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[[240,13,15]] d ≤
n
240
k
13
d
15
kd²/n
12.188
w
8

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Distance

d_X 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[7, 12, 69, 73, 80, 81, 89, 127, 130, 131, 136, 152, 166, 185, 231]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[1, 5, 21, 46, 68, 81, 84, 93, 100, 128, 130, 137, 152, 156, 212, 236]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty new parameter set claimed by submitter
construction 2BGA on the symmetric group S5 (order 120): H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T] with supports a=[1,34,61,111], b=[30,66,69,71] (indices into the deterministic BFS element order of research/kit/group_algebra.sym(5); identity is index 0). Found by a random support sweep over exotic groups.
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2306.16400 (2BGA family; exhaustive W<=8 tables stop at n<=200), arXiv:2606.17268 (coset-code searches to n=306: no n=240, all k even)
date 2026-07-08
notes Odd k=13 -- impossible for abelian two-block codes, so the BB/GB literature is excluded by parity. Claimed d is the lightest logical found by deep self-refutation: RIS flat at 15 across 1M/4M/16M trials/side x 2 seeds (gf2_fast, ~66M side-trials total), weight-15 X witness re-validated by the pinned python stack, BP+OSD (1M trials) found nothing lighter. Upper bound only. Novelty check 2026-07-08 against both references found no [[240,13,*]] and no odd-k code in any published 2BGA/coset enumeration.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

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