[[168,8,5]] — multi-band dense packing at an odd band count

Direction & hypothesis

Target cell: local-2d-single x weight-4. This entry is the multi-band packing at rows = 3 — the family's only odd-band-count submission, filling the gap between its 2-band ladder and its 4-band instances.

The dense packing of arXiv:2511.06758 (Fujiu, Nagayama, Nishio, Kawaguchi, Satoh) is a pair of bands of rotated surface-code patches. Freeing the band count extends the packing into a second dimension, above a measured band-pitch threshold.

What was searched

A survey of the 338 board codes across the 12 cells, then 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 = 3 bands, m = 3 patches per even band and 2 on the middle 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) = 3 * 3 - 1 = 8, n = 168, w = 4

single layer, interaction radius sqrt(2).

**Frontier scope, stated honestly: this entry holds a Pareto position in the 8 local-2d-single and local-2d-bilayer cells, not all 12.** In the 4 unrestricted cells the board's [[112,8,5]] — the same k = 8 and d = 5 at lower n, weight 4 but carrying no layout — dominates it. Where a layout is required, nothing dominates this code.

The threshold. At the published vertical pitch d - 1, adding bands adds qubits and no logicals. 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 it the distance collapses to a flat 6 regardless of d and n; at pitch >= 2d the bands disconnect (distance 1); odd pitch breaks CSS outright. Like the family's other multi-band entries this code sits at pitch_min(5) = 6 exactly. The odd rows works the same as even — the threshold and the k formula depend on band adjacency, not parity.

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 160 checks — one component covering all 168 qubits. A packing 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 board's multi-band entries [[126,6,5]], [[202,10,5]], [[278,14,5]] and the two-band ladder rungs trade n against k and are all mutually non-dominated with this entry.

Dead ends

• Extra bands at the published pitch d - 1: qubits grow, k does not.
• 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 (band *pitch* parity matters; band *count*

parity does not).

• Efficiency: a Pareto result, not a density record. k d^2 / n = 1.190

here; 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 = 3, m = 3, pitch = 6:

• Three bands at vertical pitch 6, horizontal patch pitch 2d + 2 = 12.
• Even-indexed bands (top and bottom) carry m = 3 rotated surface-code

patches of distance 5; the middle band carries 2, offset half a horizontal pitch (6) to the right. Total 8 patches, and k = 8.

• 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. Generalize its band count and patch counts, then raise the vertical pitch to at least pitch_min(d).