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[[320,10,28]] d ≤
n
320
k
10
d
28
kd²/n
24.5
w
8
X/Z
1.14

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Distance

X/Z asymmetry 1.14 · d_X ≤ 28, d_Z ≤ 32 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 28 · witness weight 28 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-11
witness operator (support, 28 qubits)
[4, 11, 13, 35, 37, 40, 42, 50, 59, 66, 75, 76, 85, 88, 90, 92, 95, 133, 148, 155, 189, 200, 211, 226, 243, 279, 286, 315]
d_Z 32 · witness weight 32 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-11
witness operator (support, 32 qubits)
[12, 14, 15, 31, 58, 60, 75, 80, 108, 134, 136, 137, 144, 151, 173, 180, 201, 219, 220, 229, 230, 233, 235, 239, 240, 242, 271, 284, 285, 291, 292, 295]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×320 (2,4)×160 (3,6)×7040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 320 (2,4): 160 (2,6): 4160 (3,6): 7040 (3,8): 75200 (3,10): 7040
trapping sets H_Z (1,4)×320 (2,4)×160 (3,6)×7040 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 320 (2,4): 160 (2,6): 4160 (3,6): 7040 (3,8): 75200 (3,10): 7040

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Bivariate bicycle Z_16xZ_10.
model Beam 1.0 (claimed, not verified)
date 2026-11-11
family bivariate bicycle (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

[[320,10,28]] — bivariate bicycle code found by random search

Direction & hypothesis

Track cell weight-8 × unrestricted, CSS board. Random bivariate bicycle codes on larger tori (l up to 16, m up to 12) were screened for codes whose parameters are not dominated by existing entries, especially high-distance points at moderate n.

What was searched

A sweep of 200 random bivariate bicycle candidates with four monomials per polynomial, torus sizes l in [8,16], m in [6,12], n in [100,700], max check weight 8, screened at 300 RIS trials per side. Survivors were then re-screened at higher trial counts and submitted to the trusted validation gate. The gate's refutation pass found a lighter logical than the initial witness, and the candidate was repackaged at that depth.

Evidence trail

  • Surrogate screen: d <= 40 at 300 trials.
  • Gate refutation: found a logical of weight 28; final claim d <= 28.
  • Final claim: upper bound d <= 28, witness-backed, not exact.

Dead ends

Most random candidates were either board duplicates, WL-equivalent to existing entries, or dominated after the gate's refutation search. The initial witness over-claimed the distance and had to be tightened.

Tools

Model: Beam 1.0. Agent harness: OpenCode. Repo tooling: research/kit/search.py, research/kit/surrogate.py, research/kit/coordination.py, verify/validate_candidate.py.

Reproduction

Bivariate bicycle code on Z_16 x Z_10 with polynomials A and B as in the staged candidate's repro manifest; rebuild with research/kit/bb.py::build_bb.

Parity checks

X-checks 160 (max weight 8) · Z-checks 160 (max weight 8)
H_X (160 checks, sparse supports)
[52, 114, 129, 146, 203, 213, 301, 318] [53, 115, 120, 147, 204, 214, 302, 319] [54, 116, 121, 148, 205, 215, 303, 310] [55, 117, 122, 149, 206, 216, 304, 311] [56, 118, 123, 140, 207, 217, 305, 312] [57, 119, 124, 141, 208, 218, 306, 313] [58, 110, 125, 142, 209, 219, 307, 314] [59, 111, 126, 143, 200, 210, 308, 315] [50, 112, 127, 144, 201, 211, 309, 316] [51, 113, 128, 145, 202, 212, 300, 317] [62, 124, 139, 156, 168, 213, 223, 311] [63, 125, 130, 157, 169, 214, 224, 312] [64, 126, 131, 158, 160, 215, 225, 313] [65, 127, 132, 159, 161, 216, 226, 314] [66, 128, 133, 150, 162, 217, 227, 315] [67, 129, 134, 151, 163, 218, 228, 316] [68, 120, 135, 152, 164, 219, 229, 317] [69, 121, 136, 153, 165, 210, 220, 318] [60, 122, 137, 154, 166, 211, 221, 319] [61, 123, 138, 155, 167, 212, 222, 310] [6, 72, 134, 149, 161, 178, 223, 233] [7, 73, 135, 140, 162, 179, 224, 234] [8, 74, 136, 141, 163, 170, 225, 235] [9, 75, 137, 142, 164, 171, 226, 236] [0, 76, 138, 143, 165, 172, 227, 237] [1, 77, 139, 144, 166, 173, 228, 238] [2, 78, 130, 145, 167, 174, 229, 239] [3, 79, 131, 146, 168, 175, 220, 230] [4, 70, 132, 147, 169, 176, 221, 231] [5, 71, 133, 148, 160, 177, 222, 232] [16, 82, 144, 159, 171, 188, 233, 243] [17, 83, 145, 150, 172, 189, 234, 244] [18, 84, 146, 151, 173, 180, 235, 245] [19, 85, 147, 152, 174, 181, 236, 246] [10, 86, 148, 153, 175, 182, 237, 247] [11, 87, 149, 154, 176, 183, 238, 248] [12, 88, 140, 155, 177, 184, 239, 249] [13, 89, 141, 156, 178, 185, 230, 240] [14, 80, 142, 157, 179, 186, 231, 241] [15, 81, 143, 158, 170, 187, 232, 242] [9, 26, 92, 154, 181, 198, 243, 253] [0, 27, 93, 155, 182, 199, 244, 254] [1, 28, 94, 156, 183, 190, 245, 255] [2, 29, 95, 157, 184, 191, 246, 256] [3, 20, 96, 158, 185, 192, 247, 257] [4, 21, 97, 159, 186, 193, 248, 258] [5, 22, 98, 150, 187, 194, 249, 259] [6, 23, 99, 151, 188, 195, 240, 250] [7, 24, 90, 152, 189, 196, 241, 251] [8, 25, 91, 153, 180, 197, 242, 252] [4, 19, 36, 102, 191, 208, 253, 263] [5, 10, 37, 103, 192, 209, 254, 264] [6, 11, 38, 104, 193, 200, 255, 265] [7, 12, 39, 105, 194, 201, 256, 266] [8, 13, 30, 106, 195, 202, 257, 267] [9, 14, 31, 107, 196, 203, 258, 268] [0, 15, 32, 108, 197, 204, 259, 269] [1, 16, 33, 109, 198, 205, 250, 260] [2, 17, 34, 100, 199, 206, 251, 261] [3, 18, 35, 101, 190, 207, 252, 262] [14, 29, 46, 112, 201, 218, 263, 273] [15, 20, 47, 113, 202, 219, 264, 274] [16, 21, 48, 114, 203, 210, 265, 275] [17, 22, 49, 115, 204, 211, 266, 276] [18, 23, 40, 116, 205, 212, 267, 277] [19, 24, 41, 117, 206, 213, 268, 278] [10, 25, 42, 118, 207, 214, 269, 279] [11, 26, 43, 119, 208, 215, 260, 270] [12, 27, 44, 110, 209, 216, 261, 271] [13, 28, 45, 111, 200, 217, 262, 272] [24, 39, 56, 122, 211, 228, 273, 283] [25, 30, 57, 123, 212, 229, 274, 284] [26, 31, 58, 124, 213, 220, 275, 285] [27, 32, 59, 125, 214, 221, 276, 286] [28, 33, 50, 126, 215, 222, 277, 287] [29, 34, 51, 127, 216, 223, 278, 288] [20, 35, 52, 128, 217, 224, 279, 289] [21, 36, 53, 129, 218, 225, 270, 280] [22, 37, 54, 120, 219, 226, 271, 281] [23, 38, 55, 121, 210, 227, 272, 282] [34, 49, 66, 132, 221, 238, 283, 293] [35, 40, 67, 133, 222, 239, 284, 294] [36, 41, 68, 134, 223, 230, 285, 295] [37, 42, 69, 135, 224, 231, 286, 296] [38, 43, 60, 136, 225, 232, 287, 297] [39, 44, 61, 137, 226, 233, 288, 298] [30, 45, 62, 138, 227, 234, 289, 299] [31, 46, 63, 139, 228, 235, 280, 290] [32, 47, 64, 130, 229, 236, 281, 291] [33, 48, 65, 131, 220, 237, 282, 292] [44, 59, 76, 142, 231, 248, 293, 303] [45, 50, 77, 143, 232, 249, 294, 304] [46, 51, 78, 144, 233, 240, 295, 305] [47, 52, 79, 145, 234, 241, 296, 306] [48, 53, 70, 146, 235, 242, 297, 307] [49, 54, 71, 147, 236, 243, 298, 308] [40, 55, 72, 148, 237, 244, 299, 309] [41, 56, 73, 149, 238, 245, 290, 300] [42, 57, 74, 140, 239, 246, 291, 301] [43, 58, 75, 141, 230, 247, 292, 302] [54, 69, 86, 152, 241, 258, 303, 313] [55, 60, 87, 153, 242, 259, 304, 314] [56, 61, 88, 154, 243, 250, 305, 315] [57, 62, 89, 155, 244, 251, 306, 316] [58, 63, 80, 156, 245, 252, 307, 317] [59, 64, 81, 157, 246, 253, 308, 318] [50, 65, 82, 158, 247, 254, 309, 319] [51, 66, 83, 159, 248, 255, 300, 310] [52, 67, 84, 150, 249, 256, 301, 311] [53, 68, 85, 151, 240, 257, 302, 312] [2, 64, 79, 96, 163, 251, 268, 313] [3, 65, 70, 97, 164, 252, 269, 314] [4, 66, 71, 98, 165, 253, 260, 315] [5, 67, 72, 99, 166, 254, 261, 316] [6, 68, 73, 90, 167, 255, 262, 317] [7, 69, 74, 91, 168, 256, 263, 318] [8, 60, 75, 92, 169, 257, 264, 319] [9, 61, 76, 93, 160, 258, 265, 310] [0, 62, 77, 94, 161, 259, 266, 311] [1, 63, 78, 95, 162, 250, 267, 312] [12, 74, 89, 106, 163, 173, 261, 278] [13, 75, 80, 107, 164, 174, 262, 279] [14, 76, 81, 108, 165, 175, 263, 270] [15, 77, 82, 109, 166, 176, 264, 271] [16, 78, 83, 100, 167, 177, 265, 272] [17, 79, 84, 101, 168, 178, 266, 273] [18, 70, 85, 102, 169, 179, 267, 274] [19, 71, 86, 103, 160, 170, 268, 275] [10, 72, 87, 104, 161, 171, 269, 276] [11, 73, 88, 105, 162, 172, 260, 277] [22, 84, 99, 116, 173, 183, 271, 288] [23, 85, 90, 117, 174, 184, 272, 289] [24, 86, 91, 118, 175, 185, 273, 280] [25, 87, 92, 119, 176, 186, 274, 281] [26, 88, 93, 110, 177, 187, 275, 282] [27, 89, 94, 111, 178, 188, 276, 283] [28, 80, 95, 112, 179, 189, 277, 284] [29, 81, 96, 113, 170, 180, 278, 285] [20, 82, 97, 114, 171, 181, 279, 286] [21, 83, 98, 115, 172, 182, 270, 287] [32, 94, 109, 126, 183, 193, 281, 298] [33, 95, 100, 127, 184, 194, 282, 299] [34, 96, 101, 128, 185, 195, 283, 290] [35, 97, 102, 129, 186, 196, 284, 291] [36, 98, 103, 120, 187, 197, 285, 292] [37, 99, 104, 121, 188, 198, 286, 293] [38, 90, 105, 122, 189, 199, 287, 294] [39, 91, 106, 123, 180, 190, 288, 295] [30, 92, 107, 124, 181, 191, 289, 296] [31, 93, 108, 125, 182, 192, 280, 297] [42, 104, 119, 136, 193, 203, 291, 308] [43, 105, 110, 137, 194, 204, 292, 309] [44, 106, 111, 138, 195, 205, 293, 300] [45, 107, 112, 139, 196, 206, 294, 301] [46, 108, 113, 130, 197, 207, 295, 302] [47, 109, 114, 131, 198, 208, 296, 303] [48, 100, 115, 132, 199, 209, 297, 304] [49, 101, 116, 133, 190, 200, 298, 305] [40, 102, 117, 134, 191, 201, 299, 306] [41, 103, 118, 135, 192, 202, 290, 307]
H_Z (160 checks, sparse supports)
[12, 29, 117, 127, 184, 201, 216, 278] [13, 20, 118, 128, 185, 202, 217, 279] [14, 21, 119, 129, 186, 203, 218, 270] [15, 22, 110, 120, 187, 204, 219, 271] [16, 23, 111, 121, 188, 205, 210, 272] [17, 24, 112, 122, 189, 206, 211, 273] [18, 25, 113, 123, 180, 207, 212, 274] [19, 26, 114, 124, 181, 208, 213, 275] [10, 27, 115, 125, 182, 209, 214, 276] [11, 28, 116, 126, 183, 200, 215, 277] [22, 39, 127, 137, 194, 211, 226, 288] [23, 30, 128, 138, 195, 212, 227, 289] [24, 31, 129, 139, 196, 213, 228, 280] [25, 32, 120, 130, 197, 214, 229, 281] [26, 33, 121, 131, 198, 215, 220, 282] [27, 34, 122, 132, 199, 216, 221, 283] [28, 35, 123, 133, 190, 217, 222, 284] [29, 36, 124, 134, 191, 218, 223, 285] [20, 37, 125, 135, 192, 219, 224, 286] [21, 38, 126, 136, 193, 210, 225, 287] [32, 49, 137, 147, 204, 221, 236, 298] [33, 40, 138, 148, 205, 222, 237, 299] [34, 41, 139, 149, 206, 223, 238, 290] [35, 42, 130, 140, 207, 224, 239, 291] [36, 43, 131, 141, 208, 225, 230, 292] [37, 44, 132, 142, 209, 226, 231, 293] [38, 45, 133, 143, 200, 227, 232, 294] [39, 46, 134, 144, 201, 228, 233, 295] [30, 47, 135, 145, 202, 229, 234, 296] [31, 48, 136, 146, 203, 220, 235, 297] [42, 59, 147, 157, 214, 231, 246, 308] [43, 50, 148, 158, 215, 232, 247, 309] [44, 51, 149, 159, 216, 233, 248, 300] [45, 52, 140, 150, 217, 234, 249, 301] [46, 53, 141, 151, 218, 235, 240, 302] [47, 54, 142, 152, 219, 236, 241, 303] [48, 55, 143, 153, 210, 237, 242, 304] [49, 56, 144, 154, 211, 238, 243, 305] [40, 57, 145, 155, 212, 239, 244, 306] [41, 58, 146, 156, 213, 230, 245, 307] [7, 52, 69, 157, 224, 241, 256, 318] [8, 53, 60, 158, 225, 242, 257, 319] [9, 54, 61, 159, 226, 243, 258, 310] [0, 55, 62, 150, 227, 244, 259, 311] [1, 56, 63, 151, 228, 245, 250, 312] [2, 57, 64, 152, 229, 246, 251, 313] [3, 58, 65, 153, 220, 247, 252, 314] [4, 59, 66, 154, 221, 248, 253, 315] [5, 50, 67, 155, 222, 249, 254, 316] [6, 51, 68, 156, 223, 240, 255, 317] [7, 17, 62, 79, 168, 234, 251, 266] [8, 18, 63, 70, 169, 235, 252, 267] [9, 19, 64, 71, 160, 236, 253, 268] [0, 10, 65, 72, 161, 237, 254, 269] [1, 11, 66, 73, 162, 238, 255, 260] [2, 12, 67, 74, 163, 239, 256, 261] [3, 13, 68, 75, 164, 230, 257, 262] [4, 14, 69, 76, 165, 231, 258, 263] [5, 15, 60, 77, 166, 232, 259, 264] [6, 16, 61, 78, 167, 233, 250, 265] [17, 27, 72, 89, 178, 244, 261, 276] [18, 28, 73, 80, 179, 245, 262, 277] [19, 29, 74, 81, 170, 246, 263, 278] [10, 20, 75, 82, 171, 247, 264, 279] [11, 21, 76, 83, 172, 248, 265, 270] [12, 22, 77, 84, 173, 249, 266, 271] [13, 23, 78, 85, 174, 240, 267, 272] [14, 24, 79, 86, 175, 241, 268, 273] [15, 25, 70, 87, 176, 242, 269, 274] [16, 26, 71, 88, 177, 243, 260, 275] [27, 37, 82, 99, 188, 254, 271, 286] [28, 38, 83, 90, 189, 255, 272, 287] [29, 39, 84, 91, 180, 256, 273, 288] [20, 30, 85, 92, 181, 257, 274, 289] [21, 31, 86, 93, 182, 258, 275, 280] [22, 32, 87, 94, 183, 259, 276, 281] [23, 33, 88, 95, 184, 250, 277, 282] [24, 34, 89, 96, 185, 251, 278, 283] [25, 35, 80, 97, 186, 252, 279, 284] [26, 36, 81, 98, 187, 253, 270, 285] [37, 47, 92, 109, 198, 264, 281, 296] [38, 48, 93, 100, 199, 265, 282, 297] [39, 49, 94, 101, 190, 266, 283, 298] [30, 40, 95, 102, 191, 267, 284, 299] [31, 41, 96, 103, 192, 268, 285, 290] [32, 42, 97, 104, 193, 269, 286, 291] [33, 43, 98, 105, 194, 260, 287, 292] [34, 44, 99, 106, 195, 261, 288, 293] [35, 45, 90, 107, 196, 262, 289, 294] [36, 46, 91, 108, 197, 263, 280, 295] [47, 57, 102, 119, 208, 274, 291, 306] [48, 58, 103, 110, 209, 275, 292, 307] [49, 59, 104, 111, 200, 276, 293, 308] [40, 50, 105, 112, 201, 277, 294, 309] [41, 51, 106, 113, 202, 278, 295, 300] [42, 52, 107, 114, 203, 279, 296, 301] [43, 53, 108, 115, 204, 270, 297, 302] [44, 54, 109, 116, 205, 271, 298, 303] [45, 55, 100, 117, 206, 272, 299, 304] [46, 56, 101, 118, 207, 273, 290, 305] [57, 67, 112, 129, 218, 284, 301, 316] [58, 68, 113, 120, 219, 285, 302, 317] [59, 69, 114, 121, 210, 286, 303, 318] [50, 60, 115, 122, 211, 287, 304, 319] [51, 61, 116, 123, 212, 288, 305, 310] [52, 62, 117, 124, 213, 289, 306, 311] [53, 63, 118, 125, 214, 280, 307, 312] [54, 64, 119, 126, 215, 281, 308, 313] [55, 65, 110, 127, 216, 282, 309, 314] [56, 66, 111, 128, 217, 283, 300, 315] [67, 77, 122, 139, 166, 228, 294, 311] [68, 78, 123, 130, 167, 229, 295, 312] [69, 79, 124, 131, 168, 220, 296, 313] [60, 70, 125, 132, 169, 221, 297, 314] [61, 71, 126, 133, 160, 222, 298, 315] [62, 72, 127, 134, 161, 223, 299, 316] [63, 73, 128, 135, 162, 224, 290, 317] [64, 74, 129, 136, 163, 225, 291, 318] [65, 75, 120, 137, 164, 226, 292, 319] [66, 76, 121, 138, 165, 227, 293, 310] [77, 87, 132, 149, 161, 176, 238, 304] [78, 88, 133, 140, 162, 177, 239, 305] [79, 89, 134, 141, 163, 178, 230, 306] [70, 80, 135, 142, 164, 179, 231, 307] [71, 81, 136, 143, 165, 170, 232, 308] [72, 82, 137, 144, 166, 171, 233, 309] [73, 83, 138, 145, 167, 172, 234, 300] [74, 84, 139, 146, 168, 173, 235, 301] [75, 85, 130, 147, 169, 174, 236, 302] [76, 86, 131, 148, 160, 175, 237, 303] [87, 97, 142, 159, 171, 186, 248, 314] [88, 98, 143, 150, 172, 187, 249, 315] [89, 99, 144, 151, 173, 188, 240, 316] [80, 90, 145, 152, 174, 189, 241, 317] [81, 91, 146, 153, 175, 180, 242, 318] [82, 92, 147, 154, 176, 181, 243, 319] [83, 93, 148, 155, 177, 182, 244, 310] [84, 94, 149, 156, 178, 183, 245, 311] [85, 95, 140, 157, 179, 184, 246, 312] [86, 96, 141, 158, 170, 185, 247, 313] [9, 97, 107, 152, 164, 181, 196, 258] [0, 98, 108, 153, 165, 182, 197, 259] [1, 99, 109, 154, 166, 183, 198, 250] [2, 90, 100, 155, 167, 184, 199, 251] [3, 91, 101, 156, 168, 185, 190, 252] [4, 92, 102, 157, 169, 186, 191, 253] [5, 93, 103, 158, 160, 187, 192, 254] [6, 94, 104, 159, 161, 188, 193, 255] [7, 95, 105, 150, 162, 189, 194, 256] [8, 96, 106, 151, 163, 180, 195, 257] [2, 19, 107, 117, 174, 191, 206, 268] [3, 10, 108, 118, 175, 192, 207, 269] [4, 11, 109, 119, 176, 193, 208, 260] [5, 12, 100, 110, 177, 194, 209, 261] [6, 13, 101, 111, 178, 195, 200, 262] [7, 14, 102, 112, 179, 196, 201, 263] [8, 15, 103, 113, 170, 197, 202, 264] [9, 16, 104, 114, 171, 198, 203, 265] [0, 17, 105, 115, 172, 199, 204, 266] [1, 18, 106, 116, 173, 190, 205, 267]
Code ID 320-10-28 · download JSON · raw on GitHub