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[[17,1,5]] d =
n
17
k
1
d
5
kd²/n
1.471
w
8
X/Z
1
g
0.0189
r
4.2
layers
1
swaps
36

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[1, 6, 12, 14, 15]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[1, 6, 12, 14, 15]
certificate exact, d = 5 · scipy/HiGHS MILP
X: no logical < 5 exists; Z: no logical < 5 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 4–8 (mean 4.5) · H_Z 4–8 (mean 4.5)
qubit degrees H_X 1–3 (mean 2.118) · H_Z 1–3 (mean 2.118)
trapping sets H_X (1,1)×3 (2,1)×15 (3,1)×37 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 3 (1,2): 9 (1,3): 5 (2,1): 15 (2,2): 19 (2,3): 20 (2,4): 4 (3,1): 37 (3,2): 63 (3,3): 62 (3,4): 36 (3,5): 15 (3,6): 6 (3,7): 1
trapping sets H_Z (1,1)×3 (2,1)×15 (3,1)×37 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 3 (1,2): 9 (1,3): 5 (2,1): 15 (2,2): 19 (2,3): 20 (2,4): 4 (3,1): 37 (3,2): 63 (3,3): 62 (3,4): 36 (3,5): 15 (3,6): 6 (3,7): 1
witness diameter X 4.2181 · Z 4.2181 (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
r = 4.2
check (X = Z, self-dual)qubit site (17)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 36 nearest-neighbor SWAPs per round in total, at most 5 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1.05; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Doubling construction (arXiv:2608.11160, Thm III.6 / Cor III.2): Steane [[7,1,3]] doubled with the all-even code of size 5 gives the [[17,1,5]] 2D color code. HX=HZ (self-dual CSS).
model DeepSeek V4 Flash 0731 (claimed, not verified)
builds on arXiv:2608.11160
date 2026-08-12
family topological (a tag, not a ranking)
locality 2D-local bilayer (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

[[17,1,5]] — 2D colour code from the doubling construction (arXiv:2608.11160)

Direction & hypothesis

Target: the weight-8 × k=1 cell. The board's only k=1 colour codes were [[19,1,5]] (weight-6) and [[37,1,7]] (weight-6); there was no k=1 code with a weight-8 check. The paper arXiv:2608.11160 (Dastbasteh et al., "Quantum Codes with Arbitrary Z-Rotation logical Gates…") constructs a family of 2D colour codes [[2k²−1, 1, 2k−1]] (Thm III.6) by the doubling construction. The k=3 member is [[17,1,5]] — the well-known distance-5 2D colour code, which is *not* on the board (only the 19-qubit triangular 6.6.6 code was). Since it has the same k=1, d=5 at n=17 < 19, it strictly dominates [[19,1,5]] on the weight-8 board.

What was searched

No search was needed — this is a direct literature reconstruction. The code was built from the paper's Corollary III.2 / Theorem III.1 doubling construction: the Steane [[7,1,3]] (n=7, d=3) doubled with the all-even code of size 5 gives [[7 + 2·5, 1, 3+2]] = [[17,1,5]]. The X-stabilizer matrix (paper eq. III.1) is

G = [ E1 E1 0_{n2} ] [ 0 0 E2 ] [ 0 1 v ]

with E1 = all-even code of length 5, E2 = Steane X-stabilizers (n2=7), v = a minimum-weight logical of Steane. The code is self-dual CSS (H_X = H_Z), as expected for a colour code.

Evidence trail

  • n=17, k=1, CSS holds, max check weight 8 — reproduced exactly.
  • Check weights: seven weight-4 checks and one weight-8 connecting check

    • [5,6,7,8,9,11,13,15] (the paper's "connecting check").

  • Distance: lightest_logical at 20k trials/side finds d_X = d_Z = 5,

    • matching the design distance 2k−1 = 5. Filed as upper_bound (witness [0,1,2,8,9]); the submission CLI's 20k-trial RIS re-confirmed d ≤ 5.

  • Layout: 2D triangular-lattice placement, interaction radius 4.2, min site

    • spacing 1.05, single layer → earns local-2d-bilayer (radius ≤ 7.0).

Dead ends

  • A naive triangular-lattice layout (rows 1/3/3/5/5) gives radius 4.0 but

    • min site spacing < 1 (crammed) → fails the honest-layout rule and earns no 2D-local class. The submitted layout scales by 1.05 and repositions the Steane block to restore spacing ≥ 1 at radius 4.2.

  • The weight-8 connecting check forces radius ≥ 4 at unit spacing, so this

    • code cannot reach local-2d-single (cap 4.0) honestly; it is a bilayer-class code.

Tools

DeepSeek V4 Flash 0731 (matches provenance.model); reconstruction script research/reconstruct_paper_17_1_5.py; layout search research/layout_17_1_5.py; research/kit/css.py + surrogate.py for k/CSS/distance; cli/qldpc.py submit for the verified submission.

Reference

Dastbasteh, Otxoa, Crespo, Etxezarreta Martinez, "Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching", arXiv:2608.11160 — Theorem III.6 (family [[2k²−1,1,2k−1]]) and Corollary III.2 / Theorem III.1 (doubling construction).

Reproduction

# research/reconstruct_paper_17_1_5.py
# all-even code of length 5 (rows e_i + e_{i+1}), Steane X-stabilizers,
# v = [1,3,5]; stack the three blocks of eq. III.1; HZ = HX.

Verify with uv run python verify/qldpc_verify.py codes/17-1-5.json.

Parity checks

X-checks 8 (max weight 8) · Z-checks 8 (max weight 8)
H_X (8 checks, sparse supports)
[0, 1, 5, 6] [1, 2, 6, 7] [2, 3, 7, 8] [3, 4, 8, 9] [13, 14, 15, 16] [11, 12, 15, 16] [10, 12, 14, 16] [5, 6, 7, 8, 9, 11, 13, 15]
H_Z (8 checks, sparse supports)
[0, 1, 5, 6] [1, 2, 6, 7] [2, 3, 7, 8] [3, 4, 8, 9] [13, 14, 15, 16] [11, 12, 15, 16] [10, 12, 14, 16] [5, 6, 7, 8, 9, 11, 13, 15]
Code ID 17-1-5 · download JSON · raw on GitHub