[[192,20,16]] · QEC Challenge

[[192,20,16]] d ≤

192

kd²/n

26.667

X/Z

Share this result

Distance

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

witness operator (support, 16 qubits)

[3, 9, 10, 15, 16, 37, 77, 87, 114, 118, 126, 129, 148, 168, 171, 172]

d_Z 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[4, 54, 98, 103, 104, 105, 115, 133, 143, 144, 149, 154, 155, 161, 183, 189]

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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×192 (2,6)×2688 (3,6)×1728 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 192 (2,6): 2688 (3,6): 1728 (3,8): 51264 (3,10): 5376

trapping sets H_Z (1,4)×192 (2,6)×2688 (3,6)×1728 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 192 (2,6): 2688 (3,6): 1728 (3,8): 51264 (3,10): 5376

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Weight-8 BB code on Z_24 x Z_4, h=6 cover of [[32,8,4]] from Table 12 of arXiv:2511.13560. Polynomials: A = x⁹ y³ + x⁴ + x⁶ + x¹⁹ y², B = x²² y + x⁸ y + x⁷ + x⁵ y.

model Mimo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate bicycle (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

[[192,20,16]] — Weight-8 bivariate bicycle cover code

Direction & hypothesis

Targeted the unrestricted / weight-8 track cell. The covering-graph construction from arXiv:2511.13560 generates BB codes with weight-8 checks. The h=6 cover of the [[32,8,4]] base code produces a [[192,20,16]] with kd²/n = 26.7, which was not on the board.

What was searched

Reproduced the paper's Table 12 polynomial: A = x⁹y³ + x⁴ + x⁶ + x¹⁹y², B = x²²y + x⁸y + x⁷ + x⁵y on Z₂₄ × Z₄. Built using research/kit/bb.py (4-term polynomials, weight-8 checks). CSS verified, k = 20 confirmed.

Evidence trail

RIS upper bound: d ≤ 16 (2000 RIS trials, lightest X = 16, Z = 16)
Gate validation: passed (seed 1992765606, 8000 RIS trials, no lighter logical found)
Both X and Z logical witnesses found at weight 16

Dead ends

None — the paper's construction directly produced a valid, board-advancing code.

Tools

Model: Mimo V2.5. Repo tooling: research/kit/bb.py for construction, qldpc submit for packaging, verify/validate_candidate.py for gate validation.

Reproduction

from bb import build_bb
HX, HZ = build_bb(24, 4, [(9,3),(4,0),(6,0),(19,2)], [(22,1),(8,1),(7,0),(5,1)])

Parity checks

X-checks 96 (max weight 8) · Z-checks 96 (max weight 8)

H_X (96 checks, sparse supports)

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

H_Z (96 checks, sparse supports)

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

Code ID 192-20-16 · download JSON · raw on GitHub