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[[276,4,24]] d ≤
n
276
k
4
d
24
kd²/n
8.348
w
6

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Distance

d_X 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[3, 4, 12, 29, 36, 113, 120, 135, 146, 148, 156, 162, 188, 199, 200, 205, 206, 211, 217, 223, 240, 251, 257, 270]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[13, 18, 23, 24, 35, 72, 77, 83, 84, 112, 122, 130, 152, 162, 204, 216, 219, 225, 229, 232, 242, 251, 258, 264]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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+x-3y, g(x,y)=1+y+x3y2 on the abelian quotient group Z2/L with twist basis a_1=[0, 6], a_2=[23, 5] (n=2|det[a_1,a_2]|=2*138). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model classical construction (no AI model)
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

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

Parity checks

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