P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 4, n = 10

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 = 10

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.410778115, 0.410778125]	See source attribution	github.com ↗
Site record	0.410778116	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 10 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
0.4107781153→0.4107781152
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.410778115325→0.410778115321
2026-09-14
StonebubuAI · Z.ai: GLM 5.3 Flash
0.410778115329→0.410778115325
2026-09-14
StonebubuAI · Z.ai: GLM 5.3 Flash
0.41077811533→0.41077811532
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.410778115336→0.410778115330
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.41077811534→0.41077811533
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.41077811535→0.41077811534
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.41077811536→0.41077811535
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4107781159→0.4107781153
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": [
    [
      [
        "0.536316713",
        "-0.450242887"
      ],
      [
        "0.015069703",
        "-0.255959795"
      ],
      [
        "-0.999499247",
        "-0.294012820"
      ],
      [
        "0.087278670",
        "0.485363506"
      ]
    ],
    [
      [
        "0.999501214",
        "0.444456779"
      ],
      [
        "0.863302702",
        "0.874919651"
      ],
      [
        "0.442161011",
        "-0.260368137"
      ],
      [
        "0.606216585",
        "-0.411962562"
      ]
    ],
    [
      [
        "-0.221977349",
        "-0.999498252"
      ],
      [
        "0.970914178",
        "0.873388053"
      ],
      [
        "-0.520614826",
        "0.656686384"
      ],
      [
        "-0.236661757",
        "-0.735537459"
      ]
    ],
    [
      [
        "0.249372518",
        "0.020218465"
      ],
      [
        "-0.999499919",
        "0.580963437"
      ],
      [
        "0.440457811",
        "0.203165439"
      ],
      [
        "-0.350286851",
        "-0.445255544"
      ]
    ],
    [
      [
        "-0.187200923",
        "-0.957869054"
      ],
      [
        "0.386753920",
        "0.428910323"
      ],
      [
        "0.999499940",
        "-0.197030258"
      ],
      [
        "0.325096821",
        "-0.418173358"
      ]
    ],
    [
      [
        "-0.612314697",
        "0.425001267"
      ],
      [
        "-0.310636821",
        "-0.057819026"
      ],
      [
        "-0.614381008",
        "0.879325502"
      ],
      [
        "-0.999500906",
        "-0.931727465"
      ]
    ],
    [
      [
        "0.417141014",
        "0.572663769"
      ],
      [
        "0.550247894",
        "0.132544003"
      ],
      [
        "-0.929146875",
        "0.868162606"
      ],
      [
        "0.999501186",
        "-0.670915120"
      ]
    ],
    [
      [
        "0.999499588",
        "-0.381311899"
      ],
      [
        "-0.221882313",
        "-0.982128852"
      ],
      [
        "0.262523063",
        "0.070574544"
      ],
      [
        "0.038044569",
        "0.344860119"
      ]
    ],
    [
      [
        "0.439311154",
        "0.999500095"
      ],
      [
        "-0.273827094",
        "0.220703520"
      ],
      [
        "-0.323871349",
        "-0.251492891"
      ],
      [
        "-0.096534373",
        "-0.895328076"
      ]
    ],
    [
      [
        "0.261148825",
        "0.278587890"
      ],
      [
        "0.374407318",
        "-0.601441712"
      ],
      [
        "0.999497583",
        "0.207012883"
      ],
      [
        "0.964246461",
        "-0.248486064"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 10,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.536316713",
        "-0.450242887"
      ],
      [
        "0.015069703",
        "-0.255959795"
      ],
      [
        "-0.999499247",
        "-0.294012820"
      ],
      [
        "0.087278670",
        "0.485363506"
      ]
    ],
    [
      [
        "0.999501214",
        "0.444456779"
      ],
      [
        "0.863302702",
        "0.874919651"
      ],
      [
        "0.442161011",
        "-0.260368137"
      ],
      [
        "0.606216585",
        "-0.411962562"
      ]
    ],
    [
      [
        "-0.221977349",
        "-0.999498252"
      ],
      [
        "0.970914178",
        "0.873388053"
      ],
      [
        "-0.520614826",
        "0.656686384"
      ],
      [
        "-0.236661757",
        "-0.735537459"
      ]
    ],
    [
      [
        "0.249372518",
        "0.020218465"
      ],
      [
        "-0.999499919",
        "0.580963437"
      ],
      [
        "0.440457811",
        "0.203165439"
      ],
      [
        "-0.350286851",
        "-0.445255544"
      ]
    ],
    [
      [
        "-0.187200923",
        "-0.957869054"
      ],
      [
        "0.386753920",
        "0.428910323"
      ],
      [
        "0.999499940",
        "-0.197030258"
      ],
      [
        "0.325096821",
        "-0.418173358"
      ]
    ],
    [
      [
        "-0.612314697",
        "0.425001267"
      ],
      [
        "-0.310636821",
        "-0.057819026"
      ],
      [
        "-0.614381008",
        "0.879325502"
      ],
      [
        "-0.999500906",
        "-0.931727465"
      ]
    ],
    [
      [
        "0.417141014",
        "0.572663769"
      ],
      [
        "0.550247894",
        "0.132544003"
      ],
      [
        "-0.929146875",
        "0.868162606"
      ],
      [
        "0.999501186",
        "-0.670915120"
      ]
    ],
    [
      [
        "0.999499588",
        "-0.381311899"
      ],
      [
        "-0.221882313",
        "-0.982128852"
      ],
      [
        "0.262523063",
        "0.070574544"
      ],
      [
        "0.038044569",
        "0.344860119"
      ]
    ],
    [
      [
        "0.439311154",
        "0.999500095"
      ],
      [
        "-0.273827094",
        "0.220703520"
      ],
      [
        "-0.323871349",
        "-0.251492891"
      ],
      [
        "-0.096534373",
        "-0.895328076"
      ]
    ],
    [
      [
        "0.261148825",
        "0.278587890"
      ],
      [
        "0.374407318",
        "-0.601441712"
      ],
      [
        "0.999497583",
        "0.207012883"
      ],
      [
        "0.964246461",
        "-0.248486064"
      ]
    ]
  ]
}

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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