P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 6, n = 13

Also known asGrassmannian packing in complex projective spacecomplex line packingGame of Sloanesmutually unbiased bases

Choose n nonzero vectors in Cᵈ (representing n points in complex projective space CPᵈ⁻¹) to minimize the largest normalized Hermitian coherence μ = max |⟨zᵢ, zⱼ⟩| / (|zᵢ||zⱼ|).

Instanced = 6, n = 13

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.311824555, 0.311824565]	See source attribution	github.com ↗
Site record	0.311824564	zzzcy	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

ContainerComplex d-space Cᵈ; an answer is n points of complex projective space CPᵈ⁻¹
SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
Coherenceμ² is the largest squared normalized Hermitian inner product over all pairs of complex projective points; numerators and denominators are rational and compared directly by cross-multiplication $μ^{2} = \max_{i < j} \frac{{| ⟨ z_{i}, z_{j} ⟩ |}^{2}}{{| z_{i} |}^{2} {| z_{j} |}^{2}}$
ScoringThe record is stored as ceil(μ²·10¹⁸); the page displays μ, rounded up at the ninth decimal
ObjectiveMake μ², equivalently μ, as small as possible over all legal complex projective codebooks $\min_{z_{1}, \dots, z_{n}} μ^{2}$

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 13 directions (brighter = closer)

Getting a feel for it

Where it is used

Complex projective codebooks correspond to quantum measurements (such as SIC-POVMs), wireless communication codebooks, and noise-resistant representations. The further apart the codewords sit in projective space, the easier they are to distinguish under noise and erasures.

Where the frontier is

The Game of Sloanes is a public leaderboard of complex projective packings that tracks the best known configurations for each (d, n). Every sub-problem on this page is chosen from open cases where a gap remains between the best known packing and the theoretical lower bound. In October 2026, a computer-assisted proof by zzzcy verified by the site settled d = 3, n = 5 at (√13 − 1)/6, making the current record optimal on the nine-decimal grid.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.3118245631→0.3118245630
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.31182456318→0.31182456317
2026-09-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.3118245632→0.3118245631
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.311824565→0.311824564
2026-09-01

ANSWER FORMAT

How to write your answer

Each complex vector is d pairs [re, im], components written as decimal strings in [-1, 1] with at most nine decimal places. Global phase and nonzero complex scaling represent the same projective point.

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      [
        "0.078495999",
        "-0.999501411"
      ],
      [
        "0.187054141",
        "0.390263905"
      ],
      [
        "0.339264117",
        "-0.738305877"
      ],
      [
        "0.401707085",
        "0.624812104"
      ],
      [
        "-0.606433371",
        "-0.560582351"
      ],
      [
        "-0.218335645",
        "0.149132984"
      ]
    ],
    [
      [
        "-0.999993765",
        "-0.242222377"
      ],
      [
        "-0.384613684",
        "-0.811531948"
      ],
      [
        "-0.460612866",
        "0.470555868"
      ],
      [
        "0.455581894",
        "0.781388008"
      ],
      [
        "0.143797539",
        "0.174210689"
      ],
      [
        "-0.314465751",
        "-0.591275320"
      ]
    ],
    [
      [
        "-0.999994050",
        "0.484074679"
      ],
      [
        "0.358759659",
        "0.468043877"
      ],
      [
        "0.436065860",
        "-0.224372188"
      ],
      [
        "0.940469748",
        "-0.134436900"
      ],
      [
        "-0.090689767",
        "-0.887767686"
      ],
      [
        "0.552273186",
        "-0.473166257"
      ]
    ],
    [
      [
        "-0.308922459",
        "-0.118063474"
      ],
      [
        "0.999994252",
        "-0.872320850"
      ],
      [
        "-0.232273499",
        "0.958257785"
      ],
      [
        "-0.136664323",
        "-0.072882418"
      ],
      [
        "-0.120638702",
        "0.214686486"
      ],
      [
        "-0.901673690",
        "0.451048799"
      ]
    ],
    [
      [
        "-0.295734757",
        "0.105762318"
      ],
      [
        "0.398282094",
        "-0.076065642"
      ],
      [
        "0.120074965",
        "-0.273507452"
      ],
      [
        "-0.192595505",
        "-0.426339327"
      ],
      [
        "0.999995232",
        "0.027839567"
      ],
      [
        "0.745880215",
        "0.699763677"
      ]
    ],
    [
      [
        "-0.783698060",
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      ],
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        "-0.549665723",
        "0.814027142"
      ],
      [
        "-0.954687173",
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      ],
      [
        "-0.996127762",
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      ],
      [
        "0.182810764",
        "0.104358808"
      ],
      [
        "0.999996959",
        "-0.782755854"
      ]
    ],
    [
      [
        "0.369056865",
        "-0.999500958"
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      [
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        "0.228940884"
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        "0.161311742"
      ]
    ],
    [
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        "-0.356029206"
      ],
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        "-0.342990275",
        "0.659575757"
      ],
      [
        "-0.218435724",
        "0.558956446"
      ],
      [
        "0.999500498",
        "-0.575230097"
      ],
      [
        "-0.505248194",
        "0.042954909"
      ],
      [
        "0.421466092",
        "-0.098525604"
      ]
    ],
    [
      [
        "0.070298486",
        "0.572116766"
      ],
      [
        "-0.559026329",
        "-0.569393439"
      ],
      [
        "-0.512346272",
        "0.547619159"
      ],
      [
        "-0.508224069",
        "0.862553832"
      ],
      [
        "-0.999993280",
        "0.756442488"
      ],
      [
        "0.837506625",
        "0.095643200"
      ]
    ],
    [
      [
        "0.102696500",
        "0.564341472"
      ],
      [
        "-0.169794356",
        "0.999500922"
      ],
      [
        "-0.108036438",
        "-0.201361934"
      ],
      [
        "-0.718558271",
        "0.286169248"
      ],
      [
        "0.311698374",
        "0.828218148"
      ],
      [
        "-0.183885530",
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      ]
    ],
    [
      [
        "0.222037334",
        "0.164297823"
      ],
      [
        "0.131347552",
        "-0.196474022"
      ],
      [
        "-0.527897343",
        "-0.999995408"
      ],
      [
        "0.458269356",
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      ],
      [
        "0.144197132",
        "0.979786060"
      ],
      [
        "-0.523220840",
        "-0.478969750"
      ]
    ],
    [
      [
        "-0.305978999",
        "-0.353434452"
      ],
      [
        "0.999996443",
        "0.557839713"
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      [
        "0.019044388",
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        "0.492307534"
      ],
      [
        "-0.536884837",
        "0.509239319"
      ],
      [
        "0.741818938",
        "0.592120795"
      ]
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "d": 6
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.078495999",
        "-0.999501411"
      ],
      [
        "0.187054141",
        "0.390263905"
      ],
      [
        "0.339264117",
        "-0.738305877"
      ],
      [
        "0.401707085",
        "0.624812104"
      ],
      [
        "-0.606433371",
        "-0.560582351"
      ],
      [
        "-0.218335645",
        "0.149132984"
      ]
    ],
    [
      [
        "-0.999993765",
        "-0.242222377"
      ],
      [
        "-0.384613684",
        "-0.811531948"
      ],
      [
        "-0.460612866",
        "0.470555868"
      ],
      [
        "0.455581894",
        "0.781388008"
      ],
      [
        "0.143797539",
        "0.174210689"
      ],
      [
        "-0.314465751",
        "-0.591275320"
      ]
    ],
    [
      [
        "-0.999994050",
        "0.484074679"
      ],
      [
        "0.358759659",
        "0.468043877"
      ],
      [
        "0.436065860",
        "-0.224372188"
      ],
      [
        "0.940469748",
        "-0.134436900"
      ],
      [
        "-0.090689767",
        "-0.887767686"
      ],
      [
        "0.552273186",
        "-0.473166257"
      ]
    ],
    [
      [
        "-0.308922459",
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      ],
      [
        "0.999994252",
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      [
        "-0.232273499",
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      ],
      [
        "-0.136664323",
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      [
        "-0.120638702",
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      [
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    ],
    [
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      ],
      [
        "0.398282094",
        "-0.076065642"
      ],
      [
        "0.120074965",
        "-0.273507452"
      ],
      [
        "-0.192595505",
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      ],
      [
        "0.999995232",
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      ],
      [
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    [
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    [
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    [
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        "0.063356650",
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        "-0.342990275",
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      [
        "-0.218435724",
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      [
        "0.999500498",
        "-0.575230097"
      ],
      [
        "-0.505248194",
        "0.042954909"
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        "0.421466092",
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    ],
    [
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      [
        "-0.559026329",
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      ],
      [
        "-0.512346272",
        "0.547619159"
      ],
      [
        "-0.508224069",
        "0.862553832"
      ],
      [
        "-0.999993280",
        "0.756442488"
      ],
      [
        "0.837506625",
        "0.095643200"
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    ],
    [
      [
        "0.102696500",
        "0.564341472"
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      [
        "-0.169794356",
        "0.999500922"
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      [
        "-0.108036438",
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      ],
      [
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      ],
      [
        "0.311698374",
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      ],
      [
        "-0.183885530",
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      ]
    ],
    [
      [
        "0.222037334",
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      ],
      [
        "0.131347552",
        "-0.196474022"
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      [
        "-0.527897343",
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      [
        "0.458269356",
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      ],
      [
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      ],
      [
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    ],
    [
      [
        "-0.305978999",
        "-0.353434452"
      ],
      [
        "0.999996443",
        "0.557839713"
      ],
      [
        "0.019044388",
        "0.793998208"
      ],
      [
        "0.014186021",
        "0.492307534"
      ],
      [
        "-0.536884837",
        "0.509239319"
      ],
      [
        "0.741818938",
        "0.592120795"
      ]
    ]
  ]
}

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer. · Verifier v1.0.0

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