[[96,20,8]] · QEC Challenge

[[96,20,8]] d =

kd²/n

13.333

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[29, 32, 40, 44, 63, 73, 81, 94]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[12, 14, 20, 22, 49, 51, 88, 90]

certificate exact, d = 8 · CryptoMiniSat 5.14 SAT

X: no logical < 8 exists; Z: no logical < 8 exists

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)×96 (2,4)×48 (3,6)×2688 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,4): 48 (2,6): 1248 (3,6): 2688 (3,8): 20832 (3,10): 2112

trapping sets H_Z (1,4)×96 (2,4)×48 (3,6)×2688 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 96 (2,4): 48 (2,6): 1248 (3,6): 2688 (3,8): 20832 (3,10): 2112

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Weight-8 BB code on Z_12 x Z_4, h=3 cover of [[32,8,4]] from Table 12 of arXiv:2511.13560. Polynomials: A = xy³ + x⁴ + x² + x¹¹y², B = x⁶y + x⁸y + x¹¹ + x⁹y.

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

[[96,20,8]] — 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=3 cover of the [[32,8,4]] base code produces a [[96,20,8]] with kd²/n = 13.3, which was not on the board.

What was searched

Reproduced the paper's Table 12 polynomial: A = xy³ + x⁴ + x² + x¹¹y², B = x⁶y + x⁸y + x¹¹ + x⁹y on Z₁₂ × Z₄. Built using research/kit/bb.py (4-term polynomials, weight-8 checks). CSS verified, k = 20 confirmed.

Evidence trail

RIS upper bound: d ≤ 8 (3000 RIS trials, lightest X = 8, Z = 8)
Gate validation: passed (seed 582073792, 6340 RIS trials, no lighter logical found)
Both X and Z logical witnesses found at weight 8

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(12, 4, [(1,3),(4,0),(2,0),(11,2)], [(6,1),(8,1),(11,0),(9,1)])

Parity checks

X-checks 48 (max weight 8) · Z-checks 48 (max weight 8)

H_X (48 checks, sparse supports)

[7, 8, 16, 46, 73, 81, 85, 92] [4, 9, 17, 47, 74, 82, 86, 93] [5, 10, 18, 44, 75, 83, 87, 94] [6, 11, 19, 45, 72, 80, 84, 95] [2, 11, 12, 20, 48, 77, 85, 89] [3, 8, 13, 21, 49, 78, 86, 90] [0, 9, 14, 22, 50, 79, 87, 91] [1, 10, 15, 23, 51, 76, 84, 88] [6, 15, 16, 24, 52, 81, 89, 93] [7, 12, 17, 25, 53, 82, 90, 94] [4, 13, 18, 26, 54, 83, 91, 95] [5, 14, 19, 27, 55, 80, 88, 92] [10, 19, 20, 28, 49, 56, 85, 93] [11, 16, 21, 29, 50, 57, 86, 94] [8, 17, 22, 30, 51, 58, 87, 95] [9, 18, 23, 31, 48, 59, 84, 92] [14, 23, 24, 32, 49, 53, 60, 89] [15, 20, 25, 33, 50, 54, 61, 90] [12, 21, 26, 34, 51, 55, 62, 91] [13, 22, 27, 35, 48, 52, 63, 88] [18, 27, 28, 36, 53, 57, 64, 93] [19, 24, 29, 37, 54, 58, 65, 94] [16, 25, 30, 38, 55, 59, 66, 95] [17, 26, 31, 39, 52, 56, 67, 92] [22, 31, 32, 40, 49, 57, 61, 68] [23, 28, 33, 41, 50, 58, 62, 69] [20, 29, 34, 42, 51, 59, 63, 70] [21, 30, 35, 43, 48, 56, 60, 71] [26, 35, 36, 44, 53, 61, 65, 72] [27, 32, 37, 45, 54, 62, 66, 73] [24, 33, 38, 46, 55, 63, 67, 74] [25, 34, 39, 47, 52, 60, 64, 75] [0, 30, 39, 40, 57, 65, 69, 76] [1, 31, 36, 41, 58, 66, 70, 77] [2, 28, 37, 42, 59, 67, 71, 78] [3, 29, 38, 43, 56, 64, 68, 79] [4, 34, 43, 44, 61, 69, 73, 80] [5, 35, 40, 45, 62, 70, 74, 81] [6, 32, 41, 46, 63, 71, 75, 82] [7, 33, 42, 47, 60, 68, 72, 83] [0, 8, 38, 47, 65, 73, 77, 84] [1, 9, 39, 44, 66, 74, 78, 85] [2, 10, 36, 45, 67, 75, 79, 86] [3, 11, 37, 46, 64, 72, 76, 87] [3, 4, 12, 42, 69, 77, 81, 88] [0, 5, 13, 43, 70, 78, 82, 89] [1, 6, 14, 40, 71, 79, 83, 90] [2, 7, 15, 41, 68, 76, 80, 91]

H_Z (48 checks, sparse supports)

[4, 15, 19, 27, 54, 80, 88, 93] [5, 12, 16, 24, 55, 81, 89, 94] [6, 13, 17, 25, 52, 82, 90, 95] [7, 14, 18, 26, 53, 83, 91, 92] [8, 19, 23, 31, 49, 58, 84, 92] [9, 16, 20, 28, 50, 59, 85, 93] [10, 17, 21, 29, 51, 56, 86, 94] [11, 18, 22, 30, 48, 57, 87, 95] [12, 23, 27, 35, 48, 53, 62, 88] [13, 20, 24, 32, 49, 54, 63, 89] [14, 21, 25, 33, 50, 55, 60, 90] [15, 22, 26, 34, 51, 52, 61, 91] [16, 27, 31, 39, 52, 57, 66, 92] [17, 24, 28, 36, 53, 58, 67, 93] [18, 25, 29, 37, 54, 59, 64, 94] [19, 26, 30, 38, 55, 56, 65, 95] [20, 31, 35, 43, 48, 56, 61, 70] [21, 28, 32, 40, 49, 57, 62, 71] [22, 29, 33, 41, 50, 58, 63, 68] [23, 30, 34, 42, 51, 59, 60, 69] [24, 35, 39, 47, 52, 60, 65, 74] [25, 32, 36, 44, 53, 61, 66, 75] [26, 33, 37, 45, 54, 62, 67, 72] [27, 34, 38, 46, 55, 63, 64, 73] [3, 28, 39, 43, 56, 64, 69, 78] [0, 29, 36, 40, 57, 65, 70, 79] [1, 30, 37, 41, 58, 66, 71, 76] [2, 31, 38, 42, 59, 67, 68, 77] [7, 32, 43, 47, 60, 68, 73, 82] [4, 33, 40, 44, 61, 69, 74, 83] [5, 34, 41, 45, 62, 70, 75, 80] [6, 35, 42, 46, 63, 71, 72, 81] [3, 11, 36, 47, 64, 72, 77, 86] [0, 8, 37, 44, 65, 73, 78, 87] [1, 9, 38, 45, 66, 74, 79, 84] [2, 10, 39, 46, 67, 75, 76, 85] [3, 7, 15, 40, 68, 76, 81, 90] [0, 4, 12, 41, 69, 77, 82, 91] [1, 5, 13, 42, 70, 78, 83, 88] [2, 6, 14, 43, 71, 79, 80, 89] [7, 11, 19, 44, 72, 80, 85, 94] [4, 8, 16, 45, 73, 81, 86, 95] [5, 9, 17, 46, 74, 82, 87, 92] [6, 10, 18, 47, 75, 83, 84, 93] [0, 11, 15, 23, 50, 76, 84, 89] [1, 8, 12, 20, 51, 77, 85, 90] [2, 9, 13, 21, 48, 78, 86, 91] [3, 10, 14, 22, 49, 79, 87, 88]

Code ID 96-20-8 · download JSON · raw on GitHub