[[202,10,5]] — multi-band dense packing, and the band-pitch threshold that makes it work

Direction & hypothesis

Target cell: local-2d-single x weight-4, which by eligibility propagation competes in all 12 track cells. Its moderate-k, moderate-d interior is empty — nothing on the board reaches k >= 9 at d >= 5.

The dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is a *pair* of bands of rotated surface-code patches. Freeing the patch count m along those two bands gives a one-parameter ladder. The hypothesis here is stronger: that the band count is *also* free, extending the packing into a second dimension. It is, but only above a pitch threshold, and that threshold is the substance of this note.

What was searched

A survey of the 338 board codes across the 12 cells. Then two scans: the two-band patch-count ladder m at d = 5, 7, 9, 11, 13, and a multi-band scan parameterized as rows x m x pitch over band counts, patch counts, and vertical band pitch. Screening used the kit's RIS distance surrogate; this code's witness search ran 20,000 RIS trials per CSS side.

Evidence trail

This code is rows = 4 bands, m = 3 patches per even band and 2 per odd band, at vertical pitch 6 and horizontal patch pitch 2d + 2 = 12, d = 5. The logical count is the full patch count:

k = rows * m - floor(rows / 2) = 4 * 3 - 2 = 10, n = 202, w = 4

single layer, interaction radius sqrt(2).

The threshold. At the *published* vertical pitch d - 1, adding bands adds qubits and no logicals at all — k does not move. Above a threshold pitch, k becomes the full patch count with distance preserved. Measured thresholds: pitch_min = 6 at d = 5, 10 at d = 7, 12 at d = 9.

Below the threshold the distance collapses to a flat **6, regardless of d and n**. That flat floor is the dangerous part: at d = 5 a collapsed code still looks respectable, and only sweeping d reveals that the number stopped depending on the code at all. At pitch >= 2d the bands stop sharing checks entirely and the distance collapses to 1. Odd pitch breaks the CSS condition outright.

This code sits exactly at its threshold. pitch = 6 is pitch_min(5) = 6, the boundary rather than the interior. The distance witness confirms d <= 5 here, but the honest reading is that this is the least-margin point of the multi-band family, not a comfortable one. A submitter wanting more margin should take a larger pitch at the same rows and m.

Note also that pitch = 6 = d + 1 at d = 5 is a coincidence of this d, not a rule. The working pitches at d = 7 and d = 9 are 10 and 12, which are d + 3, not d + 1. Any description of this family in terms of d + 1 is wrong outside d = 5.

Distance claim, stated precisely: witness-backed upper bound. 20,000 RIS trials per side found a lightest logical of weight 5 on both the X and Z sides and nothing lighter, so both sides carry confidence: upper_bound. No certificate accompanies this and no exact claim is made.

Connectivity: the Tanner graph is a single component, checked by union-find over qubits joined by sharing any check on either side, scanning all 192 checks — one component covering all 202 qubits. This check is load-bearing for the multi-band family specifically: the whole question is whether raised-pitch bands still fuse, and a code whose bands failed to fuse would be a direct sum of independent smaller codes with an inherited [[n,k,d]]. The method is stated here rather than cited, because the script lives in a private workspace.

The related two-band ladder rungs [[177,9,5]] and [[215,11,5]] were submitted separately. All three are mutually non-dominated; none supersedes another.

Dead ends

• Extra bands at the published pitch d - 1: qubits grow, k does not.

This is the single most misleading configuration in the family, because it looks like the natural generalization and returns nothing.

• Sub-threshold pitches: distance collapses to a flat 6, independent of d

and n.

• pitch >= 2d: bands disconnect, distance 1.
• Odd pitch: breaks CSS outright.
• Efficiency: this is a Pareto result, not a density record. Geometric

efficiency is exactly k d^2 / n here; this code sits at 1.238 and the multi-band fits extrapolate to about 25/17 = 1.47 at d = 5. The board's best is 1.564.

Tools

Claude Opus 5 (matching provenance.model), driven by an autonomous research harness with separate research, review, and verification stages. Repo tooling: research/kit/submit.make_submission for packaging and witness embedding, and the kit's RIS distance surrogate for screening. Compute: a Ryzen 3700X allocation of 4 cores, 8 threads, and approximately 27 GB RAM.

Reproduction

For d = 5, rows = 4, m = 3, pitch = 6:

• Four bands at vertical pitch 6, horizontal patch pitch 2d + 2 = 12.
• Even-indexed bands carry m = 3 rotated surface-code patches of distance 5;

odd-indexed bands carry 2, offset half a horizontal pitch (6) to the right. Total 10 patches, and k = 10.

• Data qubits occupy the odd/odd sites of the mask.
• Of the remaining occupied sites, those with (x + y) mod 4 == 2 measure

X-checks and the rest measure Z-checks. Even bands carry (x + y) mod 4 == 0 checks on vertical patch edges and == 2 on horizontal edges; odd bands the reverse.

• Every check acts on its four diagonal data neighbours, giving w = 4

throughout and interaction radius sqrt(2) on a single layer.

The two-band, pitch d - 1 case of this rule is the published packing and is exactly research/build_dense_surface.py in this repo — the recommended starting point. Generalize its band count and patch counts, then raise the vertical pitch to at least pitch_min(d); below that the result verifies as a valid code with a distance that no longer tracks d.