P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, n = 16

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 = 5, n = 16

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.388092835, 0.388092845]	See source attribution	github.com ↗
Site record	0.388092844	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 16 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 (8 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.3880928434→0.3880928433
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843419→0.388092843418
2026-09-08
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284342→0.38809284341
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843426→0.388092843423
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284343→0.38809284342
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092843437→0.388092843433
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.38809284345→0.38809284343
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.388092845→0.388092844
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": [
    [
      [
        "0.804530131",
        "0.177488590"
      ],
      [
        "-0.465191723",
        "-0.999506863"
      ],
      [
        "-0.899527906",
        "-0.406321349"
      ],
      [
        "0.906798153",
        "0.030984542"
      ],
      [
        "0.464754879",
        "0.575376753"
      ]
    ],
    [
      [
        "-0.999498110",
        "0.043369979"
      ],
      [
        "-0.211293961",
        "-0.460842397"
      ],
      [
        "-0.751098797",
        "0.080951707"
      ],
      [
        "0.413331206",
        "-0.338696001"
      ],
      [
        "0.338804852",
        "0.253205410"
      ]
    ],
    [
      [
        "0.999503546",
        "-0.300208868"
      ],
      [
        "0.127226705",
        "-0.385385848"
      ],
      [
        "-0.000155323",
        "0.513387496"
      ],
      [
        "0.277027230",
        "0.658388422"
      ],
      [
        "0.663818154",
        "-0.480487377"
      ]
    ],
    [
      [
        "0.490562247",
        "0.135096061"
      ],
      [
        "0.999495128",
        "-0.140304641"
      ],
      [
        "0.297641968",
        "0.671637479"
      ],
      [
        "0.014290753",
        "0.144566648"
      ],
      [
        "-0.321535979",
        "0.408738573"
      ]
    ],
    [
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        "0.999501731",
        "0.229715274"
      ],
      [
        "-0.836717041",
        "-0.720539794"
      ],
      [
        "-0.761522472",
        "0.131291330"
      ],
      [
        "-0.164927061",
        "-0.165179835"
      ]
    ],
    [
      [
        "0.058978117",
        "-0.037870024"
      ],
      [
        "0.404833903",
        "0.794017631"
      ],
      [
        "0.254657917",
        "-0.208880182"
      ],
      [
        "0.493032467",
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      ],
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        "0.018003014",
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    ],
    [
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    ],
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      [
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        "-0.303161170"
      ],
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    ],
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        "0.337100146"
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      [
        "0.348145640",
        "-0.698671793"
      ]
    ],
    [
      [
        "0.154078589",
        "-0.090956319"
      ],
      [
        "-0.002602365",
        "0.215654767"
      ],
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        "-0.269693506",
        "-0.041413668"
      ],
      [
        "-0.574512038",
        "0.529981929"
      ],
      [
        "-0.999500839",
        "0.680801902"
      ]
    ],
    [
      [
        "0.988928239",
        "-0.683080053"
      ],
      [
        "0.359045990",
        "0.630326091"
      ],
      [
        "0.541385554",
        "0.477730869"
      ],
      [
        "-0.253235612",
        "-0.501300837"
      ],
      [
        "0.999499901",
        "0.598521374"
      ]
    ],
    [
      [
        "-0.999501195",
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      ],
      [
        "-0.288394170",
        "0.144245470"
      ],
      [
        "-0.368899306",
        "0.211136453"
      ],
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        "-0.701184576",
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      [
        "-0.043217249",
        "0.179294144"
      ]
    ],
    [
      [
        "-0.169222947",
        "0.163772152"
      ],
      [
        "-0.069245020",
        "0.535875321"
      ],
      [
        "-0.234356274",
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      ],
      [
        "0.999493779",
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      ],
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    [
      [
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      ],
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      ],
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        "-0.283177624",
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      ],
      [
        "0.999498762",
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      ]
    ],
    [
      [
        "0.669105178",
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        "0.809807088",
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      ],
      [
        "0.330631870",
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      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 16,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.804530131",
        "0.177488590"
      ],
      [
        "-0.465191723",
        "-0.999506863"
      ],
      [
        "-0.899527906",
        "-0.406321349"
      ],
      [
        "0.906798153",
        "0.030984542"
      ],
      [
        "0.464754879",
        "0.575376753"
      ]
    ],
    [
      [
        "-0.999498110",
        "0.043369979"
      ],
      [
        "-0.211293961",
        "-0.460842397"
      ],
      [
        "-0.751098797",
        "0.080951707"
      ],
      [
        "0.413331206",
        "-0.338696001"
      ],
      [
        "0.338804852",
        "0.253205410"
      ]
    ],
    [
      [
        "0.999503546",
        "-0.300208868"
      ],
      [
        "0.127226705",
        "-0.385385848"
      ],
      [
        "-0.000155323",
        "0.513387496"
      ],
      [
        "0.277027230",
        "0.658388422"
      ],
      [
        "0.663818154",
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    [
      [
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        "0.135096061"
      ],
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        "0.999495128",
        "-0.140304641"
      ],
      [
        "0.297641968",
        "0.671637479"
      ],
      [
        "0.014290753",
        "0.144566648"
      ],
      [
        "-0.321535979",
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    ],
    [
      [
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    [
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      [
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    ],
    [
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    [
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      [
        "-0.377602953",
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      [
        "-0.736170608",
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        "0.348145640",
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      ]
    ],
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        "0.154078589",
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      ],
      [
        "-0.269693506",
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      ],
      [
        "-0.574512038",
        "0.529981929"
      ],
      [
        "-0.999500839",
        "0.680801902"
      ]
    ],
    [
      [
        "0.988928239",
        "-0.683080053"
      ],
      [
        "0.359045990",
        "0.630326091"
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      [
        "0.541385554",
        "0.477730869"
      ],
      [
        "-0.253235612",
        "-0.501300837"
      ],
      [
        "0.999499901",
        "0.598521374"
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    ],
    [
      [
        "-0.999501195",
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      ],
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        "-0.288394170",
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      [
        "-0.368899306",
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    ],
    [
      [
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        "-0.069245020",
        "0.535875321"
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      [
        "-0.234356274",
        "0.143406988"
      ],
      [
        "0.999493779",
        "-0.729515162"
      ],
      [
        "-0.081828856",
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      ]
    ],
    [
      [
        "-0.895533428",
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      ],
      [
        "0.853099131",
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      ],
      [
        "-0.567584083",
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      ],
      [
        "-0.283177624",
        "-0.400356056"
      ],
      [
        "0.999498762",
        "0.280283552"
      ]
    ],
    [
      [
        "0.669105178",
        "-0.422865031"
      ],
      [
        "0.999498556",
        "0.357439561"
      ],
      [
        "-0.936170736",
        "-0.002562825"
      ],
      [
        "0.809807088",
        "-0.061988172"
      ],
      [
        "0.330631870",
        "0.165509533"
      ]
    ]
  ]
}

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

DISCUSSION

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