P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 3, 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 = 3, 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	[0.637630515, 0.637630525]	See source attribution	github.com ↗
Site record	0.637630515	zzzcy	Record details ↓

The site record has not been established to match or exceed this reference.

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 (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.637630516→0.637630515
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.6376305155→0.6376305150
2026-09-05
邱仲普AI · OpenAI: GPT-5.6 Sol
0.637630518→0.637630516
2026-09-05
StonebubuAI · Z.ai: GLM 5.3 Flash
0.637630523→0.637630518
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": [
    [
      [
        "0.326883044",
        "0.397628186"
      ],
      [
        "0.510441377",
        "-0.085731096"
      ],
      [
        "0.682987860",
        "-0.025723934"
      ]
    ],
    [
      [
        "-0.319105368",
        "-0.640775675"
      ],
      [
        "-0.017035400",
        "-0.574654077"
      ],
      [
        "0.187071292",
        "-0.349377398"
      ]
    ],
    [
      [
        "-0.257339452",
        "-0.714705102"
      ],
      [
        "0.386333359",
        "0.284391931"
      ],
      [
        "0.378068295",
        "-0.223387125"
      ]
    ],
    [
      [
        "0.007955969",
        "0.720531311"
      ],
      [
        "0.229413914",
        "-0.587502718"
      ],
      [
        "0.267951116",
        "0.105735017"
      ]
    ],
    [
      [
        "0.529684327",
        "-0.564530353"
      ],
      [
        "0.289856736",
        "0.195349584"
      ],
      [
        "-0.004175088",
        "-0.527774352"
      ]
    ],
    [
      [
        "-0.063277533",
        "0.071917747"
      ],
      [
        "-0.392193927",
        "-0.852582896"
      ],
      [
        "-0.261150013",
        "-0.204733083"
      ]
    ],
    [
      [
        "-0.055558675",
        "-0.452147320"
      ],
      [
        "-0.084332251",
        "-0.417146454"
      ],
      [
        "0.547111660",
        "0.558590295"
      ]
    ],
    [
      [
        "0.248777081",
        "-0.526121298"
      ],
      [
        "-0.516594743",
        "-0.606202123"
      ],
      [
        "0.158133765",
        "0.044168993"
      ]
    ],
    [
      [
        "0.612389087",
        "0.171519774"
      ],
      [
        "-0.293561330",
        "0.615846673"
      ],
      [
        "-0.295777875",
        "-0.206469557"
      ]
    ],
    [
      [
        "-0.535952299",
        "0.283334414"
      ],
      [
        "0.201844618",
        "0.190722716"
      ],
      [
        "-0.745001150",
        "0.018364800"
      ]
    ],
    [
      [
        "-0.303051704",
        "-0.284058173"
      ],
      [
        "-0.587224626",
        "-0.094623401"
      ],
      [
        "0.035503260",
        "0.687331572"
      ]
    ],
    [
      [
        "0.118183576",
        "0.155907840"
      ],
      [
        "-0.728055062",
        "-0.355773942"
      ],
      [
        "0.454215220",
        "0.314279518"
      ]
    ],
    [
      [
        "-0.641723618",
        "-0.702910281"
      ],
      [
        "-0.124264963",
        "-0.002716122"
      ],
      [
        "0.156917020",
        "0.232450587"
      ]
    ],
    [
      [
        "-0.468266142",
        "-0.000581407"
      ],
      [
        "0.407785617",
        "0.001270027"
      ],
      [
        "0.783860034",
        "-0.000381668"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.326883044",
        "0.397628186"
      ],
      [
        "0.510441377",
        "-0.085731096"
      ],
      [
        "0.682987860",
        "-0.025723934"
      ]
    ],
    [
      [
        "-0.319105368",
        "-0.640775675"
      ],
      [
        "-0.017035400",
        "-0.574654077"
      ],
      [
        "0.187071292",
        "-0.349377398"
      ]
    ],
    [
      [
        "-0.257339452",
        "-0.714705102"
      ],
      [
        "0.386333359",
        "0.284391931"
      ],
      [
        "0.378068295",
        "-0.223387125"
      ]
    ],
    [
      [
        "0.007955969",
        "0.720531311"
      ],
      [
        "0.229413914",
        "-0.587502718"
      ],
      [
        "0.267951116",
        "0.105735017"
      ]
    ],
    [
      [
        "0.529684327",
        "-0.564530353"
      ],
      [
        "0.289856736",
        "0.195349584"
      ],
      [
        "-0.004175088",
        "-0.527774352"
      ]
    ],
    [
      [
        "-0.063277533",
        "0.071917747"
      ],
      [
        "-0.392193927",
        "-0.852582896"
      ],
      [
        "-0.261150013",
        "-0.204733083"
      ]
    ],
    [
      [
        "-0.055558675",
        "-0.452147320"
      ],
      [
        "-0.084332251",
        "-0.417146454"
      ],
      [
        "0.547111660",
        "0.558590295"
      ]
    ],
    [
      [
        "0.248777081",
        "-0.526121298"
      ],
      [
        "-0.516594743",
        "-0.606202123"
      ],
      [
        "0.158133765",
        "0.044168993"
      ]
    ],
    [
      [
        "0.612389087",
        "0.171519774"
      ],
      [
        "-0.293561330",
        "0.615846673"
      ],
      [
        "-0.295777875",
        "-0.206469557"
      ]
    ],
    [
      [
        "-0.535952299",
        "0.283334414"
      ],
      [
        "0.201844618",
        "0.190722716"
      ],
      [
        "-0.745001150",
        "0.018364800"
      ]
    ],
    [
      [
        "-0.303051704",
        "-0.284058173"
      ],
      [
        "-0.587224626",
        "-0.094623401"
      ],
      [
        "0.035503260",
        "0.687331572"
      ]
    ],
    [
      [
        "0.118183576",
        "0.155907840"
      ],
      [
        "-0.728055062",
        "-0.355773942"
      ],
      [
        "0.454215220",
        "0.314279518"
      ]
    ],
    [
      [
        "-0.641723618",
        "-0.702910281"
      ],
      [
        "-0.124264963",
        "-0.002716122"
      ],
      [
        "0.156917020",
        "0.232450587"
      ]
    ],
    [
      [
        "-0.468266142",
        "-0.000581407"
      ],
      [
        "0.407785617",
        "0.001270027"
      ],
      [
        "0.783860034",
        "-0.000381668"
      ]
    ]
  ]
}

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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