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[[310,16,23]] d ≤
n
310
k
16
d
23
kd²/n
27.303
w
9

Share this result

Distance

d_X 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[29, 39, 49, 66, 70, 73, 76, 93, 111, 121, 128, 136, 156, 159, 185, 217, 218, 225, 240, 247, 249, 258, 297]
d_Z 23 · witness weight 23 (claimed upper_bound)
witness operator (support, 23 qubits)
[5, 11, 12, 22, 24, 63, 67, 79, 94, 99, 103, 124, 128, 132, 153, 159, 190, 199, 200, 226, 247, 258, 270]
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 metacyclic group Z_31 x| Z_5 (order 155); n=310, k=16, max check weight 9.
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

[[310,16,23]] — 2BGA on the metacyclic group Z_31 x| Z_5

Direction & hypothesis

Advance the any-weight frontier at n=310 with a two-block group-algebra code over the non-abelian metacyclic group Z_31 x| Z_5 (order 155), a low-mined family. Moderate distance, where the RIS surrogate is reliable.

What was searched

2BGA H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over non-abelian groups (dicyclic, dihedral, metacyclic), 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: Z_31 x| Z_5, a weight 4, b weight 5, max check weight 9, k = 16.

Evidence trail

Witness-backed upper bound d <= 23 (weight-23 logical on each side, so d = 23). Confirmed on the search machine at its heavy budget (80M x 3 fresh seeds, pair-depth 20) and re-confirmed here: a fresh 600k-trial RIS search independently returns d <= 23. The verifier accepts the code at kd^2/n = 27.3.

Dead ends

High-distance variants of this construction over-estimate distance under light search and collapse under deeper search (a sibling [[672,15,33]] screen stored witnesses of weight ~90, inconsistent with the claimed distance, and was discarded). The search is confined to the moderate-distance regime.

Model & harness

Found by a continual non-abelian 2BGA search (Claude Opus 4.8) on the repo's gf2_fast RIS core; witnesses re-validated and the code run through verify/qldpc_verify.py during packaging.

Reproduction

Z_31 x| Z_5 = research/kit/group_algebra.metacyclic(31, 5, 2). Build the 2BGA via build_2bga(mul, a, b) with a = [111, 72, 128, 47], b = [1, 44, 133, 11, 69] (element indices) to obtain [[310,16,23]].

Parity checks

X-checks 155 · Z-checks 155
H_X (155 checks, sparse supports)
[63, 104, 108, 122, 159, 181, 231, 257, 309] [64, 100, 109, 123, 155, 203, 207, 300, 307] [60, 101, 105, 124, 156, 249, 258, 291, 303] [61, 102, 106, 120, 157, 180, 204, 272, 294] [62, 103, 107, 121, 158, 206, 233, 245, 270] [29, 103, 142, 148, 159, 164, 186, 236, 262] [25, 104, 143, 149, 157, 160, 208, 212, 305] [26, 100, 144, 145, 161, 254, 263, 296, 308] [27, 101, 140, 146, 162, 185, 209, 277, 299] [28, 102, 141, 147, 163, 211, 238, 250, 275] [7, 33, 109, 143, 164, 169, 191, 241, 267] [8, 34, 105, 144, 155, 162, 165, 213, 217] [9, 30, 106, 140, 158, 166, 259, 268, 301] [5, 31, 107, 141, 167, 190, 214, 282, 304] [6, 32, 108, 142, 168, 216, 243, 255, 280] [27, 28, 34, 73, 169, 174, 196, 246, 272] [28, 29, 30, 74, 160, 167, 170, 218, 222] [25, 29, 31, 70, 163, 171, 264, 273, 306] [25, 26, 32, 71, 172, 195, 219, 287, 309] [26, 27, 33, 72, 173, 221, 248, 260, 285] [47, 68, 113, 114, 174, 179, 201, 251, 277] [48, 69, 110, 114, 165, 172, 175, 223, 227] [49, 65, 110, 111, 156, 168, 176, 269, 278] [45, 66, 111, 112, 159, 177, 200, 224, 292] [46, 67, 112, 113, 178, 226, 253, 265, 290] [39, 67, 108, 153, 179, 184, 206, 256, 282] [35, 68, 109, 154, 170, 177, 180, 228, 232] [36, 69, 105, 150, 161, 173, 181, 274, 283] [37, 65, 106, 151, 164, 182, 205, 229, 297] [38, 66, 107, 152, 183, 231, 258, 270, 295] [38, 87, 119, 148, 184, 189, 211, 261, 287] [39, 88, 115, 149, 175, 182, 185, 233, 237] [35, 89, 116, 145, 166, 178, 186, 279, 288] [36, 85, 117, 146, 169, 187, 210, 234, 302] [37, 86, 118, 147, 188, 236, 263, 275, 300] [33, 44, 78, 107, 189, 194, 216, 266, 292] [34, 40, 79, 108, 180, 187, 190, 238, 242] [30, 41, 75, 109, 171, 183, 191, 284, 293] [31, 42, 76, 105, 174, 192, 215, 239, 307] [32, 43, 77, 106, 193, 241, 268, 280, 305] [73, 118, 124, 127, 194, 199, 221, 271, 297] [74, 119, 120, 128, 185, 192, 195, 243, 247] [70, 115, 121, 129, 176, 188, 196, 289, 298] [71, 116, 122, 125, 157, 179, 197, 220, 244] [72, 117, 123, 126, 155, 198, 246, 273, 285] [3, 49, 113, 147, 199, 204, 226, 276, 302] [4, 45, 114, 148, 190, 197, 200, 248, 252] [0, 46, 110, 149, 181, 193, 201, 294, 303] [1, 47, 111, 145, 162, 184, 202, 225, 249] [2, 48, 112, 146, 160, 203, 251, 278, 290] [12, 43, 129, 153, 204, 209, 231, 281, 307] [13, 44, 125, 154, 195, 202, 205, 253, 257] [14, 40, 126, 150, 186, 198, 206, 299, 308] [10, 41, 127, 151, 167, 189, 207, 230, 254] [11, 42, 128, 152, 165, 208, 256, 283, 295] [32, 38, 54, 83, 157, 209, 214, 236, 286] [33, 39, 50, 84, 200, 207, 210, 258, 262] [34, 35, 51, 80, 158, 191, 203, 211, 304] [30, 36, 52, 81, 172, 194, 212, 235, 259] [31, 37, 53, 82, 170, 213, 261, 288, 300] [52, 78, 123, 134, 162, 214, 219, 241, 291] [53, 79, 124, 130, 205, 212, 215, 263, 267] [54, 75, 120, 131, 163, 196, 208, 216, 309] [50, 76, 121, 132, 177, 199, 217, 240, 264] [51, 77, 122, 133, 175, 218, 266, 293, 305] [8, 59, 72, 118, 167, 219, 224, 246, 296] [9, 55, 73, 119, 210, 217, 220, 268, 272] [5, 56, 74, 115, 159, 168, 201, 213, 221] [6, 57, 70, 116, 182, 204, 222, 245, 269] [7, 58, 71, 117, 155, 180, 223, 271, 298] [3, 48, 92, 139, 172, 224, 229, 251, 301] [4, 49, 93, 135, 215, 222, 225, 273, 277] [0, 45, 94, 136, 164, 173, 206, 218, 226] [1, 46, 90, 137, 187, 209, 227, 250, 274] [2, 47, 91, 138, 160, 185, 228, 276, 303] [43, 64, 88, 112, 177, 229, 234, 256, 306] [44, 60, 89, 113, 220, 227, 230, 278, 282] [40, 61, 85, 114, 169, 178, 211, 223, 231] [41, 62, 86, 110, 192, 214, 232, 255, 279] [42, 63, 87, 111, 165, 190, 233, 281, 308] [83, 128, 132, 144, 156, 182, 234, 239, 261] [84, 129, 133, 140, 225, 232, 235, 283, 287] [80, 125, 134, 141, 174, 183, 216, 228, 236] [81, 126, 130, 142, 197, 219, 237, 260, 284] [82, 127, 131, 143, 158, 170, 195, 238, 286] [13, 69, 123, 152, 161, 187, 239, 244, 266] [14, 65, 124, 153, 230, 237, 240, 288, 292] [10, 66, 120, 154, 179, 188, 221, 233, 241] [11, 67, 121, 150, 202, 224, 242, 265, 289] [12, 68, 122, 151, 163, 175, 200, 243, 291] [8, 17, 53, 149, 166, 192, 244, 249, 271] [9, 18, 54, 145, 235, 242, 245, 293, 297] [5, 19, 50, 146, 184, 193, 226, 238, 246] [6, 15, 51, 147, 207, 229, 247, 270, 294] [7, 16, 52, 148, 168, 180, 205, 248, 296] [37, 48, 74, 93, 171, 197, 249, 254, 276] [38, 49, 70, 94, 240, 247, 250, 298, 302] [39, 45, 71, 90, 189, 198, 231, 243, 251] [35, 46, 72, 91, 212, 234, 252, 275, 299] [36, 47, 73, 92, 173, 185, 210, 253, 301] [57, 88, 133, 154, 176, 202, 254, 259, 281] [58, 89, 134, 150, 245, 252, 255, 303, 307] [59, 85, 130, 151, 194, 203, 236, 248, 256] [55, 86, 131, 152, 217, 239, 257, 280, 304] [56, 87, 132, 153, 178, 190, 215, 258, 306] [18, 77, 79, 128, 181, 207, 259, 264, 286] [19, 75, 78, 129, 157, 250, 257, 260, 308] [15, 76, 79, 125, 199, 208, 241, 253, 261] [16, 75, 77, 126, 222, 244, 262, 285, 309] [17, 76, 78, 127, 156, 183, 195, 220, 263] [4, 13, 58, 97, 186, 212, 264, 269, 291] [0, 14, 59, 98, 158, 162, 255, 262, 265] [1, 10, 55, 99, 204, 213, 246, 258, 266] [2, 11, 56, 95, 159, 227, 249, 267, 290] [3, 12, 57, 96, 161, 188, 200, 225, 268] [53, 84, 98, 117, 191, 217, 269, 274, 296] [54, 80, 99, 118, 163, 167, 260, 267, 270] [50, 81, 95, 119, 209, 218, 251, 263, 271] [51, 82, 96, 115, 164, 232, 254, 272, 295] [52, 83, 97, 116, 166, 193, 205, 230, 273] [9, 93, 137, 138, 196, 222, 274, 279, 301] [5, 94, 138, 139, 168, 172, 265, 272, 275] [6, 90, 135, 139, 214, 223, 256, 268, 276] [7, 91, 135, 136, 169, 237, 259, 277, 300] [8, 92, 136, 137, 171, 198, 210, 235, 278] [2, 23, 89, 133, 201, 227, 279, 284, 306] [3, 24, 85, 134, 173, 177, 270, 277, 280] [4, 20, 86, 130, 219, 228, 261, 273, 281] [0, 21, 87, 131, 174, 242, 264, 282, 305] [1, 22, 88, 132, 176, 203, 215, 240, 283] [14, 18, 22, 63, 156, 206, 232, 284, 289] [10, 19, 23, 64, 178, 182, 275, 282, 285] [11, 15, 24, 60, 224, 233, 266, 278, 286] [12, 16, 20, 61, 155, 179, 247, 269, 287] [13, 17, 21, 62, 181, 208, 220, 245, 288] [42, 58, 94, 103, 161, 211, 237, 289, 294] [43, 59, 90, 104, 183, 187, 280, 287, 290] [44, 55, 91, 100, 229, 238, 271, 283, 291] [40, 56, 92, 101, 160, 184, 252, 274, 292] [41, 57, 93, 102, 186, 213, 225, 250, 293] [19, 62, 98, 143, 166, 216, 242, 294, 299] [15, 63, 99, 144, 188, 192, 285, 292, 295] [16, 64, 95, 140, 234, 243, 276, 288, 296] [17, 60, 96, 141, 165, 189, 257, 279, 297] [18, 61, 97, 142, 191, 218, 230, 255, 298] [28, 82, 99, 138, 171, 221, 247, 299, 304] [29, 83, 95, 139, 193, 197, 290, 297, 300] [25, 84, 96, 135, 239, 248, 281, 293, 301] [26, 80, 97, 136, 170, 194, 262, 284, 302] [27, 81, 98, 137, 196, 223, 235, 260, 303] [23, 24, 68, 102, 176, 226, 252, 304, 309] [20, 24, 69, 103, 198, 202, 295, 302, 305] [20, 21, 65, 104, 244, 253, 286, 298, 306] [21, 22, 66, 100, 175, 199, 267, 289, 307] [22, 23, 67, 101, 201, 228, 240, 265, 308]
H_Z (155 checks, sparse supports)
[1, 11, 44, 69, 133, 202, 227, 266, 283] [2, 22, 80, 109, 130, 203, 228, 267, 284] [3, 6, 43, 55, 106, 204, 229, 268, 280] [4, 12, 57, 84, 111, 200, 225, 269, 281] [0, 5, 23, 67, 113, 201, 226, 265, 282] [6, 16, 49, 74, 138, 168, 222, 247, 276] [7, 27, 85, 114, 135, 169, 223, 248, 277] [8, 11, 48, 60, 111, 165, 224, 249, 278] [9, 17, 62, 89, 116, 166, 220, 245, 279] [5, 10, 28, 72, 118, 167, 221, 246, 275] [11, 21, 54, 79, 143, 208, 242, 267, 286] [12, 32, 90, 119, 140, 209, 243, 268, 287] [13, 16, 53, 65, 116, 205, 244, 269, 288] [14, 22, 67, 94, 121, 206, 240, 265, 289] [10, 15, 33, 77, 123, 207, 241, 266, 285] [16, 26, 59, 84, 148, 248, 262, 287, 296] [17, 37, 95, 124, 145, 249, 263, 288, 297] [18, 21, 58, 70, 121, 245, 264, 289, 298] [19, 27, 72, 99, 126, 246, 260, 285, 299] [15, 20, 38, 82, 128, 247, 261, 286, 295] [21, 31, 64, 89, 153, 282, 288, 306, 307] [22, 42, 100, 129, 150, 283, 289, 307, 308] [23, 26, 63, 75, 126, 284, 285, 308, 309] [24, 32, 77, 104, 131, 280, 286, 305, 309] [20, 25, 43, 87, 133, 281, 287, 305, 306] [3, 26, 36, 69, 94, 161, 172, 173, 302] [0, 27, 47, 105, 134, 162, 173, 174, 303] [28, 31, 68, 80, 131, 163, 170, 174, 304] [29, 37, 82, 109, 136, 164, 170, 171, 300] [25, 30, 48, 92, 138, 160, 171, 172, 301] [8, 31, 41, 74, 99, 167, 171, 192, 213] [5, 32, 52, 110, 139, 168, 172, 193, 214] [33, 36, 73, 85, 136, 169, 173, 194, 210] [34, 42, 87, 114, 141, 165, 174, 190, 211] [30, 35, 53, 97, 143, 166, 170, 191, 212] [13, 36, 46, 79, 104, 181, 187, 212, 253] [10, 37, 57, 115, 144, 182, 188, 213, 254] [38, 41, 78, 90, 141, 183, 189, 214, 250] [39, 47, 92, 119, 146, 184, 185, 210, 251] [35, 40, 58, 102, 148, 180, 186, 211, 252] [18, 41, 51, 84, 109, 191, 207, 232, 293] [15, 42, 62, 120, 149, 192, 208, 233, 294] [43, 46, 83, 95, 146, 193, 209, 234, 290] [44, 52, 97, 124, 151, 194, 205, 230, 291] [40, 45, 63, 107, 153, 190, 206, 231, 292] [23, 46, 56, 89, 114, 178, 201, 227, 252] [20, 47, 67, 125, 154, 179, 202, 228, 253] [48, 51, 88, 100, 151, 175, 203, 229, 254] [1, 49, 57, 102, 129, 176, 204, 225, 250] [3, 45, 50, 68, 112, 177, 200, 226, 251] [28, 51, 61, 94, 119, 211, 218, 247, 272] [4, 25, 52, 72, 130, 212, 219, 248, 273] [1, 53, 56, 93, 105, 213, 215, 249, 274] [6, 54, 62, 107, 134, 214, 216, 245, 270] [8, 50, 55, 73, 117, 210, 217, 246, 271] [33, 56, 66, 99, 124, 221, 258, 267, 292] [9, 30, 57, 77, 135, 222, 259, 268, 293] [6, 58, 61, 98, 110, 223, 255, 269, 294] [11, 59, 67, 112, 139, 224, 256, 265, 290] [13, 55, 60, 78, 122, 220, 257, 266, 291] [38, 61, 71, 104, 129, 157, 231, 287, 298] [14, 35, 62, 82, 140, 158, 232, 288, 299] [11, 63, 66, 103, 115, 159, 233, 289, 295] [16, 64, 72, 117, 144, 155, 234, 285, 296] [18, 60, 65, 83, 127, 156, 230, 286, 297] [43, 66, 76, 109, 134, 177, 183, 241, 307] [19, 40, 67, 87, 145, 178, 184, 242, 308] [16, 68, 71, 108, 120, 179, 180, 243, 309] [21, 69, 77, 122, 149, 175, 181, 244, 305] [23, 65, 70, 88, 132, 176, 182, 240, 306] [48, 71, 81, 114, 139, 172, 197, 223, 251] [24, 45, 72, 92, 150, 173, 198, 224, 252] [21, 73, 76, 113, 125, 174, 199, 220, 253] [26, 74, 82, 127, 154, 170, 195, 221, 254] [28, 70, 75, 93, 137, 171, 196, 222, 250] [53, 76, 86, 119, 144, 192, 217, 261, 263] [0, 29, 50, 77, 97, 193, 218, 262, 264] [26, 78, 81, 118, 130, 194, 219, 260, 263] [4, 31, 79, 87, 132, 190, 215, 261, 264] [33, 75, 80, 98, 142, 191, 216, 260, 262] [58, 81, 91, 124, 149, 212, 237, 271, 303] [5, 34, 55, 82, 102, 213, 238, 272, 304] [31, 83, 86, 123, 135, 214, 239, 273, 300] [9, 36, 84, 92, 137, 210, 235, 274, 301] [38, 80, 85, 103, 147, 211, 236, 270, 302] [63, 86, 96, 129, 154, 188, 232, 257, 281] [10, 39, 60, 87, 107, 189, 233, 258, 282] [36, 88, 91, 128, 140, 185, 234, 259, 283] [14, 41, 89, 97, 142, 186, 230, 255, 284] [43, 85, 90, 108, 152, 187, 231, 256, 280] [4, 68, 91, 101, 134, 228, 252, 277, 291] [15, 44, 65, 92, 112, 229, 253, 278, 292] [41, 93, 96, 133, 145, 225, 254, 279, 293] [19, 46, 94, 102, 147, 226, 250, 275, 294] [2, 48, 90, 95, 113, 227, 251, 276, 290] [9, 73, 96, 106, 139, 268, 272, 297, 301] [20, 49, 70, 97, 117, 269, 273, 298, 302] [46, 98, 101, 138, 150, 265, 274, 299, 303] [24, 51, 99, 107, 152, 266, 270, 295, 304] [7, 53, 95, 100, 118, 267, 271, 296, 300] [14, 78, 101, 111, 144, 156, 162, 292, 308] [25, 54, 75, 102, 122, 157, 163, 293, 309] [0, 51, 103, 106, 143, 158, 164, 294, 305] [2, 29, 56, 104, 112, 159, 160, 290, 306] [12, 58, 100, 105, 123, 155, 161, 291, 307] [19, 83, 106, 116, 149, 157, 166, 182, 193] [30, 59, 80, 107, 127, 158, 167, 183, 194] [5, 56, 108, 111, 148, 159, 168, 184, 190] [7, 34, 61, 109, 117, 155, 169, 180, 191] [17, 63, 105, 110, 128, 156, 165, 181, 192] [24, 88, 111, 121, 154, 176, 177, 202, 233] [35, 64, 85, 112, 132, 177, 178, 203, 234] [10, 61, 113, 116, 153, 178, 179, 204, 230] [12, 39, 66, 114, 122, 175, 179, 200, 231] [22, 68, 110, 115, 133, 175, 176, 201, 232] [4, 29, 93, 116, 126, 186, 197, 222, 273] [40, 69, 90, 117, 137, 187, 198, 223, 274] [3, 15, 66, 118, 121, 188, 199, 224, 270] [17, 44, 71, 119, 127, 189, 195, 220, 271] [27, 73, 115, 120, 138, 185, 196, 221, 272] [9, 34, 98, 121, 131, 158, 196, 217, 242] [45, 74, 95, 122, 142, 159, 197, 218, 243] [8, 20, 71, 123, 126, 155, 198, 219, 244] [22, 49, 76, 124, 132, 156, 199, 215, 240] [32, 78, 120, 125, 143, 157, 195, 216, 241] [14, 39, 103, 126, 136, 198, 206, 237, 262] [50, 79, 100, 127, 147, 199, 207, 238, 263] [13, 25, 76, 128, 131, 195, 208, 239, 264] [27, 54, 81, 129, 137, 196, 209, 235, 260] [37, 83, 125, 130, 148, 197, 205, 236, 261] [19, 44, 108, 131, 141, 216, 238, 257, 282] [55, 84, 105, 132, 152, 217, 239, 258, 283] [18, 30, 81, 133, 136, 218, 235, 259, 284] [32, 59, 86, 134, 142, 219, 236, 255, 280] [42, 88, 130, 135, 153, 215, 237, 256, 281] [24, 49, 113, 136, 146, 226, 277, 278, 302] [2, 60, 89, 110, 137, 227, 278, 279, 303] [23, 35, 86, 138, 141, 228, 275, 279, 304] [37, 64, 91, 139, 147, 229, 275, 276, 300] [3, 47, 93, 135, 140, 225, 276, 277, 301] [29, 54, 118, 141, 151, 163, 167, 236, 297] [7, 65, 94, 115, 142, 164, 168, 237, 298] [28, 40, 91, 143, 146, 160, 169, 238, 299] [42, 69, 96, 144, 152, 161, 165, 239, 295] [8, 52, 98, 140, 145, 162, 166, 235, 296] [1, 34, 59, 123, 146, 162, 187, 203, 246] [12, 70, 99, 120, 147, 163, 188, 204, 247] [33, 45, 96, 148, 151, 164, 189, 200, 248] [2, 47, 74, 101, 149, 160, 185, 201, 249] [13, 57, 103, 145, 150, 161, 186, 202, 245] [6, 39, 64, 128, 151, 182, 207, 243, 256] [17, 75, 104, 125, 152, 183, 208, 244, 257] [1, 38, 50, 101, 153, 184, 209, 240, 258] [7, 52, 79, 106, 154, 180, 205, 241, 259] [0, 18, 62, 108, 150, 181, 206, 242, 255]
raw submission JSON