[[96,4,10]] · QEC Challenge

[[96,4,10]] d =

kd²/n

4.167

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 10 · witness weight 10 (claimed exact)

witness operator (support, 10 qubits)

[7, 16, 24, 33, 37, 46, 73, 74, 78, 85]

d_Z 10 · witness weight 10 (claimed exact)

witness operator (support, 10 qubits)

[0, 1, 36, 37, 42, 43, 49, 50, 88, 89]

certificate exact, d = 10 · scipy/HiGHS MILP

X: no logical < 10 exists; Z: no logical < 10 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×48 (2,2)×48 (3,2)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 48 (1,3): 48 (2,2): 48 (2,3): 288 (2,4): 144 (3,2): 48 (3,3): 928 (3,4): 1584 (3,5): 528 (3,6): 288 (3,7): 48

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

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

(1,2): 48 (1,3): 48 (2,2): 48 (2,3): 288 (2,4): 144 (3,2): 48 (3,3): 928 (3,4): 1584 (3,5): 528 (3,6): 288 (3,7): 48

Construction & provenance

authors Lin, Hsiang-Ku and Pryadko, Leonid P.

provenance literature baseline

construction Two-block group-algebra (2BGA) code over SmallGroup(48,2), archive row "48 2 4 10 [2] [3,39]" (Lin & Pryadko, arXiv:2306.16400; 2BGA-codes archive). Reconstructed here via an equivalent bivariate-bicycle presentation over F_2[x,y]/(x¹⁶-1, y³-1): A = 1 + y + x² y², B = 1 + x³, H_X = [A|B], H_Z = [B^T|A^T]. An explicit CRT relabeling of the 48 group coordinates in each qubit block, plus an exchange of the two 48-qubit blocks, maps this presentation onto the archived row exactly -- the same CSS code up to qubit/check permutation.

model classical construction (no AI model)

reference arXiv:2306.16400, https://github.com/QEC-pages/2BGA-codes

date 2023

notes Literature baseline (arXiv:2306.16400, Lin & Pryadko, 28 Jun 2023). This [[96,4,10]] instance is not in the paper's own Table 1 (which stops at smaller cases) but in the paper's companion enumeration archive, github.com/QEC-pages/2BGA-codes, abelian row "SmallGroup(48,2), 48 2 4 10 [2] [3,39]". Reconstructed via an equivalent bivariate-bicycle presentation and independently re-verified with this repository's own tools: CSS commutation and k=4 confirmed by direct rank computation; distance re-derived by randomized search (5,000 and 50,000 trials, d=10) and BP+OSD (200,000 trials, d_heuristic=10), then certified exact by MILP (d_X=d_Z=10, no lighter logical on either side; scipy/HiGHS, tlim=600s, 78s wall time). The archive's own reported d<=10 for this row is a QDistRnd probabilistic upper bound; the MILP result above independently certifies it exact. The verifier's 'literature novelty UNVERIFIED' label is by-design (validate_candidate.py performs no literature lookup) and is intentionally not overridden.

family 2BGA coset (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 48 (max weight 5) · Z-checks 48 (max weight 5)

H_X (48 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 96-4-10 · download JSON · raw on GitHub