P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, n = 12

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 = 12

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.357389245, 0.357389255]	See source attribution	github.com ↗
Site record	0.357389252	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 12 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 (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.3573892514→0.3573892513
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.357389251437→0.357389251430
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.35738925144→0.35738925143
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3573892514469→0.3573892514468
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.357389253→0.357389252
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": [
    [
      [
        "-0.281435741",
        "0.297941873"
      ],
      [
        "-0.217290713",
        "0.706805485"
      ],
      [
        "-0.422651285",
        "0.327458210"
      ],
      [
        "0.999498842",
        "-0.492357494"
      ],
      [
        "-0.569122903",
        "-0.178376769"
      ]
    ],
    [
      [
        "0.311003188",
        "-0.308015949"
      ],
      [
        "-0.952224371",
        "-0.274754705"
      ],
      [
        "-0.256013565",
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      ],
      [
        "0.418745635",
        "-0.999500757"
      ],
      [
        "-0.168190077",
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      ]
    ],
    [
      [
        "0.245168212",
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      ],
      [
        "0.939209111",
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      [
        "-0.376959635",
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      ],
      [
        "0.416305429",
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      [
        "0.478098228",
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      ]
    ],
    [
      [
        "-0.334960798",
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      ],
      [
        "-0.232786784",
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      ],
      [
        "-0.110325288",
        "0.182929416"
      ],
      [
        "0.082093641",
        "-0.183047600"
      ],
      [
        "0.440677472",
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      ]
    ],
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      [
        "0.999501062",
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      [
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      [
        "-0.632274902",
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      [
        "-0.029668963",
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    [
      [
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      ],
      [
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    [
      [
        "0.469166320",
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      ],
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      [
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      ],
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    [
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      ],
      [
        "0.027391363",
        "0.094382325"
      ],
      [
        "0.727869273",
        "-0.673964784"
      ],
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    ],
    [
      [
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      ],
      [
        "0.772620257",
        "-0.085488247"
      ],
      [
        "0.191763698",
        "-0.429250413"
      ],
      [
        "0.145615805",
        "0.409773817"
      ],
      [
        "-0.061156900",
        "0.367893922"
      ]
    ],
    [
      [
        "-0.579037157",
        "-0.507926195"
      ],
      [
        "-0.336015299",
        "-0.516553031"
      ],
      [
        "0.517320156",
        "0.999500298"
      ],
      [
        "0.285858009",
        "0.713294250"
      ],
      [
        "0.709882578",
        "0.432449097"
      ]
    ],
    [
      [
        "-0.279599515",
        "0.119744527"
      ],
      [
        "0.999500467",
        "-0.391709867"
      ],
      [
        "-0.459883079",
        "-0.096484558"
      ],
      [
        "0.610839092",
        "0.036362775"
      ],
      [
        "-0.926260620",
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      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 12,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "-0.281435741",
        "0.297941873"
      ],
      [
        "-0.217290713",
        "0.706805485"
      ],
      [
        "-0.422651285",
        "0.327458210"
      ],
      [
        "0.999498842",
        "-0.492357494"
      ],
      [
        "-0.569122903",
        "-0.178376769"
      ]
    ],
    [
      [
        "0.311003188",
        "-0.308015949"
      ],
      [
        "-0.952224371",
        "-0.274754705"
      ],
      [
        "-0.256013565",
        "-0.324336901"
      ],
      [
        "0.418745635",
        "-0.999500757"
      ],
      [
        "-0.168190077",
        "-0.269545965"
      ]
    ],
    [
      [
        "0.245168212",
        "-0.196977436"
      ],
      [
        "0.939209111",
        "-0.295868705"
      ],
      [
        "-0.376959635",
        "-0.398833115"
      ],
      [
        "0.416305429",
        "-0.999500783"
      ],
      [
        "0.478098228",
        "-0.146137947"
      ]
    ],
    [
      [
        "-0.334960798",
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      ],
      [
        "-0.232786784",
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      ],
      [
        "-0.110325288",
        "0.182929416"
      ],
      [
        "0.082093641",
        "-0.183047600"
      ],
      [
        "0.440677472",
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      ]
    ],
    [
      [
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      ],
      [
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      ],
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        "0.151244823",
        "0.917478700"
      ],
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        "0.052257268",
        "-0.329674845"
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      [
        "0.999501062",
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      ]
    ],
    [
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      ],
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      ],
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        "0.445220120",
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      [
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        "-0.999499985",
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        "0.727869273",
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      [
        "0.285858009",
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      ],
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    [
      [
        "-0.279599515",
        "0.119744527"
      ],
      [
        "0.999500467",
        "-0.391709867"
      ],
      [
        "-0.459883079",
        "-0.096484558"
      ],
      [
        "0.610839092",
        "0.036362775"
      ],
      [
        "-0.926260620",
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      ]
    ]
  ]
}

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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