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arxiv: 1907.07848 · v1 · pith:3T4UX2XHnew · submitted 2019-07-18 · 🧮 math.MG · math.FA

Game of Sloanes: Best known packings in complex projective space

John Jasper , Emily J. King , Dustin G. Mixon This is my paper

Pith reviewed 2026-05-24 19:42 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords complex projective spacepoint packingsGrassmannian packingsnumerical optimizationminimum distanceputatively optimalequiangular lines
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The pith

Numerical algorithms produce a table of putatively optimal packings in complex projective spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the long-standing table of optimal point packings from real projective spaces to the complex setting. It applies multiple numerical search methods to identify sets of points that maximize the smallest distance between any pair. These configurations support data representations that resist noise and erasures more effectively than random selections. Several new packings are offered as candidates for global optimality, and the work calls for further improvements from other researchers.

Core claim

By running a collection of numerical optimization routines, the authors assembled an online table that records the largest known minimum distances for varying numbers of points in complex projective spaces of different dimensions, including multiple entries that improve on all previously published configurations.

What carries the argument

Numerical optimization algorithms that iteratively adjust point locations in complex projective space to increase the minimum pairwise distance.

If this is right

  • The tabulated packings supply concrete candidates for error-correcting codes that perform well under noise and erasures.
  • Direct numerical comparison between real and complex packings becomes possible for the same dimension and point count.
  • New entries can be added whenever a better configuration is found, keeping the table current.
  • The listed distances provide targets that any future analytic bound must meet or exceed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • An actively maintained public version of the table would function as a shared benchmark for testing new optimization heuristics.
  • Packings that survive further scrutiny may guide constructions in applied domains that already use complex-valued signals.
  • The same numerical pipeline could be rerun at higher dimensions or with additional symmetry constraints to generate fresh candidates.

Load-bearing premise

The numerical routines have reached configurations whose minimum distance equals or comes very close to the true global maximum.

What would settle it

Any explicit set of the same number of points in the same complex projective space whose minimum distance exceeds the value listed for that entry in the table.

Figures

Figures reproduced from arXiv: 1907.07848 by Dustin G. Mixon, Emily J. King, John Jasper.

Figure 1
Figure 1. Figure 1: The various bounds from Theorem 6 in C 5 are visualized by smoothed colored lines. The best known packings [13] for each n are denoted by black dots. The Bukh–Cox bound (4) is in green, the Welch–Rankin bound (5) in blue, the orthoplex bound (6) in pink, and the Levenstein bound (7) in red. See [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane's online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to have produced a publicly available table of putatively optimal packings of a given number of points in complex projective space, obtained via numerical search algorithms, in direct analogy to Neil Sloane's existing table for real projective spaces. It surveys the relevant literature on projective packings, describes the numerical methods employed, reports several new configurations, and explicitly invites community contributions to improve the tabulated distances.

Significance. If the reported minimum distances hold, the work supplies a practical, open resource that parallels the utility of Sloane's table and supports research in coding theory, frame theory, and quantum information. The explicit 'putatively optimal' labeling, the release of the table, and the call for competitive improvements constitute clear strengths; the numerical approach is appropriately caveated and does not rely on self-referential fitting or unstated global-optimality assumptions.

minor comments (2)
  1. [Methods] The description of the numerical algorithms in the methods section would be strengthened by stating the specific convergence tolerances and restart strategies employed, so that readers can gauge how thoroughly local minima were explored for each (N, d) pair.
  2. [Abstract] A direct URL or DOI to the online table should appear in the abstract or the first paragraph of the introduction to maximize immediate accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the referee found the numerical approach appropriately caveated and the contribution of the open table valuable.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is a table of putatively optimal point packings in complex projective space generated by independent numerical optimization routines that maximize an external minimum-distance objective. No equations, fitted parameters, or self-referential definitions appear in the derivation chain; results are labeled as putative and the work explicitly invites external improvements. The survey of prior literature and description of algorithms do not reduce to self-citation or ansatz smuggling. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The contribution rests on standard definitions of complex projective space and Euclidean distance; no new free parameters, axioms, or invented entities are introduced beyond the choice of numerical optimizer.

axioms (1)
  • standard math Complex projective space is equipped with the standard Fubini-Study metric whose chordal distance is used for the packing objective.
    Invoked when the minimum-distance objective is defined.

pith-pipeline@v0.9.0 · 5680 in / 1155 out tokens · 24397 ms · 2026-05-24T19:42:09.756346+00:00 · methodology

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