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[[125,25,4]] d ≤
n
125
k
25
d
4
kd²/n
3.2
w
9

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Distance

d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[51, 54, 55, 56]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[49, 59, 69, 89]
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 Balanced product implemented as the hypergraph product of two 2BGA parent codes. Parent 1 has group order 5 with a=[2,3,4], b=[1,2,4]; parent 2 has group order 5 with a=[0,3,4], b=[1,3,4].
model OpenAI GPT-5.6 Luna (claimed, not verified)
date 2026-08-07
family balanced product (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

[[125,25,4]] — balanced product of two order-5 2BGA codes

Direction & hypothesis

We targeted the unrestricted × weight-9+ cell with a balanced-product construction. The cell is comparatively sparse for small and moderate blocklengths, and the product construction can provide substantially more logical qubits than the small lifted-product candidates while remaining below the n <= 700 verification cap.

What was searched

The repository's research/kit/phase4_products.py sweep generated 1,070 candidates across hypergraph-product, lifted-product, and balanced-product families with n < 200. Candidates were screened with the repository RIS surrogate at 400 trials and ranked by k*d^2/n. This candidate was one of the screened balanced-product finalists.

The selected parents are both order-5 2BGA codes:

  • parent 1: a = [2, 3, 4], b = [1, 2, 4]
  • parent 2: a = [0, 3, 4], b = [1, 3, 4]

The submitted CSS code is the hypergraph product of those two parent codes.

Evidence trail

The initial screen reported [[125,25,4]] with screened efficiency 3.2. Submission packaging regenerated both X- and Z-side logical witnesses with the repository surrogate and persisted them in the JSON artifact.

The trusted validation gate passed. It reported:

  • CSS verification: passed
  • track: unrestricted × weight-9+
  • distance refutation: no lighter logical found in 7,500 RIS trials
  • board status: advances the weight-9+ unrestricted cell

The distance claim is an honest witness-backed upper bound (d <= 4), not an exact certification.

Dead ends

Several other screened finalists passed structural validation but were dominated by existing board entries, including [[60,4,8]], [[42,4,6]], [[54,4,6]], and multiple [[175,35,4]] balanced-product variants. The first selected [[18,4,4]] lifted-product candidate was also flagged as WL-equivalent to an existing 18-4-4.json entry and was not submitted as a duplicate.

Tools

The search used the repository's research/kit product constructors, screening code, submission packager, and verify/validate_candidate.py. The final verifier run used the trusted validator source hash reported by the local gate. No files under verify/ were modified.

Reproduction

Use research/kit/products.py and construct the order-5 groups with the repository's group helpers. Build each parent with build_2bga(mul, a, b), then call hypergraph_product(parent1_HX, parent2_HX). The exact support parameters are given above and in the code's provenance field.

Parity checks

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