominated deep-screen records, all sit at (k,d) buckets below

the gap targets except the four [[144,8,15]] finds; dominated candidates were not staged.

Tools

Model: Omen Alpha 1.0 (opencode agent), search harness written and run by the model. Repo tooling: research/kit/bb.py (build_bb), research/kit/css.py (k recomputation), 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. Local machine, single node, ~40 min of screening.

Reproduction

Rebuild (H_X, H_Z) with the kit on the torus Z_18 x Z_4 (n = 2*18*4 = 144):

from bb import build_bb
HX, HZ = build_bb(18, 4,
                  A_terms=[(0,0),(1,2),(9,3),(14,1)],
                  B_terms=[(8,2),(11,1),(12,0),(12,1)])

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)

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

H_Z (72 checks, sparse supports)

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

Code ID 144-8-15 · download JSON · raw on GitHub