P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, n = 14

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 = 14

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.375674755, 0.375674765]	See source attribution	github.com ↗
Site record	0.375674759	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 14 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 (9 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.3756747585→0.3756747584
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758566→0.375674758564
2026-09-23
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567475857→0.37567475856
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758575→0.375674758571
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758577→0.375674758575
2026-09-14
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567475858→0.37567475857
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.375674758589→0.375674758581
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3756747586→0.3756747585
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37567476→0.375674759
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": [
    [
      [
        "-0.265161276",
        "0.201338659"
      ],
      [
        "-0.387854670",
        "-0.139877108"
      ],
      [
        "0.538845051",
        "0.999500019"
      ],
      [
        "-0.573588113",
        "-0.408994460"
      ],
      [
        "-0.058029617",
        "0.398631280"
      ]
    ],
    [
      [
        "0.182086059",
        "-0.863854612"
      ],
      [
        "-0.334281310",
        "0.795827132"
      ],
      [
        "0.379100320",
        "-0.993301003"
      ],
      [
        "0.069413512",
        "0.124165925"
      ],
      [
        "-0.628738030",
        "0.999502577"
      ]
    ],
    [
      [
        "-0.070865576",
        "0.825757805"
      ],
      [
        "0.999503368",
        "-0.218076822"
      ],
      [
        "0.474777701",
        "-0.463804051"
      ],
      [
        "-0.154147246",
        "-0.154226976"
      ],
      [
        "0.525803430",
        "0.425526336"
      ]
    ],
    [
      [
        "0.923142499",
        "0.829614557"
      ],
      [
        "0.676099648",
        "-0.331641576"
      ],
      [
        "-0.073291816",
        "-0.451489993"
      ],
      [
        "0.490007884",
        "-0.505680490"
      ],
      [
        "-0.999499073",
        "-0.425420195"
      ]
    ],
    [
      [
        "0.440138927",
        "-0.586987674"
      ],
      [
        "0.796323342",
        "-0.556935609"
      ],
      [
        "0.999499915",
        "-0.628210053"
      ],
      [
        "-0.199224856",
        "-0.066597100"
      ],
      [
        "-0.314393221",
        "-0.308618330"
      ]
    ],
    [
      [
        "0.999499656",
        "-0.438060978"
      ],
      [
        "0.871132546",
        "-0.146292812"
      ],
      [
        "-0.528665160",
        "0.726633688"
      ],
      [
        "-0.222975780",
        "0.729075287"
      ],
      [
        "-0.719348337",
        "-0.834774469"
      ]
    ],
    [
      [
        "0.155136387",
        "-0.280561673"
      ],
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        "0.836844539",
        "0.716684289"
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        "0.058252786",
        "0.360691457"
      ],
      [
        "-0.999504685",
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        "0.400121237",
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    [
      [
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        "0.999499763"
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      [
        "-0.473065197",
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      [
        "0.007759814",
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    [
      [
        "0.998254717",
        "-0.504179881"
      ],
      [
        "0.139100742",
        "0.301291938"
      ],
      [
        "0.465144144",
        "-0.697626305"
      ],
      [
        "-0.453125644",
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      [
        "0.999500417",
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    ],
    [
      [
        "0.345589509",
        "-0.999500750"
      ],
      [
        "0.453102948",
        "0.609799574"
      ],
      [
        "-0.174547774",
        "-0.159195252"
      ],
      [
        "-0.108686390",
        "-0.708362897"
      ],
      [
        "-0.621917379",
        "-0.387924657"
      ]
    ],
    [
      [
        "-0.214887392",
        "0.095059756"
      ],
      [
        "0.617192012",
        "0.420405772"
      ],
      [
        "0.373107052",
        "-0.325033813"
      ],
      [
        "-0.302179562",
        "0.749439950"
      ],
      [
        "-0.999501641",
        "-0.140992404"
      ]
    ],
    [
      [
        "-0.335352620",
        "-0.691282375"
      ],
      [
        "0.383711439",
        "0.176960311"
      ],
      [
        "0.065705619",
        "0.777310025"
      ],
      [
        "0.721740628",
        "-0.164262632"
      ],
      [
        "0.212897941",
        "-0.999499844"
      ]
    ],
    [
      [
        "-0.058625843",
        "-0.119224209"
      ],
      [
        "-0.090666271",
        "0.449099663"
      ],
      [
        "0.322301343",
        "-0.019954466"
      ],
      [
        "0.999503014",
        "0.582333167"
      ],
      [
        "0.109110758",
        "0.289348636"
      ]
    ],
    [
      [
        "0.588540973",
        "-0.586441537"
      ],
      [
        "0.771132285",
        "0.582067653"
      ],
      [
        "-0.395442001",
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      [
        "0.614254507",
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      ],
      [
        "-0.118640444",
        "0.999500244"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "-0.265161276",
        "0.201338659"
      ],
      [
        "-0.387854670",
        "-0.139877108"
      ],
      [
        "0.538845051",
        "0.999500019"
      ],
      [
        "-0.573588113",
        "-0.408994460"
      ],
      [
        "-0.058029617",
        "0.398631280"
      ]
    ],
    [
      [
        "0.182086059",
        "-0.863854612"
      ],
      [
        "-0.334281310",
        "0.795827132"
      ],
      [
        "0.379100320",
        "-0.993301003"
      ],
      [
        "0.069413512",
        "0.124165925"
      ],
      [
        "-0.628738030",
        "0.999502577"
      ]
    ],
    [
      [
        "-0.070865576",
        "0.825757805"
      ],
      [
        "0.999503368",
        "-0.218076822"
      ],
      [
        "0.474777701",
        "-0.463804051"
      ],
      [
        "-0.154147246",
        "-0.154226976"
      ],
      [
        "0.525803430",
        "0.425526336"
      ]
    ],
    [
      [
        "0.923142499",
        "0.829614557"
      ],
      [
        "0.676099648",
        "-0.331641576"
      ],
      [
        "-0.073291816",
        "-0.451489993"
      ],
      [
        "0.490007884",
        "-0.505680490"
      ],
      [
        "-0.999499073",
        "-0.425420195"
      ]
    ],
    [
      [
        "0.440138927",
        "-0.586987674"
      ],
      [
        "0.796323342",
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      [
        "0.999499915",
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      [
        "-0.199224856",
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      ],
      [
        "-0.314393221",
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      ]
    ],
    [
      [
        "0.999499656",
        "-0.438060978"
      ],
      [
        "0.871132546",
        "-0.146292812"
      ],
      [
        "-0.528665160",
        "0.726633688"
      ],
      [
        "-0.222975780",
        "0.729075287"
      ],
      [
        "-0.719348337",
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      ]
    ],
    [
      [
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      [
        "-0.999504685",
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      [
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    [
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        "-0.294391117",
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      [
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    [
      [
        "0.998254717",
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      [
        "0.139100742",
        "0.301291938"
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      [
        "0.465144144",
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      [
        "-0.453125644",
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      [
        "0.999500417",
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    ],
    [
      [
        "0.345589509",
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      [
        "0.453102948",
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      [
        "-0.174547774",
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      ],
      [
        "-0.108686390",
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      [
        "-0.621917379",
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      ]
    ],
    [
      [
        "-0.214887392",
        "0.095059756"
      ],
      [
        "0.617192012",
        "0.420405772"
      ],
      [
        "0.373107052",
        "-0.325033813"
      ],
      [
        "-0.302179562",
        "0.749439950"
      ],
      [
        "-0.999501641",
        "-0.140992404"
      ]
    ],
    [
      [
        "-0.335352620",
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      ],
      [
        "0.383711439",
        "0.176960311"
      ],
      [
        "0.065705619",
        "0.777310025"
      ],
      [
        "0.721740628",
        "-0.164262632"
      ],
      [
        "0.212897941",
        "-0.999499844"
      ]
    ],
    [
      [
        "-0.058625843",
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      ],
      [
        "-0.090666271",
        "0.449099663"
      ],
      [
        "0.322301343",
        "-0.019954466"
      ],
      [
        "0.999503014",
        "0.582333167"
      ],
      [
        "0.109110758",
        "0.289348636"
      ]
    ],
    [
      [
        "0.588540973",
        "-0.586441537"
      ],
      [
        "0.771132285",
        "0.582067653"
      ],
      [
        "-0.395442001",
        "0.778223693"
      ],
      [
        "0.614254507",
        "-0.210806482"
      ],
      [
        "-0.118640444",
        "0.999500244"
      ]
    ]
  ]
}

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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