P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, 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 = 5, 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	[0.320117135, 0.320117145]	See source attribution	github.com ↗
Site record	0.320117145	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
0.3201171446→0.3201171445
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320117144644→0.320117144641
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.32011714466→0.32011714464
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.320117146→0.320117145
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": [
    [
      [
        "0.212445571",
        "-0.424357357"
      ],
      [
        "-0.083376378",
        "-0.927177744"
      ],
      [
        "-0.156351429",
        "0.134218508"
      ],
      [
        "-0.287789494",
        "-0.999500045"
      ],
      [
        "-0.198169172",
        "-0.569409535"
      ]
    ],
    [
      [
        "0.085987912",
        "0.359824704"
      ],
      [
        "0.149761995",
        "-0.009811995"
      ],
      [
        "-0.999499810",
        "0.183150703"
      ],
      [
        "0.046116363",
        "0.312309741"
      ],
      [
        "0.426535822",
        "-0.340518416"
      ]
    ],
    [
      [
        "0.032736923",
        "-0.733628679"
      ],
      [
        "0.332376859",
        "-0.299319436"
      ],
      [
        "0.968534387",
        "0.339793725"
      ],
      [
        "-0.241265957",
        "-0.132002979"
      ],
      [
        "0.999500228",
        "-0.547778629"
      ]
    ],
    [
      [
        "-0.717933902",
        "-0.460039618"
      ],
      [
        "-0.353084562",
        "0.778400676"
      ],
      [
        "0.660247109",
        "0.090704554"
      ],
      [
        "0.999500046",
        "-0.126945123"
      ],
      [
        "0.348752023",
        "-0.266561302"
      ]
    ],
    [
      [
        "0.530796784",
        "0.106794541"
      ],
      [
        "-0.128441011",
        "-0.999995074"
      ],
      [
        "0.192525279",
        "0.054448266"
      ],
      [
        "0.871877844",
        "-0.117107792"
      ],
      [
        "-0.302564874",
        "0.813243688"
      ]
    ],
    [
      [
        "-0.508644253",
        "-0.707763208"
      ],
      [
        "0.790785879",
        "0.227047913"
      ],
      [
        "-0.295519694",
        "-0.339774314"
      ],
      [
        "0.876328011",
        "-0.425004128"
      ],
      [
        "-0.530058955",
        "0.999995213"
      ]
    ],
    [
      [
        "-0.308733186",
        "-0.652599116"
      ],
      [
        "0.999499512",
        "-0.297039219"
      ],
      [
        "0.268877875",
        "0.510712424"
      ],
      [
        "-0.888623760",
        "-0.077608915"
      ],
      [
        "-0.434382960",
        "0.345543456"
      ]
    ],
    [
      [
        "0.268286488",
        "-0.856150635"
      ],
      [
        "-0.470826388",
        "-0.118753141"
      ],
      [
        "-0.999995194",
        "-0.533929176"
      ],
      [
        "0.092635376",
        "-0.040263747"
      ],
      [
        "-0.195664710",
        "-0.067834578"
      ]
    ],
    [
      [
        "-0.999994950",
        "-0.539361506"
      ],
      [
        "0.396867047",
        "-0.827914458"
      ],
      [
        "-0.478436170",
        "0.151422024"
      ],
      [
        "0.979183687",
        "0.491094734"
      ],
      [
        "0.130515393",
        "-0.513704725"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 9,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.212445571",
        "-0.424357357"
      ],
      [
        "-0.083376378",
        "-0.927177744"
      ],
      [
        "-0.156351429",
        "0.134218508"
      ],
      [
        "-0.287789494",
        "-0.999500045"
      ],
      [
        "-0.198169172",
        "-0.569409535"
      ]
    ],
    [
      [
        "0.085987912",
        "0.359824704"
      ],
      [
        "0.149761995",
        "-0.009811995"
      ],
      [
        "-0.999499810",
        "0.183150703"
      ],
      [
        "0.046116363",
        "0.312309741"
      ],
      [
        "0.426535822",
        "-0.340518416"
      ]
    ],
    [
      [
        "0.032736923",
        "-0.733628679"
      ],
      [
        "0.332376859",
        "-0.299319436"
      ],
      [
        "0.968534387",
        "0.339793725"
      ],
      [
        "-0.241265957",
        "-0.132002979"
      ],
      [
        "0.999500228",
        "-0.547778629"
      ]
    ],
    [
      [
        "-0.717933902",
        "-0.460039618"
      ],
      [
        "-0.353084562",
        "0.778400676"
      ],
      [
        "0.660247109",
        "0.090704554"
      ],
      [
        "0.999500046",
        "-0.126945123"
      ],
      [
        "0.348752023",
        "-0.266561302"
      ]
    ],
    [
      [
        "0.530796784",
        "0.106794541"
      ],
      [
        "-0.128441011",
        "-0.999995074"
      ],
      [
        "0.192525279",
        "0.054448266"
      ],
      [
        "0.871877844",
        "-0.117107792"
      ],
      [
        "-0.302564874",
        "0.813243688"
      ]
    ],
    [
      [
        "-0.508644253",
        "-0.707763208"
      ],
      [
        "0.790785879",
        "0.227047913"
      ],
      [
        "-0.295519694",
        "-0.339774314"
      ],
      [
        "0.876328011",
        "-0.425004128"
      ],
      [
        "-0.530058955",
        "0.999995213"
      ]
    ],
    [
      [
        "-0.308733186",
        "-0.652599116"
      ],
      [
        "0.999499512",
        "-0.297039219"
      ],
      [
        "0.268877875",
        "0.510712424"
      ],
      [
        "-0.888623760",
        "-0.077608915"
      ],
      [
        "-0.434382960",
        "0.345543456"
      ]
    ],
    [
      [
        "0.268286488",
        "-0.856150635"
      ],
      [
        "-0.470826388",
        "-0.118753141"
      ],
      [
        "-0.999995194",
        "-0.533929176"
      ],
      [
        "0.092635376",
        "-0.040263747"
      ],
      [
        "-0.195664710",
        "-0.067834578"
      ]
    ],
    [
      [
        "-0.999994950",
        "-0.539361506"
      ],
      [
        "0.396867047",
        "-0.827914458"
      ],
      [
        "-0.478436170",
        "0.151422024"
      ],
      [
        "0.979183687",
        "0.491094734"
      ],
      [
        "0.130515393",
        "-0.513704725"
      ]
    ]
  ]
}

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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