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[[318,4,26]] d ≤
n
318
k
4
d
26
kd²/n
8.503
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 26, d_Z ≤ 26 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 26 · witness weight 26 (claimed upper_bound)
witness operator (support, 26 qubits)
[5, 14, 17, 26, 31, 34, 40, 43, 49, 81, 87, 98, 107, 113, 122, 131, 145, 179, 183, 217, 237, 257, 294, 304, 309, 313]
d_Z 26 · witness weight 26 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 3×108 trials · survived 3×108 trials · 2026-09-25
witness operator (support, 26 qubits)
[3, 6, 9, 13, 18, 19, 25, 30, 37, 41, 49, 54, 61, 146, 150, 151, 154, 163, 165, 167, 172, 174, 181, 190, 215, 315]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×318 (2,4)×2385 (3,3)×318 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 318 (2,4): 2385 (3,3): 318 (3,5): 22896 (3,7): 3180
trapping sets H_Z (1,3)×318 (2,4)×2385 (3,3)×318 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 318 (2,4): 2385 (3,3): 318 (3,5): 22896 (3,7): 3180

Construction & provenance

authors Zijian Liang and Ke Liu and Hao Song and Yu-An Chen
provenance literature baseline
construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x3y-4, g(x,y)=1+y+x-1y-3 on the abelian quotient group Z2/L with twist basis a_1=[0, 159], a_2=[1, 17] (n=2|det[a_1,a_2]|=2*159). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model human (claimed, not verified)
reference arXiv:2503.03827
date 2025
notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[318,4,26]]: distance revision of the board's [[318,4,26]] entry

Revision history

The entry keeps its parameters [[318,4,26]]. The code, its checks, its layout, and its original provenance are unchanged; only the distance block is corrected. A GPU random-information-set search (verify/ris_gpu.py, recover mode, 300,000,000 trials per side, pair depth 0, seed 4101) exhibits a weight-26 Z logical, so the previous witness-backed bound was overstated on that side; the overall distance d = 26 is unchanged. Each lighter witness is carried in the entry with the budget it was found at; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

Every GPU proposal was re-validated on the CPU against the committed check matrices (in the kernel of the opposite side's checks, outside the row space of its own side) before it was recorded. The pass is the weekly fresh-seed re-measurement of the board's cell leaders (issue 2025, run of 2026-09-24); every recorded reading is the lightest CPU-verified logical at that budget.

| side | seed | claimed | GPU best weight | CPU-verified witness | trials | |---|---|---|---|---|---| | X | 4101 | 26 | 26 | 26 | 300,000,000 | | Z | 4101 | 28 | 26 | 26 | 300,000,000 | | X | 4102 | 26 | 26 | 26 | 300,000,000 | | Z | 4102 | 28 | 26 | 26 | 300,000,000 |

Parity checks

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