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[[288,8,24]] d ≤
n
288
k
8
d
24
kd²/n
16.0
w
8

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Distance

d_X 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[2, 13, 19, 32, 42, 43, 59, 66, 79, 83, 103, 117, 126, 136, 139, 143, 144, 184, 189, 208, 213, 253, 268, 273]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[1, 8, 41, 48, 61, 72, 121, 132, 159, 162, 163, 172, 183, 187, 199, 212, 222, 223, 229, 243, 247, 267, 269, 283]
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 Dic36; n=288, k=8, max check weight 8.
model Claude Claude Opus 4.8 (claimed, not verified)
date 2026-08-02
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

[[288,8,24]] — 2BGA on the dicyclic group Dic_36

Direction & hypothesis

Advance the weight-8 frontier with a two-block group-algebra code over a non-abelian dicyclic (generalized-quaternion) group, 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, screened by RIS distance and kept only when the code strictly dominates a board entry in its cell. This code: Dic_36, max check weight 8, k = 8.

Evidence trail

Witness-backed upper bound d <= 24 (weight-24 witness on each side). Confirmed on the search machine at its heavy budget (deep RIS, pair-depth 20), and rebuilt and 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, 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_36 (order 144): dicyclic presentation a^{144}=1, b^2=a^{72}, 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 recorded in the code file to obtain [[288,8,24]].

Parity checks

X-checks 144 · Z-checks 144
H_X (144 checks, sparse supports)
[14, 24, 33, 98, 137, 171, 175, 275] [15, 25, 64, 99, 104, 186, 190, 230] [16, 26, 31, 100, 135, 173, 177, 277] [17, 27, 62, 101, 102, 188, 192, 232] [18, 28, 29, 102, 133, 175, 179, 279] [19, 29, 60, 100, 103, 190, 194, 234] [20, 27, 30, 104, 131, 177, 181, 281] [21, 31, 58, 98, 105, 192, 196, 236] [22, 25, 32, 106, 129, 179, 183, 283] [23, 33, 56, 96, 107, 194, 198, 238] [23, 24, 34, 108, 127, 181, 185, 285] [25, 35, 54, 94, 109, 196, 200, 240] [21, 26, 36, 110, 125, 183, 187, 287] [27, 37, 52, 92, 111, 198, 202, 242] [19, 28, 38, 112, 123, 145, 185, 189] [29, 39, 50, 90, 113, 200, 204, 244] [17, 30, 40, 114, 121, 147, 187, 191] [31, 41, 48, 88, 115, 202, 206, 246] [15, 32, 42, 116, 119, 149, 189, 193] [33, 43, 46, 86, 117, 204, 208, 248] [13, 34, 44, 117, 118, 151, 191, 195] [35, 44, 45, 84, 119, 206, 210, 250] [11, 36, 46, 115, 120, 153, 193, 197] [37, 42, 47, 82, 121, 208, 212, 252] [9, 38, 48, 113, 122, 155, 195, 199] [39, 40, 49, 80, 123, 210, 214, 254] [7, 40, 50, 111, 124, 157, 197, 201] [38, 41, 51, 78, 125, 212, 216, 256] [5, 42, 52, 109, 126, 159, 199, 203] [36, 43, 53, 76, 127, 214, 218, 258] [3, 44, 54, 107, 128, 161, 201, 205] [34, 45, 55, 74, 129, 216, 220, 260] [1, 46, 56, 105, 130, 163, 203, 207] [32, 47, 57, 72, 131, 218, 222, 262] [48, 58, 103, 132, 143, 165, 205, 209] [30, 49, 59, 70, 133, 220, 224, 264] [50, 60, 101, 134, 141, 167, 207, 211] [28, 51, 61, 68, 135, 222, 226, 266] [52, 62, 99, 136, 139, 169, 209, 213] [26, 53, 63, 66, 137, 224, 228, 268] [54, 64, 97, 137, 138, 171, 211, 215] [24, 55, 64, 65, 139, 226, 230, 270] [56, 66, 95, 135, 140, 173, 213, 217] [22, 57, 62, 67, 141, 228, 232, 272] [58, 68, 93, 133, 142, 175, 215, 219] [20, 59, 60, 69, 143, 230, 234, 274] [0, 60, 70, 91, 131, 177, 217, 221] [1, 18, 58, 61, 71, 232, 236, 276] [2, 62, 72, 89, 129, 179, 219, 223] [3, 16, 56, 63, 73, 234, 238, 278] [4, 64, 74, 87, 127, 181, 221, 225] [5, 14, 54, 65, 75, 236, 240, 280] [6, 66, 76, 85, 125, 183, 223, 227] [7, 12, 52, 67, 77, 238, 242, 282] [8, 68, 78, 83, 123, 185, 225, 229] [9, 10, 50, 69, 79, 240, 244, 284] [10, 70, 80, 81, 121, 187, 227, 231] [8, 11, 48, 71, 81, 242, 246, 286] [12, 72, 79, 82, 119, 189, 229, 233] [6, 13, 46, 73, 83, 144, 244, 248] [14, 74, 77, 84, 117, 191, 231, 235] [4, 15, 44, 75, 85, 146, 246, 250] [16, 75, 76, 86, 115, 193, 233, 237] [2, 17, 42, 77, 87, 148, 248, 252] [18, 73, 78, 88, 113, 195, 235, 239] [0, 19, 40, 79, 89, 150, 250, 254] [20, 71, 80, 90, 111, 197, 237, 241] [21, 38, 81, 91, 142, 152, 252, 256] [22, 69, 82, 92, 109, 199, 239, 243] [23, 36, 83, 93, 140, 154, 254, 258] [24, 67, 84, 94, 107, 201, 241, 245] [25, 34, 85, 95, 138, 156, 256, 260] [26, 65, 86, 96, 105, 203, 243, 247] [27, 32, 87, 97, 136, 158, 258, 262] [28, 63, 88, 98, 103, 205, 245, 249] [29, 30, 89, 99, 134, 160, 260, 264] [30, 61, 90, 100, 101, 207, 247, 251] [28, 31, 91, 101, 132, 162, 262, 266] [32, 59, 92, 99, 102, 209, 249, 253] [26, 33, 93, 103, 130, 164, 264, 268] [34, 57, 94, 97, 104, 211, 251, 255] [24, 35, 95, 105, 128, 166, 266, 270] [36, 55, 95, 96, 106, 213, 253, 257] [22, 37, 97, 107, 126, 168, 268, 272] [38, 53, 93, 98, 108, 215, 255, 259] [20, 39, 99, 109, 124, 170, 270, 274] [40, 51, 91, 100, 110, 217, 257, 261] [18, 41, 101, 111, 122, 172, 272, 276] [42, 49, 89, 102, 112, 219, 259, 263] [16, 43, 103, 113, 120, 174, 274, 278] [44, 47, 87, 104, 114, 221, 261, 265] [14, 45, 105, 115, 118, 176, 276, 280] [45, 46, 85, 106, 116, 223, 263, 267] [12, 47, 107, 116, 117, 178, 278, 282] [43, 48, 83, 108, 118, 225, 265, 269] [10, 49, 109, 114, 119, 180, 280, 284] [41, 50, 81, 110, 120, 227, 267, 271] [8, 51, 111, 112, 121, 182, 282, 286] [39, 52, 79, 112, 122, 229, 269, 273] [6, 53, 110, 113, 123, 144, 184, 284] [37, 54, 77, 114, 124, 231, 271, 275] [4, 55, 108, 115, 125, 146, 186, 286] [35, 56, 75, 116, 126, 233, 273, 277] [2, 57, 106, 117, 127, 144, 148, 188] [33, 58, 73, 118, 128, 235, 275, 279] [0, 59, 104, 119, 129, 146, 150, 190] [31, 60, 71, 120, 130, 237, 277, 281] [61, 102, 121, 131, 142, 148, 152, 192] [29, 62, 69, 122, 132, 239, 279, 283] [63, 100, 123, 133, 140, 150, 154, 194] [27, 64, 67, 124, 134, 241, 281, 285] [65, 98, 125, 135, 138, 152, 156, 196] [25, 65, 66, 126, 136, 243, 283, 287] [67, 96, 127, 136, 137, 154, 158, 198] [23, 63, 68, 128, 138, 145, 245, 285] [69, 94, 129, 134, 139, 156, 160, 200] [21, 61, 70, 130, 140, 147, 247, 287] [71, 92, 131, 132, 141, 158, 162, 202] [19, 59, 72, 132, 142, 145, 149, 249] [73, 90, 130, 133, 143, 160, 164, 204] [0, 17, 57, 74, 134, 147, 151, 251] [1, 75, 88, 128, 135, 162, 166, 206] [2, 15, 55, 76, 136, 149, 153, 253] [3, 77, 86, 126, 137, 164, 168, 208] [4, 13, 53, 78, 138, 151, 155, 255] [5, 79, 84, 124, 139, 166, 170, 210] [6, 11, 51, 80, 140, 153, 157, 257] [7, 81, 82, 122, 141, 168, 172, 212] [8, 9, 49, 82, 142, 155, 159, 259] [9, 80, 83, 120, 143, 170, 174, 214] [0, 7, 10, 47, 84, 157, 161, 261] [1, 11, 78, 85, 118, 172, 176, 216] [2, 5, 12, 45, 86, 159, 163, 263] [3, 13, 76, 87, 116, 174, 178, 218] [3, 4, 14, 43, 88, 161, 165, 265] [5, 15, 74, 89, 114, 176, 180, 220] [1, 6, 16, 41, 90, 163, 167, 267] [7, 17, 72, 91, 112, 178, 182, 222] [8, 18, 39, 92, 143, 165, 169, 269] [9, 19, 70, 93, 110, 180, 184, 224] [10, 20, 37, 94, 141, 167, 171, 271] [11, 21, 68, 95, 108, 182, 186, 226] [12, 22, 35, 96, 139, 169, 173, 273] [13, 23, 66, 97, 106, 184, 188, 228]
H_Z (144 checks, sparse supports)
[59, 99, 103, 190, 209, 249, 264, 274] [14, 114, 118, 176, 191, 265, 275, 280] [61, 101, 105, 192, 207, 247, 266, 276] [16, 116, 120, 174, 193, 267, 277, 278] [63, 103, 107, 194, 205, 245, 268, 278] [18, 118, 122, 172, 195, 269, 276, 279] [65, 105, 109, 196, 203, 243, 270, 280] [20, 120, 124, 170, 197, 271, 274, 281] [67, 107, 111, 198, 201, 241, 272, 282] [22, 122, 126, 168, 199, 272, 273, 283] [69, 109, 113, 199, 200, 239, 274, 284] [24, 124, 128, 166, 201, 270, 275, 285] [71, 111, 115, 197, 202, 237, 276, 286] [26, 126, 130, 164, 203, 268, 277, 287] [73, 113, 117, 144, 195, 204, 235, 278] [28, 128, 132, 145, 162, 205, 266, 279] [75, 115, 119, 146, 193, 206, 233, 280] [30, 130, 134, 147, 160, 207, 264, 281] [77, 117, 121, 148, 191, 208, 231, 282] [32, 132, 136, 149, 158, 209, 262, 283] [79, 119, 123, 150, 189, 210, 229, 284] [34, 134, 138, 151, 156, 211, 260, 285] [81, 121, 125, 152, 187, 212, 227, 286] [36, 136, 140, 153, 154, 213, 258, 287] [83, 123, 127, 144, 154, 185, 214, 225] [38, 138, 142, 145, 152, 155, 215, 256] [85, 125, 129, 146, 156, 183, 216, 223] [0, 40, 140, 147, 150, 157, 217, 254] [87, 127, 131, 148, 158, 181, 218, 221] [2, 42, 142, 148, 149, 159, 219, 252] [89, 129, 133, 150, 160, 179, 219, 220] [0, 4, 44, 146, 151, 161, 221, 250] [91, 131, 135, 152, 162, 177, 217, 222] [2, 6, 46, 144, 153, 163, 223, 248] [93, 133, 137, 154, 164, 175, 215, 224] [4, 8, 48, 155, 165, 225, 246, 286] [95, 135, 139, 156, 166, 173, 213, 226] [6, 10, 50, 157, 167, 227, 244, 284] [97, 137, 141, 158, 168, 171, 211, 228] [8, 12, 52, 159, 169, 229, 242, 282] [99, 139, 143, 160, 169, 170, 209, 230] [10, 14, 54, 161, 171, 231, 240, 280] [1, 101, 141, 162, 167, 172, 207, 232] [12, 16, 56, 163, 173, 233, 238, 278] [3, 103, 143, 164, 165, 174, 205, 234] [14, 18, 58, 165, 175, 235, 236, 276] [1, 5, 105, 163, 166, 176, 203, 236] [16, 20, 60, 167, 177, 234, 237, 274] [3, 7, 107, 161, 168, 178, 201, 238] [18, 22, 62, 169, 179, 232, 239, 272] [5, 9, 109, 159, 170, 180, 199, 240] [20, 24, 64, 171, 181, 230, 241, 270] [7, 11, 111, 157, 172, 182, 197, 242] [22, 26, 66, 173, 183, 228, 243, 268] [9, 13, 113, 155, 174, 184, 195, 244] [24, 28, 68, 175, 185, 226, 245, 266] [11, 15, 115, 153, 176, 186, 193, 246] [26, 30, 70, 177, 187, 224, 247, 264] [13, 17, 117, 151, 178, 188, 191, 248] [28, 32, 72, 179, 189, 222, 249, 262] [15, 19, 119, 149, 180, 189, 190, 250] [30, 34, 74, 181, 191, 220, 251, 260] [17, 21, 121, 147, 182, 187, 192, 252] [32, 36, 76, 183, 193, 218, 253, 258] [19, 23, 123, 145, 184, 185, 194, 254] [34, 38, 78, 185, 195, 216, 255, 256] [21, 25, 125, 183, 186, 196, 256, 287] [36, 40, 80, 187, 197, 214, 254, 257] [23, 27, 127, 181, 188, 198, 258, 285] [38, 42, 82, 189, 199, 212, 252, 259] [25, 29, 129, 179, 190, 200, 260, 283] [40, 44, 84, 191, 201, 210, 250, 261] [27, 31, 131, 177, 192, 202, 262, 281] [42, 46, 86, 193, 203, 208, 248, 263] [29, 33, 133, 175, 194, 204, 264, 279] [44, 48, 88, 195, 205, 206, 246, 265] [31, 35, 135, 173, 196, 206, 266, 277] [46, 50, 90, 197, 204, 207, 244, 267] [33, 37, 137, 171, 198, 208, 268, 275] [48, 52, 92, 199, 202, 209, 242, 269] [35, 39, 139, 169, 200, 210, 270, 273] [50, 54, 94, 200, 201, 211, 240, 271] [37, 41, 141, 167, 202, 212, 271, 272] [52, 56, 96, 198, 203, 213, 238, 273] [39, 43, 143, 165, 204, 214, 269, 274] [54, 58, 98, 196, 205, 215, 236, 275] [1, 41, 45, 163, 206, 216, 267, 276] [56, 60, 100, 194, 207, 217, 234, 277] [3, 43, 47, 161, 208, 218, 265, 278] [58, 62, 102, 192, 209, 219, 232, 279] [5, 45, 49, 159, 210, 220, 263, 280] [60, 64, 104, 190, 211, 221, 230, 281] [7, 47, 51, 157, 212, 222, 261, 282] [62, 66, 106, 188, 213, 223, 228, 283] [9, 49, 53, 155, 214, 224, 259, 284] [64, 68, 108, 186, 215, 225, 226, 285] [11, 51, 55, 153, 216, 226, 257, 286] [66, 70, 110, 184, 217, 224, 227, 287] [13, 53, 57, 144, 151, 218, 228, 255] [68, 72, 112, 145, 182, 219, 222, 229] [15, 55, 59, 146, 149, 220, 230, 253] [70, 74, 114, 147, 180, 220, 221, 231] [17, 57, 61, 147, 148, 222, 232, 251] [72, 76, 116, 149, 178, 218, 223, 233] [19, 59, 63, 145, 150, 224, 234, 249] [74, 78, 118, 151, 176, 216, 225, 235] [21, 61, 65, 152, 226, 236, 247, 287] [76, 80, 120, 153, 174, 214, 227, 237] [23, 63, 67, 154, 228, 238, 245, 285] [78, 82, 122, 155, 172, 212, 229, 239] [25, 65, 69, 156, 230, 240, 243, 283] [80, 84, 124, 157, 170, 210, 231, 241] [27, 67, 71, 158, 232, 241, 242, 281] [82, 86, 126, 159, 168, 208, 233, 243] [29, 69, 73, 160, 234, 239, 244, 279] [84, 88, 128, 161, 166, 206, 235, 245] [31, 71, 75, 162, 236, 237, 246, 277] [86, 90, 130, 163, 164, 204, 237, 247] [33, 73, 77, 164, 235, 238, 248, 275] [88, 92, 132, 162, 165, 202, 239, 249] [35, 75, 79, 166, 233, 240, 250, 273] [90, 94, 134, 160, 167, 200, 241, 251] [37, 77, 81, 168, 231, 242, 252, 271] [92, 96, 136, 158, 169, 198, 243, 253] [39, 79, 83, 170, 229, 244, 254, 269] [94, 98, 138, 156, 171, 196, 245, 255] [41, 81, 85, 172, 227, 246, 256, 267] [96, 100, 140, 154, 173, 194, 247, 257] [43, 83, 87, 174, 225, 248, 258, 265] [98, 102, 142, 152, 175, 192, 249, 259] [45, 85, 89, 176, 223, 250, 260, 263] [0, 100, 104, 150, 177, 190, 251, 261] [47, 87, 91, 178, 221, 252, 261, 262] [2, 102, 106, 148, 179, 188, 253, 263] [49, 89, 93, 180, 219, 254, 259, 264] [4, 104, 108, 146, 181, 186, 255, 265] [51, 91, 95, 182, 217, 256, 257, 266] [6, 106, 110, 144, 183, 184, 257, 267] [53, 93, 97, 184, 215, 255, 258, 268] [8, 108, 112, 182, 185, 259, 269, 286] [55, 95, 99, 186, 213, 253, 260, 270] [10, 110, 114, 180, 187, 261, 271, 284] [57, 97, 101, 188, 211, 251, 262, 272] [12, 112, 116, 178, 189, 263, 273, 282]
raw submission JSON