[[116,21,2]] · QEC Challenge

[[116,21,2]] d ≤

116

kd²/n

0.724138

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 4, w_Z = 4 (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 found by @mathysrennela · gf2_fast · found at 2000 trials · 2026-11-13

witness operator (support, 2 qubits)

[10, 14]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[2, 5]

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 2–4 (mean 3.489) · H_Z 2–4 (mean 3.438)

qubit degrees H_X 1–2 (mean 1.414) · H_Z 1–2 (mean 1.422)

trapping sets H_X (1,1)×68 (2,0)×24 (3,0)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 68 (1,2): 48 (2,0): 24 (2,1): 128 (2,2): 67 (3,0): 90 (3,1): 177 (3,2): 125 (3,3): 80 (3,4): 16

trapping sets H_Z (1,1)×67 (2,0)×21 (3,0)×87 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 67 (1,2): 49 (2,0): 21 (2,1): 128 (2,2): 68 (3,0): 87 (3,1): 177 (3,2): 123 (3,3): 81 (3,4): 16

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction code-calculus v0.2rc E-PUNCTURE

model Beam 1.0 (claimed, not verified)

date 2026-11-13

notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger.

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 4 (computed)

How this code was found

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

[[116,21,2]] greedy 3 from 120-17-2

Direction & hypothesis

Unrestricted-weight CSS parents with 99 <= n <= 400, searched with the priced construction calculus (code-calculus v0.2rc) greedy multi-step composition. The rule used on this step is E-PUNCTURE.

What was searched

Greedy multi-step composition of C-DELETE-CHECK / E-PEEL / E-PUNCTURE with sector-targeted planning, sector swap, bounded support search, and Pareto ranking. This candidate is step 3 of a chain from codes/120-17-2.json (target sector X).

Evidence trail

Staged deep-verify (2k -> 20k -> 200k -> 2M trials, multi-seed) kept the screened distance unchanged; the witness-backed upper bound is d_X <= 2, d_Z <= 2. No downgrade at any depth.

Dead ends

Parents whose display carries a dependent generator produce no derivable child; candidates whose deep-verified slug collides with an existing board entry are skipped.

Tools

Model: Beam 1.0. Harness: code-calculus v0.2rc greedy gated campaign with the verifier's staged deep-verify used before submission.

Reproduction

Start from codes/120-17-2.json, apply E-PUNCTURE on the target sector, and repeat greedily with Pareto ranking and staged deep-verify.

Parity checks

X-checks 47 (max weight 4) · Z-checks 48 (max weight 4)

H_X (47 checks, sparse supports)

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

H_Z (48 checks, sparse supports)

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

Code ID 116-21-2 · download JSON · raw on GitHub