[[354,4,28]] · QEC Challenge

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[[354,4,28]] d ≤

354

kd²/n

8.859

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Distance

d_X 28 · witness weight 28 (claimed upper_bound)

witness operator (support, 28 qubits)

[22, 30, 54, 70, 86, 95, 99, 102, 107, 118, 123, 127, 133, 137, 142, 157, 167, 171, 240, 250, 258, 274, 300, 302, 306, 308, 315, 340]

d_Z 28 · witness weight 28 (claimed upper_bound)

witness operator (support, 28 qubits)

[11, 17, 21, 23, 27, 32, 48, 121, 124, 127, 129, 130, 131, 133, 153, 183, 189, 216, 236, 247, 254, 279, 302, 305, 308, 331, 337, 343]

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^-2y², g(x,y)=1+y+x^-1y^-2 on the abelian quotient group Z²/L with twist basis a_1=[0, 177], a_2=[1, -53] (n=2|det[a_1,a_2]|=2*177). 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 177 · Z-checks 177

H_X (177 checks, sparse supports)

[0, 1, 68, 177, 179, 325] [1, 3, 78, 178, 181, 332] [2, 4, 79, 179, 182, 222] [3, 5, 6, 180, 184, 338] [4, 7, 89, 181, 185, 232] [5, 8, 90, 182, 186, 233] [6, 8, 10, 183, 188, 343] [7, 9, 11, 184, 189, 243] [8, 12, 100, 177, 185, 190] [9, 13, 101, 186, 191, 244] [10, 12, 15, 187, 193, 347] [11, 13, 16, 188, 194, 254] [12, 14, 17, 178, 189, 195] [13, 18, 111, 179, 190, 196] [14, 19, 112, 191, 197, 255] [15, 17, 21, 192, 199, 350] [16, 18, 22, 193, 200, 265] [17, 19, 23, 180, 194, 201] [18, 20, 24, 181, 195, 202] [19, 25, 122, 182, 196, 203] [20, 26, 123, 197, 204, 266] [21, 23, 28, 198, 206, 352] [22, 24, 29, 199, 207, 276] [23, 25, 30, 183, 200, 208] [24, 26, 31, 184, 201, 209] [25, 27, 32, 185, 202, 210] [26, 33, 132, 186, 203, 211] [27, 34, 133, 204, 212, 277] [28, 30, 36, 205, 214, 353] [29, 31, 37, 206, 215, 287] [30, 32, 38, 187, 207, 216] [31, 33, 39, 188, 208, 217] [32, 34, 40, 189, 209, 218] [33, 35, 41, 190, 210, 219] [34, 42, 141, 191, 211, 220] [35, 43, 142, 212, 221, 288] [36, 38, 45, 213, 223, 297] [37, 39, 46, 214, 224, 298] [38, 40, 47, 192, 215, 225] [39, 41, 48, 193, 216, 226] [40, 42, 49, 194, 217, 227] [41, 43, 50, 195, 218, 228] [42, 44, 51, 196, 219, 229] [43, 52, 149, 197, 220, 230] [44, 53, 150, 221, 231, 299] [45, 47, 55, 222, 233, 307] [46, 48, 56, 223, 234, 308] [47, 49, 57, 198, 224, 235] [48, 50, 58, 199, 225, 236] [49, 51, 59, 200, 226, 237] [50, 52, 60, 201, 227, 238] [51, 53, 61, 202, 228, 239] [52, 54, 62, 203, 229, 240] [53, 63, 156, 204, 230, 241] [54, 64, 157, 231, 242, 309] [55, 57, 66, 177, 232, 316] [0, 56, 58, 233, 244, 317] [57, 59, 67, 205, 234, 245] [58, 60, 68, 206, 235, 246] [59, 61, 69, 207, 236, 247] [60, 62, 70, 208, 237, 248] [61, 63, 71, 209, 238, 249] [62, 64, 72, 210, 239, 250] [63, 65, 73, 211, 240, 251] [64, 74, 162, 212, 241, 252] [65, 75, 163, 242, 253, 318] [66, 67, 77, 178, 243, 324] [2, 67, 69, 213, 244, 255] [68, 70, 78, 214, 245, 256] [69, 71, 79, 215, 246, 257] [70, 72, 80, 216, 247, 258] [71, 73, 81, 217, 248, 259] [72, 74, 82, 218, 249, 260] [73, 75, 83, 219, 250, 261] [74, 76, 84, 220, 251, 262] [75, 85, 167, 221, 252, 263] [76, 86, 168, 253, 264, 326] [2, 77, 88, 180, 254, 331] [5, 78, 80, 223, 255, 266] [79, 81, 89, 224, 256, 267] [80, 82, 90, 225, 257, 268] [81, 83, 91, 226, 258, 269] [82, 84, 92, 227, 259, 270] [83, 85, 93, 228, 260, 271] [84, 86, 94, 229, 261, 272] [85, 87, 95, 230, 262, 273] [86, 96, 171, 231, 263, 274] [87, 97, 172, 264, 275, 333] [4, 88, 99, 183, 265, 337] [9, 89, 91, 234, 266, 277] [90, 92, 100, 235, 267, 278] [91, 93, 101, 236, 268, 279] [92, 94, 102, 237, 269, 280] [93, 95, 103, 238, 270, 281] [94, 96, 104, 239, 271, 282] [95, 97, 105, 240, 272, 283] [96, 98, 106, 241, 273, 284] [97, 107, 174, 242, 274, 285] [98, 108, 175, 275, 286, 339] [7, 99, 110, 187, 276, 342] [14, 100, 102, 245, 277, 288] [101, 103, 111, 246, 278, 289] [102, 104, 112, 247, 279, 290] [103, 105, 113, 248, 280, 291] [104, 106, 114, 249, 281, 292] [105, 107, 115, 250, 282, 293] [106, 108, 116, 251, 283, 294] [107, 109, 117, 252, 284, 295] [108, 118, 176, 253, 285, 296] [109, 110, 119, 286, 297, 344] [11, 110, 121, 192, 287, 346] [20, 111, 113, 256, 288, 299] [112, 114, 122, 257, 289, 300] [113, 115, 123, 258, 290, 301] [114, 116, 124, 259, 291, 302] [115, 117, 125, 260, 292, 303] [116, 118, 126, 261, 293, 304] [117, 119, 127, 262, 294, 305] [118, 120, 128, 263, 295, 306] [119, 121, 129, 264, 296, 307] [15, 120, 130, 297, 308, 348] [16, 121, 131, 198, 298, 349] [27, 122, 124, 267, 299, 309] [123, 125, 132, 268, 300, 310] [124, 126, 133, 269, 301, 311] [125, 127, 134, 270, 302, 312] [126, 128, 135, 271, 303, 313] [127, 129, 136, 272, 304, 314] [128, 130, 137, 273, 305, 315] [129, 131, 138, 274, 306, 316] [21, 130, 139, 275, 307, 317] [22, 131, 140, 205, 308, 351] [35, 132, 134, 278, 309, 318] [133, 135, 141, 279, 310, 319] [134, 136, 142, 280, 311, 320] [135, 137, 143, 281, 312, 321] [136, 138, 144, 282, 313, 322] [137, 139, 145, 283, 314, 323] [138, 140, 146, 284, 315, 324] [28, 139, 147, 285, 316, 325] [29, 140, 148, 213, 286, 317] [44, 141, 143, 289, 318, 326] [142, 144, 149, 290, 319, 327] [143, 145, 150, 291, 320, 328] [144, 146, 151, 292, 321, 329] [145, 147, 152, 293, 322, 330] [146, 148, 153, 294, 323, 331] [36, 147, 154, 295, 324, 332] [37, 148, 155, 222, 296, 325] [54, 149, 151, 300, 326, 333] [150, 152, 156, 301, 327, 334] [151, 153, 157, 302, 328, 335] [152, 154, 158, 303, 329, 336] [153, 155, 159, 304, 330, 337] [45, 154, 160, 305, 331, 338] [46, 155, 161, 232, 306, 332] [65, 156, 158, 310, 333, 339] [157, 159, 162, 311, 334, 340] [158, 160, 163, 312, 335, 341] [159, 161, 164, 313, 336, 342] [55, 160, 165, 314, 337, 343] [56, 161, 166, 243, 315, 338] [76, 162, 164, 319, 339, 344] [163, 165, 167, 320, 340, 345] [164, 166, 168, 321, 341, 346] [66, 165, 169, 322, 342, 347] [0, 166, 170, 254, 323, 343] [87, 167, 169, 327, 344, 348] [168, 170, 171, 328, 345, 349] [77, 169, 172, 329, 346, 350] [1, 170, 173, 265, 330, 347] [98, 171, 173, 334, 348, 351] [88, 172, 174, 335, 349, 352] [3, 173, 175, 276, 336, 350] [99, 109, 174, 340, 351, 353] [6, 175, 176, 287, 341, 352] [10, 120, 176, 298, 345, 353]

H_Z (177 checks, sparse supports)

[0, 8, 55, 177, 233, 343] [1, 12, 66, 177, 178, 347] [0, 2, 13, 179, 244, 254] [3, 17, 77, 178, 180, 350] [1, 4, 18, 179, 181, 265] [2, 5, 19, 180, 182, 255] [6, 23, 88, 180, 183, 352] [3, 7, 24, 181, 184, 276] [4, 8, 25, 182, 183, 185] [5, 9, 26, 184, 186, 266] [10, 30, 99, 183, 187, 353] [6, 11, 31, 184, 188, 287] [7, 12, 32, 185, 187, 189] [8, 13, 33, 186, 188, 190] [9, 14, 34, 189, 191, 277] [15, 38, 110, 187, 192, 297] [10, 16, 39, 188, 193, 298] [11, 17, 40, 189, 192, 194] [12, 18, 41, 190, 193, 195] [13, 19, 42, 191, 194, 196] [14, 20, 43, 195, 197, 288] [21, 47, 121, 192, 198, 307] [15, 22, 48, 193, 199, 308] [16, 23, 49, 194, 198, 200] [17, 24, 50, 195, 199, 201] [18, 25, 51, 196, 200, 202] [19, 26, 52, 197, 201, 203] [20, 27, 53, 202, 204, 299] [28, 57, 131, 198, 205, 316] [21, 29, 58, 199, 206, 317] [22, 30, 59, 200, 205, 207] [23, 31, 60, 201, 206, 208] [24, 32, 61, 202, 207, 209] [25, 33, 62, 203, 208, 210] [26, 34, 63, 204, 209, 211] [27, 35, 64, 210, 212, 309] [36, 67, 140, 205, 213, 324] [28, 37, 68, 206, 214, 325] [29, 38, 69, 207, 213, 215] [30, 39, 70, 208, 214, 216] [31, 40, 71, 209, 215, 217] [32, 41, 72, 210, 216, 218] [33, 42, 73, 211, 217, 219] [34, 43, 74, 212, 218, 220] [35, 44, 75, 219, 221, 318] [2, 45, 148, 213, 222, 331] [36, 46, 78, 214, 223, 332] [37, 47, 79, 215, 222, 224] [38, 48, 80, 216, 223, 225] [39, 49, 81, 217, 224, 226] [40, 50, 82, 218, 225, 227] [41, 51, 83, 219, 226, 228] [42, 52, 84, 220, 227, 229] [43, 53, 85, 221, 228, 230] [44, 54, 86, 229, 231, 326] [4, 55, 155, 222, 232, 337] [5, 45, 56, 223, 233, 338] [46, 57, 89, 224, 232, 234] [47, 58, 90, 225, 233, 235] [48, 59, 91, 226, 234, 236] [49, 60, 92, 227, 235, 237] [50, 61, 93, 228, 236, 238] [51, 62, 94, 229, 237, 239] [52, 63, 95, 230, 238, 240] [53, 64, 96, 231, 239, 241] [54, 65, 97, 240, 242, 333] [7, 66, 161, 232, 243, 342] [9, 56, 67, 234, 243, 244] [57, 68, 100, 177, 235, 245] [58, 69, 101, 236, 244, 246] [59, 70, 102, 237, 245, 247] [60, 71, 103, 238, 246, 248] [61, 72, 104, 239, 247, 249] [62, 73, 105, 240, 248, 250] [63, 74, 106, 241, 249, 251] [64, 75, 107, 242, 250, 252] [65, 76, 108, 251, 253, 339] [11, 77, 166, 243, 254, 346] [14, 67, 78, 178, 245, 255] [68, 79, 111, 179, 246, 256] [69, 80, 112, 247, 255, 257] [70, 81, 113, 248, 256, 258] [71, 82, 114, 249, 257, 259] [72, 83, 115, 250, 258, 260] [73, 84, 116, 251, 259, 261] [74, 85, 117, 252, 260, 262] [75, 86, 118, 253, 261, 263] [76, 87, 119, 262, 264, 344] [16, 88, 170, 254, 265, 349] [20, 78, 89, 181, 256, 266] [79, 90, 122, 182, 257, 267] [80, 91, 123, 258, 266, 268] [81, 92, 124, 259, 267, 269] [82, 93, 125, 260, 268, 270] [83, 94, 126, 261, 269, 271] [84, 95, 127, 262, 270, 272] [85, 96, 128, 263, 271, 273] [86, 97, 129, 264, 272, 274] [87, 98, 130, 273, 275, 348] [22, 99, 173, 265, 276, 351] [27, 89, 100, 185, 267, 277] [90, 101, 132, 186, 268, 278] [91, 102, 133, 269, 277, 279] [92, 103, 134, 270, 278, 280] [93, 104, 135, 271, 279, 281] [94, 105, 136, 272, 280, 282] [95, 106, 137, 273, 281, 283] [96, 107, 138, 274, 282, 284] [97, 108, 139, 275, 283, 285] [98, 109, 140, 284, 286, 351] [29, 110, 175, 276, 286, 287] [35, 100, 111, 190, 278, 288] [101, 112, 141, 191, 279, 289] [102, 113, 142, 280, 288, 290] [103, 114, 143, 281, 289, 291] [104, 115, 144, 282, 290, 292] [105, 116, 145, 283, 291, 293] [106, 117, 146, 284, 292, 294] [107, 118, 147, 285, 293, 295] [108, 119, 148, 286, 294, 296] [36, 109, 120, 295, 297, 353] [37, 121, 176, 287, 296, 298] [44, 111, 122, 196, 289, 299] [112, 123, 149, 197, 290, 300] [113, 124, 150, 291, 299, 301] [114, 125, 151, 292, 300, 302] [115, 126, 152, 293, 301, 303] [116, 127, 153, 294, 302, 304] [117, 128, 154, 295, 303, 305] [118, 129, 155, 296, 304, 306] [45, 119, 130, 297, 305, 307] [46, 120, 131, 298, 306, 308] [54, 122, 132, 203, 300, 309] [123, 133, 156, 204, 301, 310] [124, 134, 157, 302, 309, 311] [125, 135, 158, 303, 310, 312] [126, 136, 159, 304, 311, 313] [127, 137, 160, 305, 312, 314] [128, 138, 161, 306, 313, 315] [55, 129, 139, 307, 314, 316] [56, 130, 140, 308, 315, 317] [65, 132, 141, 211, 310, 318] [133, 142, 162, 212, 311, 319] [134, 143, 163, 312, 318, 320] [135, 144, 164, 313, 319, 321] [136, 145, 165, 314, 320, 322] [137, 146, 166, 315, 321, 323] [66, 138, 147, 316, 322, 324] [0, 139, 148, 317, 323, 325] [76, 141, 149, 220, 319, 326] [142, 150, 167, 221, 320, 327] [143, 151, 168, 321, 326, 328] [144, 152, 169, 322, 327, 329] [145, 153, 170, 323, 328, 330] [77, 146, 154, 324, 329, 331] [1, 147, 155, 325, 330, 332] [87, 149, 156, 230, 327, 333] [150, 157, 171, 231, 328, 334] [151, 158, 172, 329, 333, 335] [152, 159, 173, 330, 334, 336] [88, 153, 160, 331, 335, 337] [3, 154, 161, 332, 336, 338] [98, 156, 162, 241, 334, 339] [157, 163, 174, 242, 335, 340] [158, 164, 175, 336, 339, 341] [99, 159, 165, 337, 340, 342] [6, 160, 166, 338, 341, 343] [109, 162, 167, 252, 340, 344] [163, 168, 176, 253, 341, 345] [110, 164, 169, 342, 344, 346] [10, 165, 170, 343, 345, 347] [120, 167, 171, 263, 345, 348] [121, 168, 172, 264, 346, 349] [15, 169, 173, 347, 348, 350] [131, 171, 174, 274, 349, 351] [21, 172, 175, 275, 350, 352] [28, 174, 176, 285, 352, 353]

raw submission JSON