P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 3, 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 = 3, 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	[0.647754475, 0.647754485]	See source attribution	github.com ↗
Site record	0.647754476	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 (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.64775447599→0.64775447592
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754475991→0.647754475990
2026-09-16
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754475992→0.647754475991
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.647754477→0.647754476
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": [
    [
      [
        "0.442702527",
        "0.999499130"
      ],
      [
        "-0.742755331",
        "-0.479293139"
      ],
      [
        "-0.020566721",
        "0.512216072"
      ]
    ],
    [
      [
        "-0.155159493",
        "-0.062065736"
      ],
      [
        "0.087503697",
        "-0.305542109"
      ],
      [
        "0.999500371",
        "0.167457292"
      ]
    ],
    [
      [
        "0.375060956",
        "0.807899933"
      ],
      [
        "0.999498210",
        "0.860129560"
      ],
      [
        "0.236683568",
        "0.358323167"
      ]
    ],
    [
      [
        "-0.217677852",
        "-0.386636177"
      ],
      [
        "0.999498944",
        "-0.486935207"
      ],
      [
        "0.118177655",
        "0.463130369"
      ]
    ],
    [
      [
        "0.072049680",
        "-0.309835588"
      ],
      [
        "0.999497862",
        "-0.671880793"
      ],
      [
        "-0.664168731",
        "-0.332761377"
      ]
    ],
    [
      [
        "-0.803527080",
        "0.376819099"
      ],
      [
        "-0.104950806",
        "0.127737016"
      ],
      [
        "-0.999497586",
        "0.359780732"
      ]
    ],
    [
      [
        "-0.035502895",
        "0.999499663"
      ],
      [
        "-0.248262962",
        "-0.288497585"
      ],
      [
        "-0.078451686",
        "-0.587284258"
      ]
    ],
    [
      [
        "-0.999500608",
        "0.690942842"
      ],
      [
        "0.740245769",
        "-0.334102433"
      ],
      [
        "0.392657504",
        "0.973398717"
      ]
    ],
    [
      [
        "0.558221088",
        "-0.236489180"
      ],
      [
        "-0.539206820",
        "-0.184397561"
      ],
      [
        "-0.119078804",
        "0.999500186"
      ]
    ],
    [
      [
        "0.377266144",
        "-0.551691049"
      ],
      [
        "0.923314911",
        "0.616613213"
      ],
      [
        "0.999500241",
        "0.434823719"
      ]
    ],
    [
      [
        "-0.566166158",
        "-0.030997411"
      ],
      [
        "0.669922855",
        "0.999500098"
      ],
      [
        "-0.925403323",
        "-0.674857490"
      ]
    ],
    [
      [
        "-0.999502238",
        "-0.021438890"
      ],
      [
        "-0.869091418",
        "0.117861055"
      ],
      [
        "0.808165826",
        "-0.290343961"
      ]
    ],
    [
      [
        "-0.303196170",
        "0.999499268"
      ],
      [
        "0.463626619",
        "0.072015908"
      ],
      [
        "-0.180856249",
        "0.020087334"
      ]
    ],
    [
      [
        "0.213445604",
        "0.462407841"
      ],
      [
        "0.191950095",
        "-0.999500221"
      ],
      [
        "-0.058555924",
        "-0.289723577"
      ]
    ],
    [
      [
        "0.999500800",
        "0.161102348"
      ],
      [
        "0.253566239",
        "0.316747975"
      ],
      [
        "0.147715976",
        "-0.330807868"
      ]
    ],
    [
      [
        "0.259988026",
        "-0.591876124"
      ],
      [
        "-0.366401386",
        "-0.420353471"
      ],
      [
        "-0.999499805",
        "-0.013591214"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 16,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.442702527",
        "0.999499130"
      ],
      [
        "-0.742755331",
        "-0.479293139"
      ],
      [
        "-0.020566721",
        "0.512216072"
      ]
    ],
    [
      [
        "-0.155159493",
        "-0.062065736"
      ],
      [
        "0.087503697",
        "-0.305542109"
      ],
      [
        "0.999500371",
        "0.167457292"
      ]
    ],
    [
      [
        "0.375060956",
        "0.807899933"
      ],
      [
        "0.999498210",
        "0.860129560"
      ],
      [
        "0.236683568",
        "0.358323167"
      ]
    ],
    [
      [
        "-0.217677852",
        "-0.386636177"
      ],
      [
        "0.999498944",
        "-0.486935207"
      ],
      [
        "0.118177655",
        "0.463130369"
      ]
    ],
    [
      [
        "0.072049680",
        "-0.309835588"
      ],
      [
        "0.999497862",
        "-0.671880793"
      ],
      [
        "-0.664168731",
        "-0.332761377"
      ]
    ],
    [
      [
        "-0.803527080",
        "0.376819099"
      ],
      [
        "-0.104950806",
        "0.127737016"
      ],
      [
        "-0.999497586",
        "0.359780732"
      ]
    ],
    [
      [
        "-0.035502895",
        "0.999499663"
      ],
      [
        "-0.248262962",
        "-0.288497585"
      ],
      [
        "-0.078451686",
        "-0.587284258"
      ]
    ],
    [
      [
        "-0.999500608",
        "0.690942842"
      ],
      [
        "0.740245769",
        "-0.334102433"
      ],
      [
        "0.392657504",
        "0.973398717"
      ]
    ],
    [
      [
        "0.558221088",
        "-0.236489180"
      ],
      [
        "-0.539206820",
        "-0.184397561"
      ],
      [
        "-0.119078804",
        "0.999500186"
      ]
    ],
    [
      [
        "0.377266144",
        "-0.551691049"
      ],
      [
        "0.923314911",
        "0.616613213"
      ],
      [
        "0.999500241",
        "0.434823719"
      ]
    ],
    [
      [
        "-0.566166158",
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      ],
      [
        "0.669922855",
        "0.999500098"
      ],
      [
        "-0.925403323",
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    ],
    [
      [
        "-0.999502238",
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      ],
      [
        "-0.869091418",
        "0.117861055"
      ],
      [
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    ],
    [
      [
        "-0.303196170",
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      ],
      [
        "0.463626619",
        "0.072015908"
      ],
      [
        "-0.180856249",
        "0.020087334"
      ]
    ],
    [
      [
        "0.213445604",
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      ],
      [
        "0.191950095",
        "-0.999500221"
      ],
      [
        "-0.058555924",
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      ]
    ],
    [
      [
        "0.999500800",
        "0.161102348"
      ],
      [
        "0.253566239",
        "0.316747975"
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      [
        "0.147715976",
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    ],
    [
      [
        "0.259988026",
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      ],
      [
        "-0.366401386",
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      ],
      [
        "-0.999499805",
        "-0.013591214"
      ]
    ]
  ]
}

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

Discussion

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