[[125,25,3]] · QEC Challenge

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[[125,25,3]] d ≤

125

kd²/n

1.8

X/Z

0.0011

6.4031

layers

swaps

470

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[73, 77, 78]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[29, 69, 79]

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 4.5) · H_Z 4–5 (mean 4.5)

qubit degrees H_X 1–3 (mean 1.8) · H_Z 1–3 (mean 1.8)

trapping sets H_X (1,1)×50 (2,1)×100 (3,0)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 50 (1,2): 50 (1,3): 25 (2,1): 100 (2,2): 125 (2,3): 150 (2,4): 25 (3,0): 50 (3,1): 175 (3,2): 500 (3,3): 600 (3,4): 475 (3,5): 125 (3,6): 50

trapping sets H_Z (1,1)×50 (2,1)×100 (3,0)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 50 (1,2): 50 (1,3): 25 (2,1): 100 (2,2): 125 (2,3): 150 (2,4): 25 (3,0): 50 (3,1): 175 (3,2): 500 (3,3): 600 (3,4): 475 (3,5): 125 (3,6): 50

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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (100)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 470 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

[[125,25,3]] — 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=10; seed distance d(C)=3. Code parameters [[125,625,3]].

Evidence trail

Seed recurrence c_{a+25}=c_a+c_{a+1}, m=-15, seed distance 3.
Quantum code [[125,625,3]].
Witnesses are canonical logicals of weight 3 (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^{-15x10} 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 50 (max weight 5) · Z-checks 50 (max weight 5)

H_X (50 checks, sparse supports)

[0, 10, 50, 100] [1, 11, 51, 100, 101] [2, 12, 52, 101, 102] [3, 13, 53, 102, 103] [4, 14, 54, 103, 104] [5, 15, 55, 100, 104] [6, 16, 56, 101] [7, 17, 57, 102] [8, 18, 58, 103] [9, 19, 59, 104] [10, 20, 60, 105] [11, 21, 61, 105, 106] [12, 22, 62, 106, 107] [13, 23, 63, 107, 108] [14, 24, 64, 108, 109] [15, 25, 65, 105, 109] [16, 26, 66, 106] [17, 27, 67, 107] [18, 28, 68, 108] [19, 29, 69, 109] [20, 30, 70, 110] [21, 31, 71, 110, 111] [22, 32, 72, 111, 112] [23, 33, 73, 112, 113] [24, 34, 74, 113, 114] [25, 35, 75, 110, 114] [26, 36, 76, 111] [27, 37, 77, 112] [28, 38, 78, 113] [29, 39, 79, 114] [30, 40, 80, 115] [31, 41, 81, 115, 116] [32, 42, 82, 116, 117] [33, 43, 83, 117, 118] [34, 44, 84, 118, 119] [35, 45, 85, 115, 119] [36, 46, 86, 116] [37, 47, 87, 117] [38, 48, 88, 118] [39, 49, 89, 119] [40, 50, 90, 120] [41, 51, 91, 120, 121] [42, 52, 92, 121, 122] [43, 53, 93, 122, 123] [44, 54, 94, 123, 124] [45, 55, 95, 120, 124] [46, 56, 96, 121] [47, 57, 97, 122] [48, 58, 98, 123] [49, 59, 99, 124]

H_Z (50 checks, sparse supports)

[0, 1, 5, 100] [1, 2, 6, 101] [2, 3, 7, 102] [3, 4, 8, 103] [4, 5, 9, 104] [10, 11, 15, 100, 105] [11, 12, 16, 101, 106] [12, 13, 17, 102, 107] [13, 14, 18, 103, 108] [14, 15, 19, 104, 109] [20, 21, 25, 105, 110] [21, 22, 26, 106, 111] [22, 23, 27, 107, 112] [23, 24, 28, 108, 113] [24, 25, 29, 109, 114] [30, 31, 35, 110, 115] [31, 32, 36, 111, 116] [32, 33, 37, 112, 117] [33, 34, 38, 113, 118] [34, 35, 39, 114, 119] [40, 41, 45, 115, 120] [41, 42, 46, 116, 121] [42, 43, 47, 117, 122] [43, 44, 48, 118, 123] [44, 45, 49, 119, 124] [50, 51, 55, 100, 120] [51, 52, 56, 101, 121] [52, 53, 57, 102, 122] [53, 54, 58, 103, 123] [54, 55, 59, 104, 124] [60, 61, 65, 105] [61, 62, 66, 106] [62, 63, 67, 107] [63, 64, 68, 108] [64, 65, 69, 109] [70, 71, 75, 110] [71, 72, 76, 111] [72, 73, 77, 112] [73, 74, 78, 113] [74, 75, 79, 114] [80, 81, 85, 115] [81, 82, 86, 116] [82, 83, 87, 117] [83, 84, 88, 118] [84, 85, 89, 119] [90, 91, 95, 120] [91, 92, 96, 121] [92, 93, 97, 122] [93, 94, 98, 123] [94, 95, 99, 124]

Code ID 125-25-3 · download JSON · raw on GitHub