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[[264,8,22]] d ≤
n
264
k
8
d
22
kd²/n
14.667
w
7

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Distance

d_X 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[15, 27, 35, 45, 54, 64, 69, 72, 96, 106, 145, 148, 152, 177, 181, 191, 195, 227, 239, 246, 249, 256]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[1, 14, 15, 17, 43, 45, 47, 49, 61, 62, 90, 94, 101, 129, 174, 181, 183, 191, 203, 224, 234, 249]
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 Two-block group-algebra (2BGA) code on the non-abelian group Dic33; n=264, k=8, max check weight 7.
model Claude Claude Opus 4.8 (claimed, not verified)
date 2026-08-04
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

[[264,8,22]] — 2BGA on the dicyclic Dic_33 group

Direction & hypothesis

Advance the frontier with a two-block group-algebra code over a non-abelian dicyclic (generalized-quaternion) group, a family that was not represented on the board when this search began. The distance was deliberately kept in the range where the RIS surrogate converges, so the claim is verifiable rather than an optimistic screen.

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, sampling element sets a, b under a check-weight bound and keeping only codes that strictly dominate an existing board entry in their (locality, weight) cell. This code sits on dicyclic Dic_33 with max check weight 7 and k = 8.

Evidence trail

Witness-backed upper bound d <= 22, with an explicit weight-22 logical on each side. The code was rebuilt from its group data and re-witnessed independently of the search that found it, and the distance held at 3,000,000-trial pair-depth-20 RIS (an independent deeper pass returned the same 22, no collapse). verify/qldpc_verify.py accepts it at kd^2/n = 14.7. It strictly dominates 1 existing board entries.

Dead ends

The same search produced higher-distance candidates that did not survive scrutiny and are deliberately not submitted. A [[390,82,41]] screen resolved to d = 38 once measured at a budget calibrated against a known-answer control, matching the existing record rather than beating it, and several [[310,16,28-29]] screens fell to <= 25. Everything submitted here sits in the distance range where repeated independent passes agree.

Model & harness

Found by a continual non-abelian 2BGA dominance search (Claude Opus 4.8) built on this repository's own research/kit group-algebra constructors and the gf2_fast RIS core. Rebuilt, re-witnessed, and re-verified through verify/qldpc_verify.py during packaging.

Reproduction

Build dicyclic Dic_33 with research/kit/group_algebra (dicyclic presentation a^{2m} = 1, b^2 = a^m, b a b^-1 = a^-1), then form the 2BGA via build_2bga(mul, a, b) using the a, b element-index lists recorded in the code file to obtain [[264,8,22]].

Parity checks

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