[[210,8,16]] · QEC Challenge

[[210,8,16]] d ≤

210

kd²/n

9.752

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)

witness found by @FarLab · verify/ris_gpu --pair-depth 24 · found at 1.5×10⁷ trials · survived 3×10⁷ trials · 2026-08-19

witness operator (support, 16 qubits)

[3, 4, 21, 27, 45, 62, 74, 85, 92, 103, 123, 129, 152, 170, 181, 204]

d_Z 16 · witness weight 16 (claimed upper_bound)

witness found by @FarLab · verify/ris_gpu --pair-depth 24 · found at 1.5×10⁷ trials · survived 3×10⁷ trials · 2026-08-19

witness operator (support, 16 qubits)

[3, 8, 26, 49, 55, 61, 78, 122, 139, 156, 157, 186, 191, 197, 204, 209]

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)×210 (2,4)×1575 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 210 (2,4): 1575 (3,3): 420 (3,5): 14490 (3,7): 2100

trapping sets H_Z (1,3)×210 (2,4)×1575 (3,3)×420 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 210 (2,4): 1575 (3,3): 420 (3,5): 14490 (3,7): 2100

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty new parameter set claimed by submitter

construction Generalized bicycle (two-block) code on Z_105: H_X=[A|B], H_Z=[B^T|A^T], a(x)=x³¹+x⁵⁵+x⁸³, b(x)=x²⁸+x⁷⁴+x⁹²; g=gcd(a,b,x¹⁰⁵-1) irreducible of degree 4, k=2*deg(g)=8, row weight 6. Full circulant check sets kept (redundant rows carry the symmetry).

model Claude Fable 5 (claimed, not verified)

builds on arXiv:2203.17216, arXiv:2306.16400

date 2026-08-19

notes Orbit-structured GB search find; the full Tanner automorphism group acts with 2 orbits on the 8 canonical logical classes. Distance evidence: GPU deep-RIS 15M full-kernel pair trials x 2 seeds per side (CPU-verified witnesses) + sqetch 20Mx2 trials/side; nothing below 16 by any engine. Novelty basis: 2026-08-19 lit pass (Wang-Pryadko 2203.17216 = k=2 family; Lin-Pryadko 2306.16400 = n<=100; no zoo/board hits).

family generalized 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 105 (max weight 6) · Z-checks 105 (max weight 6)

H_X (105 checks, sparse supports)

[31, 55, 83, 133, 179, 197] [32, 56, 84, 134, 180, 198] [33, 57, 85, 135, 181, 199] [34, 58, 86, 136, 182, 200] [35, 59, 87, 137, 183, 201] [36, 60, 88, 138, 184, 202] [37, 61, 89, 139, 185, 203] [38, 62, 90, 140, 186, 204] [39, 63, 91, 141, 187, 205] [40, 64, 92, 142, 188, 206] [41, 65, 93, 143, 189, 207] [42, 66, 94, 144, 190, 208] [43, 67, 95, 145, 191, 209] [44, 68, 96, 105, 146, 192] [45, 69, 97, 106, 147, 193] [46, 70, 98, 107, 148, 194] [47, 71, 99, 108, 149, 195] [48, 72, 100, 109, 150, 196] [49, 73, 101, 110, 151, 197] [50, 74, 102, 111, 152, 198] [51, 75, 103, 112, 153, 199] [52, 76, 104, 113, 154, 200] [0, 53, 77, 114, 155, 201] [1, 54, 78, 115, 156, 202] [2, 55, 79, 116, 157, 203] [3, 56, 80, 117, 158, 204] [4, 57, 81, 118, 159, 205] [5, 58, 82, 119, 160, 206] [6, 59, 83, 120, 161, 207] [7, 60, 84, 121, 162, 208] [8, 61, 85, 122, 163, 209] [9, 62, 86, 105, 123, 164] [10, 63, 87, 106, 124, 165] [11, 64, 88, 107, 125, 166] [12, 65, 89, 108, 126, 167] [13, 66, 90, 109, 127, 168] [14, 67, 91, 110, 128, 169] [15, 68, 92, 111, 129, 170] [16, 69, 93, 112, 130, 171] [17, 70, 94, 113, 131, 172] [18, 71, 95, 114, 132, 173] [19, 72, 96, 115, 133, 174] [20, 73, 97, 116, 134, 175] [21, 74, 98, 117, 135, 176] [22, 75, 99, 118, 136, 177] [23, 76, 100, 119, 137, 178] [24, 77, 101, 120, 138, 179] [25, 78, 102, 121, 139, 180] [26, 79, 103, 122, 140, 181] [27, 80, 104, 123, 141, 182] [0, 28, 81, 124, 142, 183] [1, 29, 82, 125, 143, 184] [2, 30, 83, 126, 144, 185] [3, 31, 84, 127, 145, 186] [4, 32, 85, 128, 146, 187] [5, 33, 86, 129, 147, 188] [6, 34, 87, 130, 148, 189] [7, 35, 88, 131, 149, 190] [8, 36, 89, 132, 150, 191] [9, 37, 90, 133, 151, 192] [10, 38, 91, 134, 152, 193] [11, 39, 92, 135, 153, 194] [12, 40, 93, 136, 154, 195] [13, 41, 94, 137, 155, 196] [14, 42, 95, 138, 156, 197] [15, 43, 96, 139, 157, 198] [16, 44, 97, 140, 158, 199] [17, 45, 98, 141, 159, 200] [18, 46, 99, 142, 160, 201] [19, 47, 100, 143, 161, 202] [20, 48, 101, 144, 162, 203] [21, 49, 102, 145, 163, 204] [22, 50, 103, 146, 164, 205] [23, 51, 104, 147, 165, 206] [0, 24, 52, 148, 166, 207] [1, 25, 53, 149, 167, 208] [2, 26, 54, 150, 168, 209] [3, 27, 55, 105, 151, 169] [4, 28, 56, 106, 152, 170] [5, 29, 57, 107, 153, 171] [6, 30, 58, 108, 154, 172] [7, 31, 59, 109, 155, 173] [8, 32, 60, 110, 156, 174] [9, 33, 61, 111, 157, 175] [10, 34, 62, 112, 158, 176] [11, 35, 63, 113, 159, 177] [12, 36, 64, 114, 160, 178] [13, 37, 65, 115, 161, 179] [14, 38, 66, 116, 162, 180] [15, 39, 67, 117, 163, 181] [16, 40, 68, 118, 164, 182] [17, 41, 69, 119, 165, 183] [18, 42, 70, 120, 166, 184] [19, 43, 71, 121, 167, 185] [20, 44, 72, 122, 168, 186] [21, 45, 73, 123, 169, 187] [22, 46, 74, 124, 170, 188] [23, 47, 75, 125, 171, 189] [24, 48, 76, 126, 172, 190] [25, 49, 77, 127, 173, 191] [26, 50, 78, 128, 174, 192] [27, 51, 79, 129, 175, 193] [28, 52, 80, 130, 176, 194] [29, 53, 81, 131, 177, 195] [30, 54, 82, 132, 178, 196]

H_Z (105 checks, sparse supports)

[13, 31, 77, 127, 155, 179] [14, 32, 78, 128, 156, 180] [15, 33, 79, 129, 157, 181] [16, 34, 80, 130, 158, 182] [17, 35, 81, 131, 159, 183] [18, 36, 82, 132, 160, 184] [19, 37, 83, 133, 161, 185] [20, 38, 84, 134, 162, 186] [21, 39, 85, 135, 163, 187] [22, 40, 86, 136, 164, 188] [23, 41, 87, 137, 165, 189] [24, 42, 88, 138, 166, 190] [25, 43, 89, 139, 167, 191] [26, 44, 90, 140, 168, 192] [27, 45, 91, 141, 169, 193] [28, 46, 92, 142, 170, 194] [29, 47, 93, 143, 171, 195] [30, 48, 94, 144, 172, 196] [31, 49, 95, 145, 173, 197] [32, 50, 96, 146, 174, 198] [33, 51, 97, 147, 175, 199] [34, 52, 98, 148, 176, 200] [35, 53, 99, 149, 177, 201] [36, 54, 100, 150, 178, 202] [37, 55, 101, 151, 179, 203] [38, 56, 102, 152, 180, 204] [39, 57, 103, 153, 181, 205] [40, 58, 104, 154, 182, 206] [0, 41, 59, 155, 183, 207] [1, 42, 60, 156, 184, 208] [2, 43, 61, 157, 185, 209] [3, 44, 62, 105, 158, 186] [4, 45, 63, 106, 159, 187] [5, 46, 64, 107, 160, 188] [6, 47, 65, 108, 161, 189] [7, 48, 66, 109, 162, 190] [8, 49, 67, 110, 163, 191] [9, 50, 68, 111, 164, 192] [10, 51, 69, 112, 165, 193] [11, 52, 70, 113, 166, 194] [12, 53, 71, 114, 167, 195] [13, 54, 72, 115, 168, 196] [14, 55, 73, 116, 169, 197] [15, 56, 74, 117, 170, 198] [16, 57, 75, 118, 171, 199] [17, 58, 76, 119, 172, 200] [18, 59, 77, 120, 173, 201] [19, 60, 78, 121, 174, 202] [20, 61, 79, 122, 175, 203] [21, 62, 80, 123, 176, 204] [22, 63, 81, 124, 177, 205] [23, 64, 82, 125, 178, 206] [24, 65, 83, 126, 179, 207] [25, 66, 84, 127, 180, 208] [26, 67, 85, 128, 181, 209] [27, 68, 86, 105, 129, 182] [28, 69, 87, 106, 130, 183] [29, 70, 88, 107, 131, 184] [30, 71, 89, 108, 132, 185] [31, 72, 90, 109, 133, 186] [32, 73, 91, 110, 134, 187] [33, 74, 92, 111, 135, 188] [34, 75, 93, 112, 136, 189] [35, 76, 94, 113, 137, 190] [36, 77, 95, 114, 138, 191] [37, 78, 96, 115, 139, 192] [38, 79, 97, 116, 140, 193] [39, 80, 98, 117, 141, 194] [40, 81, 99, 118, 142, 195] [41, 82, 100, 119, 143, 196] [42, 83, 101, 120, 144, 197] [43, 84, 102, 121, 145, 198] [44, 85, 103, 122, 146, 199] [45, 86, 104, 123, 147, 200] [0, 46, 87, 124, 148, 201] [1, 47, 88, 125, 149, 202] [2, 48, 89, 126, 150, 203] [3, 49, 90, 127, 151, 204] [4, 50, 91, 128, 152, 205] [5, 51, 92, 129, 153, 206] [6, 52, 93, 130, 154, 207] [7, 53, 94, 131, 155, 208] [8, 54, 95, 132, 156, 209] [9, 55, 96, 105, 133, 157] [10, 56, 97, 106, 134, 158] [11, 57, 98, 107, 135, 159] [12, 58, 99, 108, 136, 160] [13, 59, 100, 109, 137, 161] [14, 60, 101, 110, 138, 162] [15, 61, 102, 111, 139, 163] [16, 62, 103, 112, 140, 164] [17, 63, 104, 113, 141, 165] [0, 18, 64, 114, 142, 166] [1, 19, 65, 115, 143, 167] [2, 20, 66, 116, 144, 168] [3, 21, 67, 117, 145, 169] [4, 22, 68, 118, 146, 170] [5, 23, 69, 119, 147, 171] [6, 24, 70, 120, 148, 172] [7, 25, 71, 121, 149, 173] [8, 26, 72, 122, 150, 174] [9, 27, 73, 123, 151, 175] [10, 28, 74, 124, 152, 176] [11, 29, 75, 125, 153, 177] [12, 30, 76, 126, 154, 178]

Code ID 210-8-16 · download JSON · raw on GitHub