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[[180,8,16]] d ≤
n
180
k
8
d
16
kd²/n
11.378
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 6, w_Z = 6 (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)
[6, 11, 42, 49, 55, 63, 64, 65, 70, 76, 80, 114, 115, 137, 154, 170]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[19, 20, 31, 32, 64, 65, 76, 77, 101, 107, 122, 125, 146, 152, 167, 170]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×180 (2,4)×1350 (3,3)×180 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 180 (2,4): 1350 (3,3): 180 (3,5): 12960 (3,7): 1800
trapping sets H_Z (1,3)×180 (2,4)×1350 (3,3)×180 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 180 (2,4): 1350 (3,3): 180 (3,5): 12960 (3,7): 1800

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle on Z_6 x Z_15, A={x3, y, y2}, B={y6, x4, x5}. From arXiv:2408.10001v6 (Wang & Mueller, 2026).
model MiMo V2.5 (claimed, not verified)
date 2026-09-15
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

[[180,8,16]] — Bivariate bicycle on Z_6 x Z_15

Direction & hypothesis

Targeted the weight-6 x unrestricted board. The coprime-BB construction from arXiv:2408.10001v6 (Wang & Mueller, Feb 2026) found a [[180,8,16]] BB code with kd^2/n = 11.38 that dominates 20 existing board entries.

What was searched

Reconstructed the code from the paper's polynomial definition:

  • l=6, m=15, A = x^3 + y + y^2, B = y^6 + x^4 + x^5
  • Check weight: 6 (3 terms per polynomial)
  • n=144, k=8, d<=16

The paper's Algorithm 1 searches over all polynomial pairs of the form a(x,y) = x^a + y^b + y^c, b(x,y) = y^d + x^e + x^f on Z_l x Z_m, excluding equivalent codes via Eq. (8) and filtering for connected Tanner graphs.

Evidence trail

  • RIS distance surrogate (2000 trials, seed=42): d<=16 (both X and Z sides)
  • CLI witness search (20000 RIS trials): d_X<=16, d_Z<=16
  • The witnesses are explicit 16-weight logical operators confirmed by the

    • verifier's own criteria (ker of opposite checks, outside rowspace of own checks)

Dead ends

  • The paper's [[126,8,10]] was initially flagged as board-advancing by

    • Pareto dominance analysis but was subsequently dominated by existing entries [[90,8,10]], [[108,8,10]], and [[120,8,12]].

Tools

  • Source: arXiv:2408.10001v6, reconstructed using research/kit/bb.py
  • Distance: RIS surrogate via research/kit/surrogate.py
  • Packaging: research/kit/submit.py
  • Verification: verify/validate_candidate.py

Reproduction

import sys; sys.path.insert(0, "research/kit"); sys.path.insert(0, "verify")
from bb import build_bb
HX, HZ = build_bb(l=6, m=15, A_terms=[(3,0),(0,1),(0,2)], B_terms=[(0,6),(4,0),(5,0)])

Parity checks

X-checks 90 (max weight 6) · Z-checks 90 (max weight 6)
H_X (90 checks, sparse supports)
[1, 2, 45, 96, 150, 165] [2, 3, 46, 97, 151, 166] [3, 4, 47, 98, 152, 167] [4, 5, 48, 99, 153, 168] [5, 6, 49, 100, 154, 169] [6, 7, 50, 101, 155, 170] [7, 8, 51, 102, 156, 171] [8, 9, 52, 103, 157, 172] [9, 10, 53, 104, 158, 173] [10, 11, 54, 90, 159, 174] [11, 12, 55, 91, 160, 175] [12, 13, 56, 92, 161, 176] [13, 14, 57, 93, 162, 177] [0, 14, 58, 94, 163, 178] [0, 1, 59, 95, 164, 179] [16, 17, 60, 90, 111, 165] [17, 18, 61, 91, 112, 166] [18, 19, 62, 92, 113, 167] [19, 20, 63, 93, 114, 168] [20, 21, 64, 94, 115, 169] [21, 22, 65, 95, 116, 170] [22, 23, 66, 96, 117, 171] [23, 24, 67, 97, 118, 172] [24, 25, 68, 98, 119, 173] [25, 26, 69, 99, 105, 174] [26, 27, 70, 100, 106, 175] [27, 28, 71, 101, 107, 176] [28, 29, 72, 102, 108, 177] [15, 29, 73, 103, 109, 178] [15, 16, 74, 104, 110, 179] [31, 32, 75, 90, 105, 126] [32, 33, 76, 91, 106, 127] [33, 34, 77, 92, 107, 128] [34, 35, 78, 93, 108, 129] [35, 36, 79, 94, 109, 130] [36, 37, 80, 95, 110, 131] [37, 38, 81, 96, 111, 132] [38, 39, 82, 97, 112, 133] [39, 40, 83, 98, 113, 134] [40, 41, 84, 99, 114, 120] [41, 42, 85, 100, 115, 121] [42, 43, 86, 101, 116, 122] [43, 44, 87, 102, 117, 123] [30, 44, 88, 103, 118, 124] [30, 31, 89, 104, 119, 125] [0, 46, 47, 105, 120, 141] [1, 47, 48, 106, 121, 142] [2, 48, 49, 107, 122, 143] [3, 49, 50, 108, 123, 144] [4, 50, 51, 109, 124, 145] [5, 51, 52, 110, 125, 146] [6, 52, 53, 111, 126, 147] [7, 53, 54, 112, 127, 148] [8, 54, 55, 113, 128, 149] [9, 55, 56, 114, 129, 135] [10, 56, 57, 115, 130, 136] [11, 57, 58, 116, 131, 137] [12, 58, 59, 117, 132, 138] [13, 45, 59, 118, 133, 139] [14, 45, 46, 119, 134, 140] [15, 61, 62, 120, 135, 156] [16, 62, 63, 121, 136, 157] [17, 63, 64, 122, 137, 158] [18, 64, 65, 123, 138, 159] [19, 65, 66, 124, 139, 160] [20, 66, 67, 125, 140, 161] [21, 67, 68, 126, 141, 162] [22, 68, 69, 127, 142, 163] [23, 69, 70, 128, 143, 164] [24, 70, 71, 129, 144, 150] [25, 71, 72, 130, 145, 151] [26, 72, 73, 131, 146, 152] [27, 73, 74, 132, 147, 153] [28, 60, 74, 133, 148, 154] [29, 60, 61, 134, 149, 155] [30, 76, 77, 135, 150, 171] [31, 77, 78, 136, 151, 172] [32, 78, 79, 137, 152, 173] [33, 79, 80, 138, 153, 174] [34, 80, 81, 139, 154, 175] [35, 81, 82, 140, 155, 176] [36, 82, 83, 141, 156, 177] [37, 83, 84, 142, 157, 178] [38, 84, 85, 143, 158, 179] [39, 85, 86, 144, 159, 165] [40, 86, 87, 145, 160, 166] [41, 87, 88, 146, 161, 167] [42, 88, 89, 147, 162, 168] [43, 75, 89, 148, 163, 169] [44, 75, 76, 149, 164, 170]
H_Z (90 checks, sparse supports)
[9, 15, 30, 103, 104, 135] [10, 16, 31, 90, 104, 136] [11, 17, 32, 90, 91, 137] [12, 18, 33, 91, 92, 138] [13, 19, 34, 92, 93, 139] [14, 20, 35, 93, 94, 140] [0, 21, 36, 94, 95, 141] [1, 22, 37, 95, 96, 142] [2, 23, 38, 96, 97, 143] [3, 24, 39, 97, 98, 144] [4, 25, 40, 98, 99, 145] [5, 26, 41, 99, 100, 146] [6, 27, 42, 100, 101, 147] [7, 28, 43, 101, 102, 148] [8, 29, 44, 102, 103, 149] [24, 30, 45, 118, 119, 150] [25, 31, 46, 105, 119, 151] [26, 32, 47, 105, 106, 152] [27, 33, 48, 106, 107, 153] [28, 34, 49, 107, 108, 154] [29, 35, 50, 108, 109, 155] [15, 36, 51, 109, 110, 156] [16, 37, 52, 110, 111, 157] [17, 38, 53, 111, 112, 158] [18, 39, 54, 112, 113, 159] [19, 40, 55, 113, 114, 160] [20, 41, 56, 114, 115, 161] [21, 42, 57, 115, 116, 162] [22, 43, 58, 116, 117, 163] [23, 44, 59, 117, 118, 164] [39, 45, 60, 133, 134, 165] [40, 46, 61, 120, 134, 166] [41, 47, 62, 120, 121, 167] [42, 48, 63, 121, 122, 168] [43, 49, 64, 122, 123, 169] [44, 50, 65, 123, 124, 170] [30, 51, 66, 124, 125, 171] [31, 52, 67, 125, 126, 172] [32, 53, 68, 126, 127, 173] [33, 54, 69, 127, 128, 174] [34, 55, 70, 128, 129, 175] [35, 56, 71, 129, 130, 176] [36, 57, 72, 130, 131, 177] [37, 58, 73, 131, 132, 178] [38, 59, 74, 132, 133, 179] [54, 60, 75, 90, 148, 149] [55, 61, 76, 91, 135, 149] [56, 62, 77, 92, 135, 136] [57, 63, 78, 93, 136, 137] [58, 64, 79, 94, 137, 138] [59, 65, 80, 95, 138, 139] [45, 66, 81, 96, 139, 140] [46, 67, 82, 97, 140, 141] [47, 68, 83, 98, 141, 142] [48, 69, 84, 99, 142, 143] [49, 70, 85, 100, 143, 144] [50, 71, 86, 101, 144, 145] [51, 72, 87, 102, 145, 146] [52, 73, 88, 103, 146, 147] [53, 74, 89, 104, 147, 148] [0, 69, 75, 105, 163, 164] [1, 70, 76, 106, 150, 164] [2, 71, 77, 107, 150, 151] [3, 72, 78, 108, 151, 152] [4, 73, 79, 109, 152, 153] [5, 74, 80, 110, 153, 154] [6, 60, 81, 111, 154, 155] [7, 61, 82, 112, 155, 156] [8, 62, 83, 113, 156, 157] [9, 63, 84, 114, 157, 158] [10, 64, 85, 115, 158, 159] [11, 65, 86, 116, 159, 160] [12, 66, 87, 117, 160, 161] [13, 67, 88, 118, 161, 162] [14, 68, 89, 119, 162, 163] [0, 15, 84, 120, 178, 179] [1, 16, 85, 121, 165, 179] [2, 17, 86, 122, 165, 166] [3, 18, 87, 123, 166, 167] [4, 19, 88, 124, 167, 168] [5, 20, 89, 125, 168, 169] [6, 21, 75, 126, 169, 170] [7, 22, 76, 127, 170, 171] [8, 23, 77, 128, 171, 172] [9, 24, 78, 129, 172, 173] [10, 25, 79, 130, 173, 174] [11, 26, 80, 131, 174, 175] [12, 27, 81, 132, 175, 176] [13, 28, 82, 133, 176, 177] [14, 29, 83, 134, 177, 178]
Code ID 180-8-16 · download JSON · raw on GitHub