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[[128,13,10]] d ≤
n
128
k
13
d
10
kd²/n
10.156
w
8

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Distance

d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[0, 9, 25, 28, 32, 40, 65, 74, 110, 119]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[8, 30, 34, 54, 58, 59, 72, 94, 98, 117]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA on nonabelian group C8:Q8 (order 64). Weight-4 supports a=[2,6,17,23], b=[6,16,37,59] give row weight 8. Odd k=13 unreachable by abelian BB. Found by systematic 2BGA sweep over nonabelian groups of order 60-200, filtering for odd k, screened at 400 RIS trials. Second-best efficiency kd²/n=10.156.
model Xiaomi Mimo-V2.5 (claimed, not verified)
date 2026-07-30
family generalized 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

[[128,13,10]] — 2BGA on C₈:Q₈ (odd k)

Direction & hypothesis

Same strategy as [[128,21,8]]: nonabelian 2BGA with odd k to reach Pareto slots abelian BB cannot. This code trades k for d — k=13 (vs 21) but d=10 (vs 8), yielding a comparable efficiency of 10.156.

What was searched

Identical sweep to [[128,21,8]] — systematic 2BGA over nonabelian groups of order 60–200, weight-4 supports, odd-k filter, 400-trial screening. This code emerged from the same search run.

  • Total candidates: ~140k
  • Screened at: 400 RIS trials
  • Group: C₈:Q₈ (order 64, GAP index 182)

Evidence trail

| Trial depth | d found | eff = kd²/n | |-------------|---------|-------------| | 400 (screen) | 10 | 10.156 | | 4,000 (submission) | 10 | 10.156 |

Distance held at 4,000 trials. Witness: X-logical weight 10, Z-logical weight 10.

Dead ends

Same as [[128,21,8]] — Phases 1, 2, 4 killed. This code was the second-best find from Phase 3, discovered at candidate #27 alongside [[128,21,8]].

Tools

Same as [[128,21,8]].

Reproduction

import sys; sys.path.insert(0, 'research/kit'); sys.path.insert(0, 'verify')
import numpy as np
from gap_bridge import all_groups_of_order
from group_algebra import build_2bga

groups = all_groups_of_order(64)
g = groups[90]  # C8:Q8, GAP index 182
cayley = np.array(g['cayley_table'], dtype=np.int64)
a, b = [2, 6, 17, 23], [6, 16, 37, 59]
HX, HZ = build_2bga(cayley, a, b)
# n=128, k=13, d≤10 (witness-backed upper bound)

Parity checks

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