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[[144,24,6]] d ≤
n
144
k
24
d
6
kd²/n
6.0
w
6

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Distance

d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[9, 21, 42, 57, 66, 69]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[77, 86, 113, 125, 134, 137]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle (A=[(0,1),(0,2),(6,0)], B=[(0,3),(2,0),(4,0)]) on (ell,m)=(12,6)
model Claude Claude Opus 4.6 + GPT-5.2 + Gemini 3 Pro Preview (claimed, not verified)
date 2026-08-05
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[144,24,6]] — x/y-swap trinomial BB, MILP-exact d=6, high rate (k/n=1/6)

Direction & hypothesis

Same x/y-swap trinomial family as [notes/360-16-14.md](360-16-14.md), targeting the high-rate end at (ell,m)=(12,6). Found by our LLM-guided evolutionary search (arXiv:2606.02418, ensemble Campaign 2-3: Claude Opus 4.6 + GPT-5.2 + Gemini 3 Pro Preview).

What was searched

Same population-based sweep as 360-16-14 (500 iterations x 1,000 population, weight-3 trinomial pairs across (ell,m) grid). This code: A(x,y) = y + y^2 + x^6, B(x,y) = y^3 + x^2 + x^4 over F_2[x,y]/(x^12-1, y^6-1).

Evidence trail

BP-OSD_0 across independent 5,000-trial batches gave estimates ranging from d<=8 to d<=12 — this run-to-run spread on this code is one of the examples we report in our paper illustrating why we treat any single BP-OSD batch as a screening signal rather than a final answer. MILP-exact (evaluation/distance_milp.py) proved d=6 exactly, all 24 logical operators verified to proven optimality, ~22s wall-clock. Claim: exact, d=6.

Dead ends

Higher-k trinomials at (12,6) and neighboring lattices collapsed to d=2 or 4 once run through MILP even when BP-OSD reported d>=8; this code was one of a handful in the (ell,m)=(12,6) family that held at d=6 under exact verification. The rate-distance envelope in our full catalog shows k>24 at this n forces d<=4 for indecomposable codes.

Tools

Claude Opus 4.6 + GPT-5.2 + Gemini 3 Pro Preview ensemble, openevolve harness, evaluation/bb_code.py + evaluation/distance_milp.py from qcode-discovery. Apple M4 Max.

Reproduction

Bivariate bicycle at (ell,m)=(12,6): A = {(0,1),(0,2),(6,0)}, B = {(0,3),(2,0),(4,0)}. Construction string matches codes/144-24-6.json.

Parity checks

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