← back to the board
[[336,20,20]] d ≤
n
336
k
20
d
20
kd²/n
23.81
w
8

Share this result

Distance

d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[52, 60, 93, 105, 111, 127, 136, 144, 171, 186, 187, 204, 219, 266, 281, 296, 312, 317, 325, 327]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[6, 46, 51, 69, 82, 84, 101, 118, 136, 144, 156, 163, 171, 193, 194, 219, 228, 274, 275, 309]
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 with supports a=[0, 88, 101, 111], b=[0, 9, 59, 98] on PSL(2,7) (order 168, the Klein-quartic/Hurwitz group) as permutations of the projective line P1(F7) with points ordered (0,...,6,inf), generated by T: z -> z+1 and S: z -> -1/z; elements enumerated by BFS closure (research/kit/group_algebra.perm_group, identity at index 0). H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T].
model Claude Claude Fable 5 (claimed, not verified)
builds on arXiv:2306.16400 (2BGA family), arXiv:2606.17268 (novelty check), arXiv:2606.24808 (novelty check)
date 2026-07-08
notes Found by simulated annealing over weight-8 supports (identity pinned in both supports by shift equivalence; screen-then-confirm fitness). Claimed d is the lightest logical found by deep self-refutation: flat across 1M/4M/16M RIS trials/side on THREE independent seeds (discovery + 2 fresh), witness re-validated by the pinned python stack, BP+OSD (1M trials) found nothing lighter. Upper bound only. Novelty due diligence 2026-07-08: outside arXiv:2306.16400 (n<=200), arXiv:2606.17268 (n<=306, all even k), arXiv:2606.24808 (no PSL(2,7)); no [[336,20/19,*]] found; first qLDPC use of PSL(2,7) to our knowledge.
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 168 · Z-checks 168
H_X (168 checks, sparse supports)
[0, 42, 95, 151, 168, 199, 258, 281] [1, 60, 124, 145, 169, 186, 195, 304] [2, 61, 68, 166, 170, 207, 224, 238] [3, 84, 90, 104, 170, 171, 206, 285] [4, 85, 117, 149, 172, 222, 242, 262] [5, 17, 94, 112, 173, 176, 180, 326] [6, 56, 113, 118, 172, 174, 221, 227] [7, 64, 76, 94, 175, 240, 266, 288] [8, 69, 114, 144, 176, 181, 188, 254] [9, 28, 123, 140, 168, 177, 233, 244] [9, 10, 82, 84, 178, 239, 256, 284] [11, 77, 141, 146, 175, 179, 239, 245] [12, 37, 134, 147, 180, 252, 257, 264] [13, 89, 103, 123, 181, 189, 199, 277] [14, 96, 131, 142, 169, 182, 185, 268] [15, 48, 115, 162, 183, 209, 263, 307] [16, 43, 67, 135, 178, 184, 200, 255] [17, 44, 122, 141, 185, 282, 302, 306] [16, 18, 111, 113, 186, 196, 209, 220] [19, 83, 98, 105, 180, 187, 263, 269] [20, 106, 125, 140, 188, 194, 275, 290] [21, 55, 65, 95, 186, 189, 200, 300] [22, 44, 67, 132, 171, 190, 193, 293] [23, 88, 122, 133, 191, 224, 289, 311] [24, 53, 125, 153, 183, 192, 215, 278] [25, 97, 101, 160, 193, 255, 305, 322] [20, 26, 62, 156, 185, 194, 262, 286] [27, 47, 63, 105, 195, 218, 223, 308] [10, 28, 83, 85, 196, 207, 210, 260] [26, 29, 71, 141, 173, 175, 197, 310] [27, 30, 126, 134, 188, 198, 289, 294] [31, 59, 70, 135, 199, 205, 256, 259] [32, 38, 135, 150, 170, 200, 215, 320] [23, 33, 91, 124, 174, 201, 204, 253] [34, 45, 54, 68, 202, 242, 258, 327] [10, 20, 35, 87, 191, 197, 203, 301] [28, 36, 47, 143, 204, 230, 278, 313] [37, 75, 150, 163, 193, 205, 210, 288] [35, 38, 128, 146, 206, 236, 241, 324] [39, 49, 116, 130, 207, 222, 225, 232] [32, 40, 86, 155, 195, 208, 294, 311] [11, 41, 87, 157, 179, 209, 295, 305] [42, 66, 88, 134, 176, 210, 227, 299] [18, 27, 43, 114, 177, 211, 214, 303] [5, 44, 59, 60, 212, 258, 269, 307] [40, 45, 83, 99, 198, 213, 226, 261] [41, 46, 50, 140, 214, 221, 275, 335] [47, 82, 97, 156, 172, 197, 215, 332] [48, 57, 156, 164, 179, 216, 219, 276] [24, 49, 105, 165, 195, 217, 266, 329] [4, 50, 104, 108, 173, 202, 218, 321] [13, 51, 119, 149, 219, 248, 251, 301] [18, 32, 52, 102, 204, 220, 225, 257] [5, 11, 53, 62, 221, 260, 265, 330] [43, 54, 66, 161, 184, 222, 240, 243] [55, 63, 93, 164, 206, 223, 281, 327] [56, 102, 148, 165, 187, 224, 230, 317] [52, 57, 98, 152, 181, 225, 245, 279] [12, 58, 143, 154, 178, 182, 226, 323] [19, 59, 102, 106, 168, 226, 227, 331] [60, 92, 95, 117, 184, 219, 228, 231] [61, 89, 93, 113, 224, 229, 284, 316] [29, 42, 62, 142, 182, 212, 230, 330] [16, 30, 63, 115, 203, 231, 261, 272] [15, 27, 64, 81, 189, 214, 232, 306] [39, 59, 65, 127, 233, 279, 283, 318] [49, 66, 111, 155, 187, 211, 234, 237] [67, 96, 112, 157, 206, 235, 252, 328] [68, 78, 133, 155, 176, 217, 236, 333] [69, 79, 157, 161, 237, 271, 274, 321] [8, 56, 70, 137, 194, 219, 238, 243] [1, 71, 76, 78, 232, 239, 280, 291] [22, 46, 69, 72, 192, 240, 264, 267] [29, 36, 48, 73, 221, 241, 304, 329] [30, 49, 74, 75, 198, 242, 248, 331] [9, 19, 75, 86, 189, 243, 269, 302] [62, 76, 92, 167, 183, 190, 244, 334] [36, 77, 129, 147, 169, 244, 245, 315] [73, 78, 110, 126, 192, 237, 246, 249] [70, 74, 79, 159, 242, 247, 286, 307] [21, 80, 121, 161, 190, 208, 248, 280] [6, 81, 132, 136, 218, 233, 249, 297] [30, 36, 38, 82, 194, 246, 250, 253] [31, 75, 77, 83, 180, 251, 298, 302] [84, 120, 124, 144, 169, 229, 252, 296] [68, 85, 121, 145, 217, 234, 253, 330] [45, 60, 86, 153, 223, 231, 254, 259] [46, 61, 87, 160, 255, 277, 301, 309] [0, 26, 61, 88, 174, 256, 270, 282] [2, 41, 42, 89, 233, 238, 257, 316] [8, 16, 58, 90, 203, 208, 258, 334] [51, 58, 82, 91, 196, 236, 259, 286] [12, 71, 92, 162, 181, 235, 260, 283] [32, 93, 126, 130, 261, 296, 299, 333] [94, 107, 149, 162, 205, 237, 262, 267] [95, 103, 108, 130, 184, 263, 287, 303] [25, 96, 106, 167, 203, 264, 290, 292] [14, 77, 97, 158, 239, 265, 285, 328] [15, 37, 98, 138, 168, 266, 271, 315] [3, 99, 103, 107, 200, 267, 294, 322] [31, 35, 96, 100, 191, 201, 268, 308] [0, 12, 63, 101, 171, 229, 268, 269] [17, 31, 102, 116, 210, 266, 270, 311] [86, 103, 120, 139, 201, 223, 271, 303] [87, 104, 121, 158, 185, 236, 272, 314] [54, 65, 74, 105, 205, 231, 273, 276] [22, 100, 106, 118, 188, 274, 319, 322] [34, 107, 127, 151, 171, 247, 275, 318] [72, 80, 108, 111, 213, 235, 276, 280] [33, 109, 148, 167, 196, 241, 249, 277] [34, 92, 110, 164, 278, 300, 321, 325] [0, 54, 57, 111, 186, 251, 279, 325] [18, 22, 24, 112, 280, 296, 312, 315] [104, 113, 119, 131, 179, 253, 270, 281] [90, 94, 114, 148, 213, 282, 291, 308] [64, 84, 115, 163, 255, 283, 298, 326] [65, 85, 116, 128, 183, 234, 284, 320] [1, 17, 40, 117, 198, 211, 285, 317] [14, 26, 80, 118, 173, 274, 285, 286] [33, 81, 112, 119, 245, 284, 287, 289] [21, 99, 120, 166, 220, 261, 288, 292] [6, 50, 121, 144, 192, 212, 289, 323] [48, 122, 151, 154, 218, 259, 290, 316] [69, 123, 136, 166, 227, 263, 291, 335] [124, 132, 137, 154, 215, 281, 292, 313] [38, 39, 125, 139, 202, 216, 293, 324] [2, 55, 79, 126, 174, 247, 293, 294] [7, 120, 127, 133, 225, 252, 295, 327] [47, 52, 125, 128, 216, 241, 296, 323] [19, 53, 97, 129, 193, 228, 260, 297] [1, 21, 88, 130, 211, 220, 249, 298] [2, 6, 43, 131, 199, 299, 309, 313] [115, 119, 132, 159, 207, 265, 272, 300] [107, 109, 116, 133, 301, 312, 320, 333] [76, 91, 101, 134, 170, 274, 302, 312] [45, 52, 55, 135, 229, 250, 303, 318] [7, 51, 136, 145, 187, 276, 295, 304] [71, 100, 109, 137, 177, 287, 305, 324] [50, 129, 138, 139, 254, 278, 306, 319] [51, 139, 158, 163, 191, 213, 307, 332] [29, 35, 37, 140, 230, 295, 308, 319] [46, 53, 56, 141, 244, 262, 297, 309] [118, 123, 129, 142, 212, 228, 264, 310] [91, 114, 143, 152, 177, 202, 310, 311] [3, 15, 28, 144, 228, 257, 293, 312] [4, 9, 39, 145, 175, 273, 279, 313] [24, 40, 109, 146, 204, 232, 246, 314] [25, 41, 110, 147, 209, 292, 315, 335] [11, 70, 131, 148, 197, 251, 304, 316] [136, 138, 143, 149, 243, 275, 317, 329] [57, 58, 150, 159, 234, 265, 318, 334] [5, 23, 25, 151, 226, 238, 272, 319] [66, 73, 150, 152, 222, 291, 297, 320] [67, 74, 146, 153, 250, 283, 321, 332] [3, 33, 117, 154, 172, 250, 299, 322] [4, 34, 89, 155, 247, 273, 282, 323] [23, 64, 73, 156, 248, 309, 317, 324] [13, 72, 90, 157, 268, 288, 314, 325] [99, 128, 138, 158, 208, 246, 290, 326] [72, 93, 159, 165, 182, 217, 326, 327] [100, 122, 147, 160, 201, 214, 277, 328] [108, 110, 142, 161, 190, 235, 254, 329] [7, 8, 44, 162, 271, 325, 330, 331] [20, 98, 101, 163, 256, 267, 328, 331] [78, 80, 160, 164, 240, 287, 314, 332] [79, 81, 152, 165, 273, 306, 310, 333] [10, 13, 14, 166, 270, 298, 305, 334] [127, 137, 153, 167, 178, 216, 300, 335]
H_Z (168 checks, sparse supports)
[0, 9, 59, 98, 168, 256, 269, 279] [1, 14, 77, 84, 169, 239, 285, 298] [2, 3, 32, 134, 170, 257, 294, 299] [3, 22, 101, 107, 171, 267, 312, 322] [4, 6, 47, 154, 172, 218, 313, 323] [5, 29, 50, 118, 173, 212, 221, 319] [6, 33, 88, 126, 174, 249, 289, 299] [7, 11, 29, 145, 175, 295, 304, 330] [5, 8, 42, 68, 176, 238, 258, 330] [9, 43, 137, 143, 177, 178, 243, 313] [10, 16, 58, 167, 178, 196, 203, 334] [11, 41, 48, 113, 179, 209, 221, 316] [5, 12, 19, 83, 180, 226, 260, 269] [8, 13, 57, 92, 181, 219, 325, 334] [14, 58, 62, 159, 182, 265, 286, 334] [15, 24, 76, 116, 183, 232, 266, 312] [16, 54, 60, 95, 184, 186, 231, 258] [14, 17, 26, 104, 173, 185, 270, 285] [1, 18, 21, 111, 186, 211, 220, 280] [19, 56, 66, 136, 187, 227, 243, 297] [8, 20, 30, 106, 188, 194, 203, 331] [13, 21, 64, 75, 189, 248, 288, 298] [22, 76, 80, 161, 190, 240, 274, 280] [23, 35, 100, 139, 191, 201, 319, 324] [24, 72, 78, 121, 192, 217, 280, 314] [22, 25, 37, 129, 193, 264, 315, 319] [20, 26, 70, 82, 194, 197, 256, 286] [1, 27, 40, 49, 195, 198, 211, 232] [18, 28, 91, 109, 177, 196, 204, 312] [29, 35, 47, 148, 197, 230, 241, 308] [30, 45, 74, 117, 198, 231, 242, 250] [0, 13, 31, 131, 199, 251, 268, 270] [16, 21, 32, 99, 200, 208, 220, 261] [33, 100, 103, 160, 201, 277, 287, 322] [34, 50, 125, 143, 202, 275, 278, 323] [35, 63, 90, 96, 203, 206, 268, 308] [33, 36, 52, 146, 204, 241, 245, 250] [31, 37, 94, 105, 180, 205, 266, 308] [3, 38, 55, 67, 200, 206, 250, 293] [2, 28, 39, 132, 207, 233, 293, 313] [40, 80, 90, 158, 208, 213, 285, 314] [15, 18, 41, 147, 209, 214, 257, 315] [28, 37, 42, 102, 168, 210, 230, 257] [43, 66, 117, 130, 184, 211, 222, 299] [44, 62, 121, 142, 185, 190, 212, 330] [45, 108, 114, 139, 202, 213, 254, 303] [43, 46, 64, 160, 214, 240, 255, 309] [24, 32, 47, 124, 195, 204, 215, 296] [48, 125, 128, 167, 183, 216, 241, 290] [49, 68, 85, 159, 207, 217, 234, 242] [27, 50, 81, 122, 214, 218, 289, 306] [48, 51, 60, 70, 219, 259, 304, 307] [18, 52, 120, 130, 220, 225, 296, 303] [6, 46, 53, 73, 192, 221, 297, 309] [4, 39, 54, 152, 202, 222, 273, 279] [27, 55, 86, 103, 189, 223, 294, 303] [2, 23, 56, 61, 174, 224, 238, 309] [39, 52, 57, 127, 216, 225, 279, 318] [45, 58, 59, 151, 226, 258, 259, 318] [6, 42, 59, 123, 199, 212, 227, 233] [60, 129, 142, 144, 169, 212, 228, 254] [61, 84, 101, 135, 170, 229, 255, 256] [36, 56, 62, 140, 194, 221, 230, 244] [60, 63, 86, 105, 195, 223, 231, 269] [39, 64, 71, 146, 175, 232, 283, 324] [9, 65, 81, 89, 189, 233, 273, 284] [66, 85, 116, 150, 210, 222, 234, 320] [67, 92, 108, 161, 184, 190, 235, 321] [38, 68, 91, 104, 170, 202, 236, 253] [66, 69, 78, 94, 176, 237, 240, 291] [2, 70, 89, 151, 199, 238, 247, 316] [10, 11, 71, 97, 197, 239, 260, 305] [7, 54, 72, 164, 240, 276, 325, 327] [38, 73, 109, 128, 241, 246, 320, 324] [4, 34, 74, 79, 242, 247, 273, 321] [54, 70, 75, 149, 205, 242, 243, 251] [9, 76, 77, 141, 175, 239, 244, 302] [11, 57, 77, 119, 179, 245, 251, 265] [78, 82, 146, 158, 236, 239, 246, 332] [79, 107, 126, 155, 237, 247, 294, 333] [51, 74, 80, 156, 248, 276, 286, 332] [78, 81, 109, 130, 232, 249, 287, 333] [82, 135, 153, 154, 178, 215, 250, 259] [51, 83, 111, 148, 187, 196, 213, 251] [12, 67, 84, 127, 171, 178, 252, 283] [33, 82, 85, 113, 172, 196, 253, 284] [8, 86, 138, 161, 208, 243, 254, 271] [16, 25, 87, 115, 203, 209, 255, 272] [10, 31, 88, 163, 191, 210, 256, 298] [12, 52, 89, 144, 181, 229, 257, 323] [0, 34, 44, 90, 171, 258, 282, 325] [31, 86, 91, 122, 201, 259, 302, 311] [28, 53, 92, 129, 228, 244, 260, 278] [45, 63, 93, 120, 223, 229, 261, 327] [4, 26, 94, 141, 173, 175, 262, 282] [15, 19, 95, 123, 168, 189, 228, 263] [12, 72, 96, 142, 182, 235, 264, 268] [53, 97, 132, 150, 193, 215, 265, 297] [7, 49, 98, 102, 187, 225, 266, 331] [72, 94, 99, 163, 213, 267, 288, 326] [14, 100, 101, 157, 268, 274, 305, 328] [19, 44, 75, 101, 193, 269, 302, 331] [88, 102, 113, 166, 220, 224, 227, 270] [69, 98, 103, 162, 181, 263, 267, 271] [63, 104, 132, 151, 171, 218, 272, 281] [105, 145, 155, 165, 187, 195, 217, 273] [69, 106, 118, 134, 188, 227, 264, 274] [20, 46, 107, 149, 262, 267, 275, 301] [48, 105, 108, 136, 218, 263, 276, 329] [13, 87, 109, 160, 277, 301, 305, 314] [24, 36, 110, 138, 246, 278, 315, 329] [57, 65, 111, 145, 186, 234, 276, 279] [71, 80, 108, 112, 173, 235, 280, 287] [0, 55, 113, 124, 174, 186, 229, 281] [17, 88, 114, 155, 176, 211, 282, 311] [65, 92, 115, 153, 183, 231, 283, 300] [10, 61, 116, 119, 207, 270, 284, 301] [3, 97, 117, 118, 172, 228, 285, 322] [26, 79, 91, 118, 174, 274, 286, 310] [95, 119, 137, 164, 219, 281, 287, 300] [7, 37, 120, 157, 252, 271, 288, 295] [23, 30, 119, 121, 248, 253, 272, 289] [20, 96, 122, 158, 185, 191, 290, 328] [71, 114, 123, 152, 177, 181, 291, 310] [96, 120, 124, 147, 169, 201, 252, 292] [22, 125, 126, 144, 188, 192, 293, 296] [30, 40, 99, 126, 198, 246, 261, 294] [41, 127, 136, 140, 233, 275, 295, 335] [84, 93, 112, 128, 206, 284, 296, 326] [81, 129, 141, 152, 245, 297, 306, 310] [83, 115, 130, 166, 207, 261, 263, 298] [42, 93, 131, 154, 182, 281, 299, 316] [21, 110, 132, 167, 190, 249, 292, 300] [35, 51, 87, 133, 191, 236, 295, 301] [17, 75, 83, 134, 180, 198, 210, 302] [43, 95, 103, 135, 184, 199, 200, 303] [1, 73, 136, 148, 249, 291, 304, 317] [25, 41, 137, 166, 238, 292, 305, 335] [17, 64, 138, 165, 266, 306, 317, 326] [15, 44, 79, 139, 271, 293, 306, 307] [27, 100, 114, 140, 177, 188, 214, 308] [87, 131, 141, 156, 179, 185, 197, 309] [29, 142, 143, 165, 182, 230, 310, 329] [23, 40, 102, 143, 204, 226, 311, 317] [112, 133, 134, 144, 176, 252, 289, 312] [36, 124, 131, 145, 169, 253, 304, 313] [104, 146, 157, 164, 179, 206, 314, 321] [77, 98, 112, 147, 180, 245, 315, 328] [61, 89, 122, 148, 224, 277, 282, 316] [56, 117, 149, 156, 172, 219, 262, 317] [65, 107, 135, 150, 200, 205, 318, 320] [106, 138, 140, 151, 168, 275, 290, 319] [32, 116, 133, 152, 225, 311, 320, 333] [50, 69, 110, 153, 192, 254, 321, 335] [25, 99, 106, 154, 226, 290, 292, 322] [58, 121, 128, 155, 208, 234, 236, 323] [38, 125, 137, 156, 194, 215, 216, 324] [110, 111, 157, 162, 209, 235, 237, 325] [5, 115, 158, 159, 265, 272, 307, 326] [34, 55, 127, 159, 247, 300, 318, 327] [67, 97, 160, 163, 193, 255, 328, 332] [49, 73, 149, 161, 222, 237, 248, 329] [53, 62, 85, 162, 183, 260, 262, 330] [59, 74, 162, 163, 205, 283, 307, 331] [47, 139, 153, 164, 216, 223, 278, 332] [68, 93, 133, 165, 217, 224, 327, 333] [76, 90, 150, 166, 170, 288, 291, 334] [46, 123, 147, 167, 244, 264, 277, 335]
raw submission JSON