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[[97,25,2]] d ≤
n
97
k
25
d
2
kd²/n
1.031
w
5
X/Z
1
g
0.0006
r
6.4031
layers
2
swaps
328

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[54, 59]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[7, 52]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–5 (mean 4.333) · H_Z 4–5 (mean 4.333)
qubit degrees H_X 1–3 (mean 1.608) · H_Z 1–3 (mean 1.608)
trapping sets H_X (1,1)×54 (2,0)×18 (3,0)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 54 (1,2): 27 (1,3): 16 (2,0): 18 (2,1): 72 (2,2): 90 (2,3): 72 (2,4): 12 (3,0): 45 (3,1): 174 (3,2): 315 (3,3): 282 (3,4): 222 (3,5): 50 (3,6): 18
trapping sets H_Z (1,1)×54 (2,0)×18 (3,0)×45 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 54 (1,2): 27 (1,3): 16 (2,0): 18 (2,1): 72 (2,2): 90 (2,3): 72 (2,4): 12 (3,0): 45 (3,1): 174 (3,2): 315 (3,3): 282 (3,4): 222 (3,5): 50 (3,6): 18
witness diameter X 5.0 · Z 5.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/97-25-2/ · canonical noise recipe, 2 rounds, stim 1.16.0
d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[6, 11]
d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[310, 311]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 6.403
X checkZ checkqubit site (81)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 328 nearest-neighbor SWAPs per round in total, at most 7 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model Beam 1.0 (claimed, not verified)
date 2026-10-18
family hypergraph product (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[97,25,2]] — La-cross hypergraph-product code (truncated seed)

Direction & hypothesis

Aimed at the weight-6 / local-2d-bilayer track cell. La-cross codes (arXiv:2610.12685) are hypergraph products of a classical recurrence-code seed with itself; stabilizers weight at most six and bilayer layout.

What was searched

Enumerated truncated La-cross seeds for k=25, seed n=9; seed distance d(C)=2. Code parameters [[97,625,2]].

Evidence trail

  • Seed recurrence c_{a+25}=c_a+c_{a+1}, m=-16, seed distance 2.
  • Quantum code [[97,625,2]].
  • Witnesses are canonical logicals of weight 2 (upper bound).
  • Gate: passed true, board_advancing true, dominated_by [].

Dead ends

Other truncated seeds pass the gate but are dominated.

Tools

Beam 1.0 model assisting reconstruction; repo kit.

Reproduction

Seed matrix H in F2^{-16x9} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H). Layout: principal (i,j) at (j,i), check (a,b) at (b+1,a), 2 layers.

Parity checks

X-checks 36 (max weight 5) · Z-checks 36 (max weight 5)
H_X (36 checks, sparse supports)
[0, 9, 45, 81] [1, 10, 46, 81, 82] [2, 11, 47, 82, 83] [3, 12, 48, 83, 84] [4, 13, 49, 84] [5, 14, 50, 81] [6, 15, 51, 82] [7, 16, 52, 83] [8, 17, 53, 84] [9, 18, 54, 85] [10, 19, 55, 85, 86] [11, 20, 56, 86, 87] [12, 21, 57, 87, 88] [13, 22, 58, 88] [14, 23, 59, 85] [15, 24, 60, 86] [16, 25, 61, 87] [17, 26, 62, 88] [18, 27, 63, 89] [19, 28, 64, 89, 90] [20, 29, 65, 90, 91] [21, 30, 66, 91, 92] [22, 31, 67, 92] [23, 32, 68, 89] [24, 33, 69, 90] [25, 34, 70, 91] [26, 35, 71, 92] [27, 36, 72, 93] [28, 37, 73, 93, 94] [29, 38, 74, 94, 95] [30, 39, 75, 95, 96] [31, 40, 76, 96] [32, 41, 77, 93] [33, 42, 78, 94] [34, 43, 79, 95] [35, 44, 80, 96]
H_Z (36 checks, sparse supports)
[0, 1, 5, 81] [1, 2, 6, 82] [2, 3, 7, 83] [3, 4, 8, 84] [9, 10, 14, 81, 85] [10, 11, 15, 82, 86] [11, 12, 16, 83, 87] [12, 13, 17, 84, 88] [18, 19, 23, 85, 89] [19, 20, 24, 86, 90] [20, 21, 25, 87, 91] [21, 22, 26, 88, 92] [27, 28, 32, 89, 93] [28, 29, 33, 90, 94] [29, 30, 34, 91, 95] [30, 31, 35, 92, 96] [36, 37, 41, 93] [37, 38, 42, 94] [38, 39, 43, 95] [39, 40, 44, 96] [45, 46, 50, 81] [46, 47, 51, 82] [47, 48, 52, 83] [48, 49, 53, 84] [54, 55, 59, 85] [55, 56, 60, 86] [56, 57, 61, 87] [57, 58, 62, 88] [63, 64, 68, 89] [64, 65, 69, 90] [65, 66, 70, 91] [66, 67, 71, 92] [72, 73, 77, 93] [73, 74, 78, 94] [74, 75, 79, 95] [75, 76, 80, 96]
Code ID 97-25-2 · download JSON · raw on GitHub