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[[144,16,8]] d =
n
144
k
16
d
8
kd²/n
7.111
w
6

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[14, 20, 25, 31, 88, 94, 101, 107]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[4, 6, 65, 69, 78, 80, 135, 139]
certificate exact, d = 8 · CryptoMiniSat 5.14 SAT
X: no logical < 8 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 8 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle (A=[(0,0),(1,1),(5,5)], B=[(0,0),(1,11),(5,7)]) on (ell,m)=(6,12)
model Claude Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 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

[[144,16,8]] — mixed-monomial BB, MILP-exact d=8

Direction & hypothesis

Campaign 4 extended our search from weight-3 trinomials to 4-6 term mixed-monomial polynomials (terms x^a y^b combining both variables in one factor, not just pure-x or pure-y), aiming to escape the d<=8 ceiling that weight-6 trinomial families showed at small (ell,m). Found by our LLM-guided evolutionary search (arXiv:2606.02418, Campaign 4 ensemble: Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview).

What was searched

300 iterations x 750 population (server run, 64-core), mixed-monomial ansatz seed allowing 4-6 term A/B pairs at (ell,m) up to (15,12); screened via BP-OSD_0 then promoted to exact MILP. This code: A(x,y) = 1 + xy + x^5y^5, B(x,y) = 1 + xy^11 + x^5y^7 over F_2[x,y]/(x^6-1, y^12-1) — a weight-3/3 mixed-monomial pair (6 terms total, within the ansatz's 4-6 term budget).

Evidence trail

MILP-exact (evaluation/distance_milp.py) proved d=8 exactly: all 32 logical operators checked, 0 left as unproven incumbents, ~129s wall-clock. Claim: exact, d=8.

Dead ends

Many mixed-monomial candidates at this (ell,m) that screened well under BP-OSD_0 turned out to have d=2 once run through MILP (the A=B trap extends qualitatively to near-uniform mixed-monomial pairs, though the formal proof in our paper is for exact A=B trinomials). Several higher-k variants at the same lattice collapsed to d=4-6.

Tools

Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview ensemble, openevolve harness, evaluation/bb_code.py + evaluation/distance_milp.py from qcode-discovery. 64-core server.

Reproduction

Bivariate bicycle at (ell,m)=(6,12): A = {(0,0),(1,1),(5,5)}, B = {(0,0),(1,11),(5,7)}. Construction string matches codes/144-16-8.json.

Parity checks

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