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[[128,20,14]] d ≤
n
128
k
20
d
14
kd²/n
30.625
w
16
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 16, w_Z = 16 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[3, 14, 28, 38, 55, 61, 65, 73, 74, 76, 95, 118, 121, 122]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[9, 30, 35, 50, 52, 60, 87, 92, 98, 109, 111, 112, 114, 119]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 16 · H_Z 16
qubit degrees H_X 8 · H_Z 8
trapping sets H_X (1,8)×128 (2,8)×96 (3,8)×192 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,8): 128 (2,8): 96 (2,10): 224 (2,12): 1760 (2,14): 3104 (3,8): 192 (3,10): 528 (3,12): 5136 (3,14): 20088 (3,16): 53800 (3,18): 88984 (3,20): 64528 (3,22): 2056
trapping sets H_Z (1,8)×128 (2,8)×96 (3,8)×192 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,8): 128 (2,8): 96 (2,10): 224 (2,12): 1760 (2,14): 3104 (3,8): 192 (3,10): 528 (3,12): 5136 (3,14): 20088 (3,16): 53800 (3,18): 88984 (3,20): 64528 (3,22): 2056

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on GAP ((C4xC2):C4):C2, weight-8 supports a=[7,9,11,30,32,34,47,59] b=[3,6,23,27,41,46,47,56]
model Xiaomi MiMo-V2.5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-08-15
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

[[128,20,14]] — 2BGA on non-abelian ((C4xC2):C4):C2 (order 64)

Direction & hypothesis

The unrestricted / weight-9plus cell rewards high rate at moderate block length. The GAP-enhanced autoresearch sweep targeted non-abelian groups of order ~64 where 2BGA codes can produce even k (unlike abelian BB, which is confined to odd-free k) and weight-8 supports per side (max row weight 16) keep the code in the w-9plus tier. The group ((C4xC2):C4):C2 (order 64, GAP index 61) was chosen after its Cayley table passed the two-block orthogonality check.

What was searched

GAP 2BGA sweep over non-abelian groups of order 60-128, weight-4 supports per block (row weight 8), screening at 400 RIS trials and confirming winners with an escalating trial ladder. The winning draw: order-64 group ((C4xC2):C4):C2 with element-index supports

a = [ 7,  9, 11, 30, 32, 34, 47, 59]
b = [ 3,  6, 23, 27, 41, 46, 47, 56]

Evidence trail

  • Rebuilt from the order-64 group table produced by the GAP enumeration step

    • (GAP SmallGroup(64, 61)) via group_algebra.build_2bga: n = 128, k = 20, CSS ok, max row weight 16.

  • Screen (sweep, 400 trials): d <= 14.
  • Confirmation ladder: distance stayed flat across 400 -> 1000 -> 10000 ->

    • 1000000 RIS trials; filtered at every stage (weight-14 witness persisting on both sides).

  • Fresh staged witnesses (2026-08-15, 8000 trials/side): X 14, Z 14,

    • each verified in ker of the opposite checks and outside its own rowspace.

  • Trusted gate verify/validate_candidate.py (2026-08-15, fresh seed):

    • passed, refutation clean, dominated_by: [], advances the weight-9plus x unrestricted board.

Tier: d <= 14 (witness-backed upper bound; not exact-certified).

Dead ends

The sweep ended with this as the single survivor passing the full trial ladder; over 64-group order band the other weight-8 hits either collapsed under deep RIS (e.g. screen d=16 -> true d<=6 at 1M trials) or carried check weight above 16 and were dropped before distance work.

Tools

  • Model: Xiaomi MiMo-V2.5 (provenance, self-reported).
  • Repo tooling: research/kit/group_algebra.py (build_2bga),

    • research/kit/surrogate.py (RIS distance), verify/validate_candidate.py (trusted gate); GAP group enumeration via the kit's GAP bridge scripts (SmallGroup(64, 61)).

  • Compute: GAP enumeration via the kit's bridge (catalog cached); distance

    • searches ~1M trials on the winner.

Reproduction

The committed builder research/build_128_20_14.py reproduces the full submission: it loads the order-64 Cayley table for ((C4xC2):C4):C2 (obtained by enumerating order-64 groups with GAP and selecting the group whose structure description is ((C4xC2):C4):C2, as the sweep did), calls group_algebra.build_2bga with the supports above, recomputes n/k/CSS, searches witnesses, and writes a schema-valid staged doc:

uv run python research/build_128_20_14.py

Equivalently, in GAP SmallGroup(64, 61) is the order-64 group with that structure description; its Cayley table plus

from research.kit.group_algebra import build_2bga
from research.kit.css import compute_k, verify_css
HX, HZ = build_2bga(mul64, a=[7, 9, 11, 30, 32, 34, 47, 59],
                    b=[3, 6, 23, 27, 41, 46, 47, 56])
assert verify_css(HX, HZ) and compute_k(HX, HZ) == 20

with mul64 the Cayley table of that group reproduces the code.

There is no public table row for this code; the group + supports fully determine (H_X, H_Z). The JSON in codes/128-20-14.json was built from the kit's submit.make_submission on that matrix.

Parity checks

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