[[150,16,8]] · QEC Challenge

[[150,16,8]] d ≤

150

kd²/n

6.827

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[9, 14, 19, 24, 49, 54, 69, 74]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[84, 89, 99, 104, 109, 119, 139, 149]

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)×150 (2,4)×1125 (3,3)×150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 150 (2,4): 1125 (3,3): 150 (3,5): 10800 (3,7): 1500

trapping sets H_Z (1,3)×150 (2,4)×1125 (3,3)×150 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 150 (2,4): 1125 (3,3): 150 (3,5): 10800 (3,7): 1500

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction BB on Z_5xZ_15, A={x⁰*y⁰, x⁰*y⁶, x⁰*y⁸}, B={x⁰*y⁵, x¹*y⁰, x⁴*y⁰}

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

[[150,16,8]] — Bivariate bicycle on Z_5 x Z_15

Direction & hypothesis

Targeted the weight-6 x unrestricted board. High-rate BB code from arXiv:2408.10001v6 (Wang & Mueller, Feb 2026) with k=16 at n=150.

What was searched

Reconstructed from the paper's polynomial definition:

l=5, m=15, A = 1 + y^6 + y^8, B = y^5 + x + x^4
Check weight: 6 (3 terms per polynomial)
n=150, k=16, d<=8

Evidence trail

RIS distance surrogate: d<=8 (both X and Z sides)
Witnesses confirmed by verifier's criteria

Dead ends

None significant.

Tools

Source: arXiv:2408.10001v6, reconstructed using research/kit/bb.py
Distance: RIS surrogate via research/kit/surrogate.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=5, m=15, A_terms=[(0,0),(0,6),(0,8)], B_terms=[(0,5),(1,0),(4,0)])

Parity checks

X-checks 75 (max weight 6) · Z-checks 75 (max weight 6)

H_X (75 checks, sparse supports)

[0, 6, 8, 80, 90, 135] [1, 7, 9, 81, 91, 136] [2, 8, 10, 82, 92, 137] [3, 9, 11, 83, 93, 138] [4, 10, 12, 84, 94, 139] [5, 11, 13, 85, 95, 140] [6, 12, 14, 86, 96, 141] [0, 7, 13, 87, 97, 142] [1, 8, 14, 88, 98, 143] [0, 2, 9, 89, 99, 144] [1, 3, 10, 75, 100, 145] [2, 4, 11, 76, 101, 146] [3, 5, 12, 77, 102, 147] [4, 6, 13, 78, 103, 148] [5, 7, 14, 79, 104, 149] [15, 21, 23, 75, 95, 105] [16, 22, 24, 76, 96, 106] [17, 23, 25, 77, 97, 107] [18, 24, 26, 78, 98, 108] [19, 25, 27, 79, 99, 109] [20, 26, 28, 80, 100, 110] [21, 27, 29, 81, 101, 111] [15, 22, 28, 82, 102, 112] [16, 23, 29, 83, 103, 113] [15, 17, 24, 84, 104, 114] [16, 18, 25, 85, 90, 115] [17, 19, 26, 86, 91, 116] [18, 20, 27, 87, 92, 117] [19, 21, 28, 88, 93, 118] [20, 22, 29, 89, 94, 119] [30, 36, 38, 90, 110, 120] [31, 37, 39, 91, 111, 121] [32, 38, 40, 92, 112, 122] [33, 39, 41, 93, 113, 123] [34, 40, 42, 94, 114, 124] [35, 41, 43, 95, 115, 125] [36, 42, 44, 96, 116, 126] [30, 37, 43, 97, 117, 127] [31, 38, 44, 98, 118, 128] [30, 32, 39, 99, 119, 129] [31, 33, 40, 100, 105, 130] [32, 34, 41, 101, 106, 131] [33, 35, 42, 102, 107, 132] [34, 36, 43, 103, 108, 133] [35, 37, 44, 104, 109, 134] [45, 51, 53, 105, 125, 135] [46, 52, 54, 106, 126, 136] [47, 53, 55, 107, 127, 137] [48, 54, 56, 108, 128, 138] [49, 55, 57, 109, 129, 139] [50, 56, 58, 110, 130, 140] [51, 57, 59, 111, 131, 141] [45, 52, 58, 112, 132, 142] [46, 53, 59, 113, 133, 143] [45, 47, 54, 114, 134, 144] [46, 48, 55, 115, 120, 145] [47, 49, 56, 116, 121, 146] [48, 50, 57, 117, 122, 147] [49, 51, 58, 118, 123, 148] [50, 52, 59, 119, 124, 149] [60, 66, 68, 75, 120, 140] [61, 67, 69, 76, 121, 141] [62, 68, 70, 77, 122, 142] [63, 69, 71, 78, 123, 143] [64, 70, 72, 79, 124, 144] [65, 71, 73, 80, 125, 145] [66, 72, 74, 81, 126, 146] [60, 67, 73, 82, 127, 147] [61, 68, 74, 83, 128, 148] [60, 62, 69, 84, 129, 149] [61, 63, 70, 85, 130, 135] [62, 64, 71, 86, 131, 136] [63, 65, 72, 87, 132, 137] [64, 66, 73, 88, 133, 138] [65, 67, 74, 89, 134, 139]

H_Z (75 checks, sparse supports)

[10, 15, 60, 75, 82, 84] [11, 16, 61, 76, 83, 85] [12, 17, 62, 77, 84, 86] [13, 18, 63, 78, 85, 87] [14, 19, 64, 79, 86, 88] [0, 20, 65, 80, 87, 89] [1, 21, 66, 75, 81, 88] [2, 22, 67, 76, 82, 89] [3, 23, 68, 75, 77, 83] [4, 24, 69, 76, 78, 84] [5, 25, 70, 77, 79, 85] [6, 26, 71, 78, 80, 86] [7, 27, 72, 79, 81, 87] [8, 28, 73, 80, 82, 88] [9, 29, 74, 81, 83, 89] [0, 25, 30, 90, 97, 99] [1, 26, 31, 91, 98, 100] [2, 27, 32, 92, 99, 101] [3, 28, 33, 93, 100, 102] [4, 29, 34, 94, 101, 103] [5, 15, 35, 95, 102, 104] [6, 16, 36, 90, 96, 103] [7, 17, 37, 91, 97, 104] [8, 18, 38, 90, 92, 98] [9, 19, 39, 91, 93, 99] [10, 20, 40, 92, 94, 100] [11, 21, 41, 93, 95, 101] [12, 22, 42, 94, 96, 102] [13, 23, 43, 95, 97, 103] [14, 24, 44, 96, 98, 104] [15, 40, 45, 105, 112, 114] [16, 41, 46, 106, 113, 115] [17, 42, 47, 107, 114, 116] [18, 43, 48, 108, 115, 117] [19, 44, 49, 109, 116, 118] [20, 30, 50, 110, 117, 119] [21, 31, 51, 105, 111, 118] [22, 32, 52, 106, 112, 119] [23, 33, 53, 105, 107, 113] [24, 34, 54, 106, 108, 114] [25, 35, 55, 107, 109, 115] [26, 36, 56, 108, 110, 116] [27, 37, 57, 109, 111, 117] [28, 38, 58, 110, 112, 118] [29, 39, 59, 111, 113, 119] [30, 55, 60, 120, 127, 129] [31, 56, 61, 121, 128, 130] [32, 57, 62, 122, 129, 131] [33, 58, 63, 123, 130, 132] [34, 59, 64, 124, 131, 133] [35, 45, 65, 125, 132, 134] [36, 46, 66, 120, 126, 133] [37, 47, 67, 121, 127, 134] [38, 48, 68, 120, 122, 128] [39, 49, 69, 121, 123, 129] [40, 50, 70, 122, 124, 130] [41, 51, 71, 123, 125, 131] [42, 52, 72, 124, 126, 132] [43, 53, 73, 125, 127, 133] [44, 54, 74, 126, 128, 134] [0, 45, 70, 135, 142, 144] [1, 46, 71, 136, 143, 145] [2, 47, 72, 137, 144, 146] [3, 48, 73, 138, 145, 147] [4, 49, 74, 139, 146, 148] [5, 50, 60, 140, 147, 149] [6, 51, 61, 135, 141, 148] [7, 52, 62, 136, 142, 149] [8, 53, 63, 135, 137, 143] [9, 54, 64, 136, 138, 144] [10, 55, 65, 137, 139, 145] [11, 56, 66, 138, 140, 146] [12, 57, 67, 139, 141, 147] [13, 58, 68, 140, 142, 148] [14, 59, 69, 141, 143, 149]

Code ID 150-16-8 · download JSON · raw on GitHub