P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 4, n = 9

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 = 4, n = 9

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.401850115, 0.401850125]	See source attribution	github.com ↗
Site record	0.401850121	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 9 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
0.40185012049→0.40185012042
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850120494→0.401850120492
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850120496→0.401850120494
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.401850122→0.401850121
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": [
    [
      [
        "0.272911897",
        "0.854746890"
      ],
      [
        "0.677574206",
        "-0.021190766"
      ],
      [
        "0.199849069",
        "0.160045053"
      ],
      [
        "0.821752371",
        "0.999499860"
      ]
    ],
    [
      [
        "-0.059838190",
        "0.743051642"
      ],
      [
        "-0.075909675",
        "-0.298217343"
      ],
      [
        "0.079743009",
        "-0.078971251"
      ],
      [
        "-0.999498657",
        "-0.541301266"
      ]
    ],
    [
      [
        "0.670265048",
        "0.427166631"
      ],
      [
        "0.999498643",
        "-0.637010890"
      ],
      [
        "0.833532178",
        "0.449710734"
      ],
      [
        "-0.298128124",
        "-0.485777262"
      ]
    ],
    [
      [
        "-0.306736402",
        "0.898068074"
      ],
      [
        "0.108607162",
        "-0.999499674"
      ],
      [
        "0.269220173",
        "-0.496953485"
      ],
      [
        "0.094374068",
        "0.190914380"
      ]
    ],
    [
      [
        "0.586335878",
        "-0.748948812"
      ],
      [
        "-0.166111888",
        "0.263051178"
      ],
      [
        "0.999499756",
        "0.033172814"
      ],
      [
        "0.233759675",
        "-0.057983304"
      ]
    ],
    [
      [
        "0.084589679",
        "0.336284223"
      ],
      [
        "-0.809781375",
        "0.350549489"
      ],
      [
        "0.598811255",
        "-0.999499929"
      ],
      [
        "-0.263388584",
        "-0.603384525"
      ]
    ],
    [
      [
        "-0.262424181",
        "0.035909886"
      ],
      [
        "-0.462267141",
        "0.295453287"
      ],
      [
        "0.223491544",
        "0.999499903"
      ],
      [
        "-0.177717346",
        "-0.589021776"
      ]
    ],
    [
      [
        "0.185814164",
        "0.794957205"
      ],
      [
        "-0.973031164",
        "0.999500362"
      ],
      [
        "0.403756641",
        "0.133374796"
      ],
      [
        "0.210526342",
        "0.518489834"
      ]
    ],
    [
      [
        "-0.402105899",
        "-0.716717987"
      ],
      [
        "-0.255029729",
        "-0.256250918"
      ],
      [
        "0.873239437",
        "0.466369335"
      ],
      [
        "-0.462423052",
        "0.999502001"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 9,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.272911897",
        "0.854746890"
      ],
      [
        "0.677574206",
        "-0.021190766"
      ],
      [
        "0.199849069",
        "0.160045053"
      ],
      [
        "0.821752371",
        "0.999499860"
      ]
    ],
    [
      [
        "-0.059838190",
        "0.743051642"
      ],
      [
        "-0.075909675",
        "-0.298217343"
      ],
      [
        "0.079743009",
        "-0.078971251"
      ],
      [
        "-0.999498657",
        "-0.541301266"
      ]
    ],
    [
      [
        "0.670265048",
        "0.427166631"
      ],
      [
        "0.999498643",
        "-0.637010890"
      ],
      [
        "0.833532178",
        "0.449710734"
      ],
      [
        "-0.298128124",
        "-0.485777262"
      ]
    ],
    [
      [
        "-0.306736402",
        "0.898068074"
      ],
      [
        "0.108607162",
        "-0.999499674"
      ],
      [
        "0.269220173",
        "-0.496953485"
      ],
      [
        "0.094374068",
        "0.190914380"
      ]
    ],
    [
      [
        "0.586335878",
        "-0.748948812"
      ],
      [
        "-0.166111888",
        "0.263051178"
      ],
      [
        "0.999499756",
        "0.033172814"
      ],
      [
        "0.233759675",
        "-0.057983304"
      ]
    ],
    [
      [
        "0.084589679",
        "0.336284223"
      ],
      [
        "-0.809781375",
        "0.350549489"
      ],
      [
        "0.598811255",
        "-0.999499929"
      ],
      [
        "-0.263388584",
        "-0.603384525"
      ]
    ],
    [
      [
        "-0.262424181",
        "0.035909886"
      ],
      [
        "-0.462267141",
        "0.295453287"
      ],
      [
        "0.223491544",
        "0.999499903"
      ],
      [
        "-0.177717346",
        "-0.589021776"
      ]
    ],
    [
      [
        "0.185814164",
        "0.794957205"
      ],
      [
        "-0.973031164",
        "0.999500362"
      ],
      [
        "0.403756641",
        "0.133374796"
      ],
      [
        "0.210526342",
        "0.518489834"
      ]
    ],
    [
      [
        "-0.402105899",
        "-0.716717987"
      ],
      [
        "-0.255029729",
        "-0.256250918"
      ],
      [
        "0.873239437",
        "0.466369335"
      ],
      [
        "-0.462423052",
        "0.999502001"
      ]
    ]
  ]
}

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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