[[308,8,28]] · QEC Challenge

[[308,8,28]] d ≤

308

kd²/n

20.364

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 28, d_Z ≤ 28 · 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 @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 3×10⁸ trials · survived 3×10⁸ trials · 2026-09-27

witness operator (support, 28 qubits)

[3, 9, 12, 26, 37, 40, 50, 53, 58, 78, 81, 106, 129, 130, 142, 150, 166, 188, 191, 198, 201, 239, 278, 294, 296, 298, 301, 307]

d_Z 28 · witness weight 28 (claimed upper_bound)

witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 3×10⁸ trials · survived 3×10⁸ trials · 2026-09-27

witness operator (support, 28 qubits)

[8, 16, 18, 22, 28, 29, 62, 70, 72, 76, 110, 122, 128, 138, 144, 146, 148, 149, 158, 167, 175, 190, 192, 224, 254, 262, 270, 303]

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)×308 (2,4)×154 (3,6)×6468 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 308 (2,4): 154 (2,6): 4004 (3,6): 6468 (3,8): 73304 (3,10): 6776

trapping sets H_Z (1,4)×308 (2,4)×154 (3,6)×6468 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,4): 308 (2,4): 154 (2,6): 4004 (3,6): 6468 (3,8): 73304 (3,10): 6776

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Periodic bivariate-bicycle code on Z_77 x Z_2 (n = 2*l*m = 308): x = S_77 tensor I_2, y = I_77 tensor S_2 (cyclic shifts), qubit index i*m + j in each block; A = x⁰y⁰ + x²⁷y⁰ + x³⁶y⁰ + x³⁹y⁰, B = x⁰y⁰ + x¹⁹y⁰ + x³⁶y⁰ + x⁴⁴y¹; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(77, 2, [[0, 0], [27, 0], [36, 0], [39, 0]], [[0, 0], [19, 0], [36, 0], [44, 1]])). gcd(77, 2) = 1, so Z_77 x Z_2 is cyclic of order 154 and the code is the cyclic generalized-bicycle code over Z_154 with a(z) = z⁰ + z³⁶ + z¹⁰⁴ + z¹¹⁶, b(z) = z⁰ + z³⁶ + z⁹⁶ + z¹²¹ (CRT relabeling x^a y^b -> z^t, t = a mod 77, t = b mod 2).

model Claude Fable 5.1 (Claude Code) (claimed, not verified)

date 2026-09-27

family other (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

[[308,8,28]] weight-8 coprime bivariate-bicycle code

Construction

Periodic bivariate-bicycle code on Z_77 x Z_2 (n = 2*l*m = 308): x = S_77 tensor I_2, y = I_77 tensor S_2 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^27y^0 + x^36y^0 + x^39y^0, B = x^0y^0 + x^19y^0 + x^36y^0 + x^44y^1; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(77, 2, [[0, 0], [27, 0], [36, 0], [39, 0]], [[0, 0], [19, 0], [36, 0], [44, 1]])). gcd(77, 2) = 1, so Z_77 x Z_2 is cyclic of order 154 and the code is the cyclic generalized-bicycle code over Z_154 with a(z) = z^0 + z^36 + z^104 + z^116, b(z) = z^0 + z^36 + z^96 + z^121 (CRT relabeling x^a y^b -> z^t, t = a mod 77, t = b mod 2).

Every check has weight exactly 8.

Distance evidence

The distance is a witness-backed upper bound. Every witness in the submission was validated against the published check matrices before filing.

| trials per side | lightest logical found | |---|---| | 300 | 35 | | 2,000 | 32 | | 20,000 | 28 | | 300,000,000 | X 28, Z 28 |

The deeper passes did not lower the claim.

What is not claimed

This code carries no circuit-level or logical-error-rate measurement. It came from a search whose objective was a memory-experiment comparison against the rotated surface code, but it did not reach that stage, so nothing here speaks to its performance under a decoder. The submission is a claim about n, k, d, and check weight only.

The distance is an upper bound, not a certificate. A lighter logical operator would refute it, and the witnesses are published so that anyone can try.

Reproduction

import sys; sys.path.insert(0, "research/kit")
from bb import build_bb
HX, HZ = build_bb(l=77, m=2, A_terms=[[0, 0], [27, 0], [36, 0], [39, 0]], B_terms=[[0, 0], [19, 0], [36, 0], [44, 1]])

Parity checks

X-checks 154 (max weight 8) · Z-checks 154 (max weight 8)

H_X (154 checks, sparse supports)

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

H_Z (154 checks, sparse supports)

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

Code ID 308-8-28 · download JSON · raw on GitHub