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[[192,12,16]] d ≤
n
192
k
12
d
16
kd²/n
16.0
w
8

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[5, 6, 41, 42, 45, 62, 65, 82, 118, 122, 127, 135, 139, 142, 146, 147]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[0, 20, 24, 61, 69, 73, 81, 92, 100, 107, 111, 131, 136, 156, 167, 176]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA on the dicyclic (generalized quaternion) group Dic_24; n=192, k=12, max check weight 8.
model Claude Claude Opus 4.8 (claimed, not verified)
date 2026-08-03
family 2BGA coset (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

[[192,12,16]] — 2BGA on the dicyclic group Dic_24

Direction & hypothesis

Advance the weight-8 frontier with a two-block group-algebra code over the non-abelian dicyclic (generalized-quaternion) group Dic_24, a low-mined family. Moderate distance, where the RIS surrogate is reliable and the claim verifiable.

What was searched

2BGA H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over a roster of dicyclic, dihedral and metacyclic groups, random a, b with bounded check weight, kept only when the code strictly dominates a board entry in its cell. This code: Dic_24, max check weight 8, k = 12.

Evidence trail

Witness-backed upper bound d <= 16 (weight-16 witness on each side). Found on the search machine at its heavy budget and rebuilt + re-witnessed here; the verifier accepts it at kd^2/n = 16.0.

Dead ends

High-distance variants of this construction over-estimate distance under light search and collapse under deeper search (a sibling [[310,16,29]] screen fell to <= 25 under 2M trials), so the search is confined to the moderate-distance regime where the bound is trustworthy.

Model & harness

Found by a continual dicyclic-2BGA dominance search (Claude Opus 4.8) on the repo's gf2_fast RIS core; rebuilt, re-witnessed, and run through verify/qldpc_verify.py during packaging.

Reproduction

Dic_24 (order 96): a^{96}=1, b^2=a^{48}, b a b^-1 = a^-1. Build the 2BGA via research/kit/group_algebra.build_2bga(mul, a, b) with the a, b element-index lists in the code file to obtain [[192,12,16]].

Parity checks

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