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[[234,28,18]] d ≤
n
234
k
28
d
18
kd²/n
38.769
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[1, 8, 26, 30, 37, 53, 56, 70, 75, 95, 112, 122, 136, 142, 171, 176, 196, 209]
d_Z 18 · witness weight 18 (claimed upper_bound)
witness operator (support, 18 qubits)
[31, 34, 45, 75, 97, 108, 113, 123, 130, 137, 150, 166, 168, 172, 189, 209, 223, 228]
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 10 · H_Z 10
qubit degrees H_X 5 · H_Z 5
trapping sets H_X (1,5)×234 (2,8)×5265 (3,9)×9282 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 234 (2,8): 5265 (3,9): 9282 (3,11): 161694 (3,13): 14040
trapping sets H_Z (1,5)×234 (2,8)×5265 (3,9)×9282 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 234 (2,8): 5265 (3,9): 9282 (3,11): 161694 (3,13): 14040

Construction & provenance

authors Dongheng Qian and Tianyi Li and @mathysrennela
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Free-action abelian 2BGA / lifted product over Z_13 x Z_9, with A support {(0,0),(0,2),(0,8),(4,4),(6,8)} and B support {(2,8),(5,4),(10,2),(11,5),(12,0)}, from the Supplementary Information construction data for arXiv:2608.08996v1.
model GPT-5.6 Luna (claimed, not verified)
builds on arXiv:2608.08996
date 2026-08-10
notes The paper reports exact d=18 by MILP. This staged record uses repository witnesses and therefore retains upper_bound confidence until trusted certification.
family lifted 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

[[234,28,18]] — abelian lifted-product reconstruction from arXiv:2608.08996v1

Direction & hypothesis

This is a direct reconstruction of the free-action abelian row in the Supplementary Information of arXiv:2608.08996v1. The paper reports [[234,28,18]] with exact MILP distance 18. The reconstruction advances the unrestricted weight-capable board cell.

What was searched

No new search was performed. The host is Z_13 x Z_9 with free action. Assemble the ordinary abelian 2BGA using A={(0,0),(0,2),(0,8),(4,4),(6,8)} and B={(2,8),(5,4),(10,2),(11,5),(12,0)}.

Evidence trail

The matrices recompute to n=234, k=28, CSS commutation, and maximum check weight 10. The submission contains explicit X- and Z-side witnesses produced by the repository builder. verify/validate_candidate.py returned passed: true, with no exact or WL-equivalent duplicate and board_advancing: true. The repository record uses witness-backed upper_bound confidence; the paper's MILP-exact distance claim is not treated as repository exact certification.

Dead ends

The already represented [[288,16,18]] row from the same paper was not duplicated. Non-normal subgroup rows were deferred because the repository kit has no tracked balanced-product/coset assembler for those actions.

Tools

Model: GPT-5.6 Luna. Repository tooling: research/kit/bb.py, research/kit/css.py, research/kit/submit.py, and verify/validate_candidate.py. Author: @mathysrennela.

Reproduction

Use the host and supports above with the free-action abelian 2BGA constructor. Source: arXiv:2608.08996v1, Supplementary Information.

Parity checks

X-checks 117 (max weight 10) · Z-checks 117 (max weight 10)
H_X (117 checks, sparse supports)
[0, 2, 8, 40, 62, 143, 166, 209, 221, 225] [0, 1, 3, 41, 54, 135, 167, 210, 222, 226] [1, 2, 4, 42, 55, 136, 168, 211, 223, 227] [2, 3, 5, 43, 56, 137, 169, 212, 224, 228] [3, 4, 6, 44, 57, 138, 170, 213, 216, 229] [4, 5, 7, 36, 58, 139, 162, 214, 217, 230] [5, 6, 8, 37, 59, 140, 163, 215, 218, 231] [0, 6, 7, 38, 60, 141, 164, 207, 219, 232] [1, 7, 8, 39, 61, 142, 165, 208, 220, 233] [9, 11, 17, 49, 71, 117, 152, 175, 218, 230] [9, 10, 12, 50, 63, 118, 144, 176, 219, 231] [10, 11, 13, 51, 64, 119, 145, 177, 220, 232] [11, 12, 14, 52, 65, 120, 146, 178, 221, 233] [12, 13, 15, 53, 66, 121, 147, 179, 222, 225] [13, 14, 16, 45, 67, 122, 148, 171, 223, 226] [14, 15, 17, 46, 68, 123, 149, 172, 224, 227] [9, 15, 16, 47, 69, 124, 150, 173, 216, 228] [10, 16, 17, 48, 70, 125, 151, 174, 217, 229] [18, 20, 26, 58, 80, 122, 126, 161, 184, 227] [18, 19, 21, 59, 72, 123, 127, 153, 185, 228] [19, 20, 22, 60, 73, 124, 128, 154, 186, 229] [20, 21, 23, 61, 74, 125, 129, 155, 187, 230] [21, 22, 24, 62, 75, 117, 130, 156, 188, 231] [22, 23, 25, 54, 76, 118, 131, 157, 180, 232] [23, 24, 26, 55, 77, 119, 132, 158, 181, 233] [18, 24, 25, 56, 78, 120, 133, 159, 182, 225] [19, 25, 26, 57, 79, 121, 134, 160, 183, 226] [27, 29, 35, 67, 89, 119, 131, 135, 170, 193] [27, 28, 30, 68, 81, 120, 132, 136, 162, 194] [28, 29, 31, 69, 82, 121, 133, 137, 163, 195] [29, 30, 32, 70, 83, 122, 134, 138, 164, 196] [30, 31, 33, 71, 84, 123, 126, 139, 165, 197] [31, 32, 34, 63, 85, 124, 127, 140, 166, 189] [32, 33, 35, 64, 86, 125, 128, 141, 167, 190] [27, 33, 34, 65, 87, 117, 129, 142, 168, 191] [28, 34, 35, 66, 88, 118, 130, 143, 169, 192] [36, 38, 44, 76, 98, 128, 140, 144, 179, 202] [36, 37, 39, 77, 90, 129, 141, 145, 171, 203] [37, 38, 40, 78, 91, 130, 142, 146, 172, 204] [38, 39, 41, 79, 92, 131, 143, 147, 173, 205] [39, 40, 42, 80, 93, 132, 135, 148, 174, 206] [40, 41, 43, 72, 94, 133, 136, 149, 175, 198] [41, 42, 44, 73, 95, 134, 137, 150, 176, 199] [36, 42, 43, 74, 96, 126, 138, 151, 177, 200] [37, 43, 44, 75, 97, 127, 139, 152, 178, 201] [45, 47, 53, 85, 107, 137, 149, 153, 188, 211] [45, 46, 48, 86, 99, 138, 150, 154, 180, 212] [46, 47, 49, 87, 100, 139, 151, 155, 181, 213] [47, 48, 50, 88, 101, 140, 152, 156, 182, 214] [48, 49, 51, 89, 102, 141, 144, 157, 183, 215] [49, 50, 52, 81, 103, 142, 145, 158, 184, 207] [50, 51, 53, 82, 104, 143, 146, 159, 185, 208] [45, 51, 52, 83, 105, 135, 147, 160, 186, 209] [46, 52, 53, 84, 106, 136, 148, 161, 187, 210] [54, 56, 62, 94, 116, 146, 158, 162, 197, 220] [54, 55, 57, 95, 108, 147, 159, 163, 189, 221] [55, 56, 58, 96, 109, 148, 160, 164, 190, 222] [56, 57, 59, 97, 110, 149, 161, 165, 191, 223] [57, 58, 60, 98, 111, 150, 153, 166, 192, 224] [58, 59, 61, 90, 112, 151, 154, 167, 193, 216] [59, 60, 62, 91, 113, 152, 155, 168, 194, 217] [54, 60, 61, 92, 114, 144, 156, 169, 195, 218] [55, 61, 62, 93, 115, 145, 157, 170, 196, 219] [8, 63, 65, 71, 103, 155, 167, 171, 206, 229] [0, 63, 64, 66, 104, 156, 168, 172, 198, 230] [1, 64, 65, 67, 105, 157, 169, 173, 199, 231] [2, 65, 66, 68, 106, 158, 170, 174, 200, 232] [3, 66, 67, 69, 107, 159, 162, 175, 201, 233] [4, 67, 68, 70, 99, 160, 163, 176, 202, 225] [5, 68, 69, 71, 100, 161, 164, 177, 203, 226] [6, 63, 69, 70, 101, 153, 165, 178, 204, 227] [7, 64, 70, 71, 102, 154, 166, 179, 205, 228] [17, 72, 74, 80, 112, 121, 164, 176, 180, 215] [9, 72, 73, 75, 113, 122, 165, 177, 181, 207] [10, 73, 74, 76, 114, 123, 166, 178, 182, 208] [11, 74, 75, 77, 115, 124, 167, 179, 183, 209] [12, 75, 76, 78, 116, 125, 168, 171, 184, 210] [13, 76, 77, 79, 108, 117, 169, 172, 185, 211] [14, 77, 78, 80, 109, 118, 170, 173, 186, 212] [15, 72, 78, 79, 110, 119, 162, 174, 187, 213] [16, 73, 79, 80, 111, 120, 163, 175, 188, 214] [4, 26, 81, 83, 89, 130, 173, 185, 189, 224] [5, 18, 81, 82, 84, 131, 174, 186, 190, 216] [6, 19, 82, 83, 85, 132, 175, 187, 191, 217] [7, 20, 83, 84, 86, 133, 176, 188, 192, 218] [8, 21, 84, 85, 87, 134, 177, 180, 193, 219] [0, 22, 85, 86, 88, 126, 178, 181, 194, 220] [1, 23, 86, 87, 89, 127, 179, 182, 195, 221] [2, 24, 81, 87, 88, 128, 171, 183, 196, 222] [3, 25, 82, 88, 89, 129, 172, 184, 197, 223] [13, 35, 90, 92, 98, 139, 182, 194, 198, 233] [14, 27, 90, 91, 93, 140, 183, 195, 199, 225] [15, 28, 91, 92, 94, 141, 184, 196, 200, 226] [16, 29, 92, 93, 95, 142, 185, 197, 201, 227] [17, 30, 93, 94, 96, 143, 186, 189, 202, 228] [9, 31, 94, 95, 97, 135, 187, 190, 203, 229] [10, 32, 95, 96, 98, 136, 188, 191, 204, 230] [11, 33, 90, 96, 97, 137, 180, 192, 205, 231] [12, 34, 91, 97, 98, 138, 181, 193, 206, 232] [22, 44, 99, 101, 107, 125, 148, 191, 203, 207] [23, 36, 99, 100, 102, 117, 149, 192, 204, 208] [24, 37, 100, 101, 103, 118, 150, 193, 205, 209] [25, 38, 101, 102, 104, 119, 151, 194, 206, 210] [26, 39, 102, 103, 105, 120, 152, 195, 198, 211] [18, 40, 103, 104, 106, 121, 144, 196, 199, 212] [19, 41, 104, 105, 107, 122, 145, 197, 200, 213] [20, 42, 99, 105, 106, 123, 146, 189, 201, 214] [21, 43, 100, 106, 107, 124, 147, 190, 202, 215] [31, 53, 108, 110, 116, 134, 157, 200, 212, 216] [32, 45, 108, 109, 111, 126, 158, 201, 213, 217] [33, 46, 109, 110, 112, 127, 159, 202, 214, 218] [34, 47, 110, 111, 113, 128, 160, 203, 215, 219] [35, 48, 111, 112, 114, 129, 161, 204, 207, 220] [27, 49, 112, 113, 115, 130, 153, 205, 208, 221] [28, 50, 113, 114, 116, 131, 154, 206, 209, 222] [29, 51, 108, 114, 115, 132, 155, 198, 210, 223] [30, 52, 109, 115, 116, 133, 156, 199, 211, 224]
H_Z (117 checks, sparse supports)
[9, 22, 34, 77, 100, 117, 118, 124, 181, 203] [10, 23, 35, 78, 101, 118, 119, 125, 182, 204] [11, 24, 27, 79, 102, 117, 119, 120, 183, 205] [12, 25, 28, 80, 103, 118, 120, 121, 184, 206] [13, 26, 29, 72, 104, 119, 121, 122, 185, 198] [14, 18, 30, 73, 105, 120, 122, 123, 186, 199] [15, 19, 31, 74, 106, 121, 123, 124, 187, 200] [16, 20, 32, 75, 107, 122, 124, 125, 188, 201] [17, 21, 33, 76, 99, 117, 123, 125, 180, 202] [18, 31, 43, 86, 109, 126, 127, 133, 190, 212] [19, 32, 44, 87, 110, 127, 128, 134, 191, 213] [20, 33, 36, 88, 111, 126, 128, 129, 192, 214] [21, 34, 37, 89, 112, 127, 129, 130, 193, 215] [22, 35, 38, 81, 113, 128, 130, 131, 194, 207] [23, 27, 39, 82, 114, 129, 131, 132, 195, 208] [24, 28, 40, 83, 115, 130, 132, 133, 196, 209] [25, 29, 41, 84, 116, 131, 133, 134, 197, 210] [26, 30, 42, 85, 108, 126, 132, 134, 189, 211] [1, 27, 40, 52, 95, 135, 136, 142, 199, 221] [2, 28, 41, 53, 96, 136, 137, 143, 200, 222] [3, 29, 42, 45, 97, 135, 137, 138, 201, 223] [4, 30, 43, 46, 98, 136, 138, 139, 202, 224] [5, 31, 44, 47, 90, 137, 139, 140, 203, 216] [6, 32, 36, 48, 91, 138, 140, 141, 204, 217] [7, 33, 37, 49, 92, 139, 141, 142, 205, 218] [8, 34, 38, 50, 93, 140, 142, 143, 206, 219] [0, 35, 39, 51, 94, 135, 141, 143, 198, 220] [10, 36, 49, 61, 104, 144, 145, 151, 208, 230] [11, 37, 50, 62, 105, 145, 146, 152, 209, 231] [12, 38, 51, 54, 106, 144, 146, 147, 210, 232] [13, 39, 52, 55, 107, 145, 147, 148, 211, 233] [14, 40, 53, 56, 99, 146, 148, 149, 212, 225] [15, 41, 45, 57, 100, 147, 149, 150, 213, 226] [16, 42, 46, 58, 101, 148, 150, 151, 214, 227] [17, 43, 47, 59, 102, 149, 151, 152, 215, 228] [9, 44, 48, 60, 103, 144, 150, 152, 207, 229] [19, 45, 58, 70, 113, 122, 153, 154, 160, 217] [20, 46, 59, 71, 114, 123, 154, 155, 161, 218] [21, 47, 60, 63, 115, 124, 153, 155, 156, 219] [22, 48, 61, 64, 116, 125, 154, 156, 157, 220] [23, 49, 62, 65, 108, 117, 155, 157, 158, 221] [24, 50, 54, 66, 109, 118, 156, 158, 159, 222] [25, 51, 55, 67, 110, 119, 157, 159, 160, 223] [26, 52, 56, 68, 111, 120, 158, 160, 161, 224] [18, 53, 57, 69, 112, 121, 153, 159, 161, 216] [5, 28, 54, 67, 79, 131, 162, 163, 169, 226] [6, 29, 55, 68, 80, 132, 163, 164, 170, 227] [7, 30, 56, 69, 72, 133, 162, 164, 165, 228] [8, 31, 57, 70, 73, 134, 163, 165, 166, 229] [0, 32, 58, 71, 74, 126, 164, 166, 167, 230] [1, 33, 59, 63, 75, 127, 165, 167, 168, 231] [2, 34, 60, 64, 76, 128, 166, 168, 169, 232] [3, 35, 61, 65, 77, 129, 167, 169, 170, 233] [4, 27, 62, 66, 78, 130, 162, 168, 170, 225] [14, 37, 63, 76, 88, 118, 140, 171, 172, 178] [15, 38, 64, 77, 89, 119, 141, 172, 173, 179] [16, 39, 65, 78, 81, 120, 142, 171, 173, 174] [17, 40, 66, 79, 82, 121, 143, 172, 174, 175] [9, 41, 67, 80, 83, 122, 135, 173, 175, 176] [10, 42, 68, 72, 84, 123, 136, 174, 176, 177] [11, 43, 69, 73, 85, 124, 137, 175, 177, 178] [12, 44, 70, 74, 86, 125, 138, 176, 178, 179] [13, 36, 71, 75, 87, 117, 139, 171, 177, 179] [23, 46, 72, 85, 97, 127, 149, 180, 181, 187] [24, 47, 73, 86, 98, 128, 150, 181, 182, 188] [25, 48, 74, 87, 90, 129, 151, 180, 182, 183] [26, 49, 75, 88, 91, 130, 152, 181, 183, 184] [18, 50, 76, 89, 92, 131, 144, 182, 184, 185] [19, 51, 77, 81, 93, 132, 145, 183, 185, 186] [20, 52, 78, 82, 94, 133, 146, 184, 186, 187] [21, 53, 79, 83, 95, 134, 147, 185, 187, 188] [22, 45, 80, 84, 96, 126, 148, 180, 186, 188] [32, 55, 81, 94, 106, 136, 158, 189, 190, 196] [33, 56, 82, 95, 107, 137, 159, 190, 191, 197] [34, 57, 83, 96, 99, 138, 160, 189, 191, 192] [35, 58, 84, 97, 100, 139, 161, 190, 192, 193] [27, 59, 85, 98, 101, 140, 153, 191, 193, 194] [28, 60, 86, 90, 102, 141, 154, 192, 194, 195] [29, 61, 87, 91, 103, 142, 155, 193, 195, 196] [30, 62, 88, 92, 104, 143, 156, 194, 196, 197] [31, 54, 89, 93, 105, 135, 157, 189, 195, 197] [41, 64, 90, 103, 115, 145, 167, 198, 199, 205] [42, 65, 91, 104, 116, 146, 168, 199, 200, 206] [43, 66, 92, 105, 108, 147, 169, 198, 200, 201] [44, 67, 93, 106, 109, 148, 170, 199, 201, 202] [36, 68, 94, 107, 110, 149, 162, 200, 202, 203] [37, 69, 95, 99, 111, 150, 163, 201, 203, 204] [38, 70, 96, 100, 112, 151, 164, 202, 204, 205] [39, 71, 97, 101, 113, 152, 165, 203, 205, 206] [40, 63, 98, 102, 114, 144, 166, 198, 204, 206] [7, 50, 73, 99, 112, 154, 176, 207, 208, 214] [8, 51, 74, 100, 113, 155, 177, 208, 209, 215] [0, 52, 75, 101, 114, 156, 178, 207, 209, 210] [1, 53, 76, 102, 115, 157, 179, 208, 210, 211] [2, 45, 77, 103, 116, 158, 171, 209, 211, 212] [3, 46, 78, 104, 108, 159, 172, 210, 212, 213] [4, 47, 79, 105, 109, 160, 173, 211, 213, 214] [5, 48, 80, 106, 110, 161, 174, 212, 214, 215] [6, 49, 72, 107, 111, 153, 175, 207, 213, 215] [4, 16, 59, 82, 108, 163, 185, 216, 217, 223] [5, 17, 60, 83, 109, 164, 186, 217, 218, 224] [6, 9, 61, 84, 110, 165, 187, 216, 218, 219] [7, 10, 62, 85, 111, 166, 188, 217, 219, 220] [8, 11, 54, 86, 112, 167, 180, 218, 220, 221] [0, 12, 55, 87, 113, 168, 181, 219, 221, 222] [1, 13, 56, 88, 114, 169, 182, 220, 222, 223] [2, 14, 57, 89, 115, 170, 183, 221, 223, 224] [3, 15, 58, 81, 116, 162, 184, 216, 222, 224] [0, 13, 25, 68, 91, 172, 194, 225, 226, 232] [1, 14, 26, 69, 92, 173, 195, 226, 227, 233] [2, 15, 18, 70, 93, 174, 196, 225, 227, 228] [3, 16, 19, 71, 94, 175, 197, 226, 228, 229] [4, 17, 20, 63, 95, 176, 189, 227, 229, 230] [5, 9, 21, 64, 96, 177, 190, 228, 230, 231] [6, 10, 22, 65, 97, 178, 191, 229, 231, 232] [7, 11, 23, 66, 98, 179, 192, 230, 232, 233] [8, 12, 24, 67, 90, 171, 193, 225, 231, 233]
Code ID 234-28-18 · download JSON · raw on GitHub