P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 6, 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 = 6, 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	[0.272203145, 0.272203155]	See source attribution	github.com ↗
Site record	0.27220315	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 (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.2722031499→0.2722031498
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149948→0.272203149945
2026-09-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.27220314995→0.27220314994
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149958→0.272203149952
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.27220314996→0.27220314995
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203149967→0.272203149962
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.272203152→0.27220315
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": [
    [
      [
        "0.999499563",
        "-0.199572763"
      ],
      [
        "0.447251077",
        "0.544147372"
      ],
      [
        "-0.201225674",
        "-0.200923148"
      ],
      [
        "0.827632643",
        "-0.009505971"
      ],
      [
        "0.183630022",
        "-0.590664826"
      ],
      [
        "-0.857751470",
        "0.840445603"
      ]
    ],
    [
      [
        "-0.631551936",
        "0.576236980"
      ],
      [
        "0.157957888",
        "-0.889323906"
      ],
      [
        "-0.064480125",
        "-0.457339132"
      ],
      [
        "0.312570108",
        "-0.219637271"
      ],
      [
        "0.999498778",
        "-0.434655746"
      ],
      [
        "-0.184185466",
        "-0.379671753"
      ]
    ],
    [
      [
        "-0.107739484",
        "-0.372929224"
      ],
      [
        "-0.682380717",
        "-0.634242056"
      ],
      [
        "-0.022807240",
        "0.299100304"
      ],
      [
        "0.522167562",
        "0.035576823"
      ],
      [
        "0.755496832",
        "0.489289283"
      ],
      [
        "-0.999500321",
        "0.408357269"
      ]
    ],
    [
      [
        "0.085609340",
        "0.334557121"
      ],
      [
        "0.212163855",
        "-0.032144926"
      ],
      [
        "-0.419079037",
        "0.999500149"
      ],
      [
        "-0.916167037",
        "0.638067645"
      ],
      [
        "-0.061269982",
        "0.041415189"
      ],
      [
        "-0.177171120",
        "-0.011015412"
      ]
    ],
    [
      [
        "0.475662476",
        "-0.129303141"
      ],
      [
        "-0.288035533",
        "0.578535209"
      ],
      [
        "0.150512997",
        "0.005309555"
      ],
      [
        "-0.029899101",
        "-0.781459188"
      ],
      [
        "0.302919024",
        "0.999499880"
      ],
      [
        "-0.415374760",
        "-0.532983748"
      ]
    ],
    [
      [
        "-0.465632395",
        "-0.321423840"
      ],
      [
        "-0.860847307",
        "0.868039111"
      ],
      [
        "-0.137602902",
        "0.020983683"
      ],
      [
        "0.486166011",
        "0.999500208"
      ],
      [
        "-0.096529674",
        "-0.234682355"
      ],
      [
        "0.260624277",
        "-0.075591580"
      ]
    ],
    [
      [
        "-0.399270392",
        "0.171133064"
      ],
      [
        "-0.521923524",
        "0.148353816"
      ],
      [
        "-0.485164438",
        "0.999500208"
      ],
      [
        "0.326058948",
        "-0.534183683"
      ],
      [
        "0.435506251",
        "-0.203446001"
      ],
      [
        "0.240046744",
        "-0.556421607"
      ]
    ],
    [
      [
        "-0.655431546",
        "-0.999500928"
      ],
      [
        "0.292935242",
        "0.246177527"
      ],
      [
        "-0.503815266",
        "0.713537922"
      ],
      [
        "0.086975686",
        "0.077165556"
      ],
      [
        "-0.219811237",
        "-0.164700604"
      ],
      [
        "-0.667966908",
        "-0.112373038"
      ]
    ],
    [
      [
        "0.548766226",
        "0.588861468"
      ],
      [
        "0.973532366",
        "0.279430519"
      ],
      [
        "-0.636976794",
        "0.334998303"
      ],
      [
        "0.372408485",
        "0.592522483"
      ],
      [
        "0.999499953",
        "-0.344122967"
      ],
      [
        "0.755743316",
        "-0.181422596"
      ]
    ],
    [
      [
        "0.754090933",
        "0.468543339"
      ],
      [
        "-0.089391185",
        "-0.250902723"
      ],
      [
        "-0.449010902",
        "0.874992037"
      ],
      [
        "0.241640636",
        "-0.305550889"
      ],
      [
        "0.121046999",
        "0.878027413"
      ],
      [
        "0.328567244",
        "0.999499996"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 10,
  "d": 6
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.999499563",
        "-0.199572763"
      ],
      [
        "0.447251077",
        "0.544147372"
      ],
      [
        "-0.201225674",
        "-0.200923148"
      ],
      [
        "0.827632643",
        "-0.009505971"
      ],
      [
        "0.183630022",
        "-0.590664826"
      ],
      [
        "-0.857751470",
        "0.840445603"
      ]
    ],
    [
      [
        "-0.631551936",
        "0.576236980"
      ],
      [
        "0.157957888",
        "-0.889323906"
      ],
      [
        "-0.064480125",
        "-0.457339132"
      ],
      [
        "0.312570108",
        "-0.219637271"
      ],
      [
        "0.999498778",
        "-0.434655746"
      ],
      [
        "-0.184185466",
        "-0.379671753"
      ]
    ],
    [
      [
        "-0.107739484",
        "-0.372929224"
      ],
      [
        "-0.682380717",
        "-0.634242056"
      ],
      [
        "-0.022807240",
        "0.299100304"
      ],
      [
        "0.522167562",
        "0.035576823"
      ],
      [
        "0.755496832",
        "0.489289283"
      ],
      [
        "-0.999500321",
        "0.408357269"
      ]
    ],
    [
      [
        "0.085609340",
        "0.334557121"
      ],
      [
        "0.212163855",
        "-0.032144926"
      ],
      [
        "-0.419079037",
        "0.999500149"
      ],
      [
        "-0.916167037",
        "0.638067645"
      ],
      [
        "-0.061269982",
        "0.041415189"
      ],
      [
        "-0.177171120",
        "-0.011015412"
      ]
    ],
    [
      [
        "0.475662476",
        "-0.129303141"
      ],
      [
        "-0.288035533",
        "0.578535209"
      ],
      [
        "0.150512997",
        "0.005309555"
      ],
      [
        "-0.029899101",
        "-0.781459188"
      ],
      [
        "0.302919024",
        "0.999499880"
      ],
      [
        "-0.415374760",
        "-0.532983748"
      ]
    ],
    [
      [
        "-0.465632395",
        "-0.321423840"
      ],
      [
        "-0.860847307",
        "0.868039111"
      ],
      [
        "-0.137602902",
        "0.020983683"
      ],
      [
        "0.486166011",
        "0.999500208"
      ],
      [
        "-0.096529674",
        "-0.234682355"
      ],
      [
        "0.260624277",
        "-0.075591580"
      ]
    ],
    [
      [
        "-0.399270392",
        "0.171133064"
      ],
      [
        "-0.521923524",
        "0.148353816"
      ],
      [
        "-0.485164438",
        "0.999500208"
      ],
      [
        "0.326058948",
        "-0.534183683"
      ],
      [
        "0.435506251",
        "-0.203446001"
      ],
      [
        "0.240046744",
        "-0.556421607"
      ]
    ],
    [
      [
        "-0.655431546",
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      ],
      [
        "0.292935242",
        "0.246177527"
      ],
      [
        "-0.503815266",
        "0.713537922"
      ],
      [
        "0.086975686",
        "0.077165556"
      ],
      [
        "-0.219811237",
        "-0.164700604"
      ],
      [
        "-0.667966908",
        "-0.112373038"
      ]
    ],
    [
      [
        "0.548766226",
        "0.588861468"
      ],
      [
        "0.973532366",
        "0.279430519"
      ],
      [
        "-0.636976794",
        "0.334998303"
      ],
      [
        "0.372408485",
        "0.592522483"
      ],
      [
        "0.999499953",
        "-0.344122967"
      ],
      [
        "0.755743316",
        "-0.181422596"
      ]
    ],
    [
      [
        "0.754090933",
        "0.468543339"
      ],
      [
        "-0.089391185",
        "-0.250902723"
      ],
      [
        "-0.449010902",
        "0.874992037"
      ],
      [
        "0.241640636",
        "-0.305550889"
      ],
      [
        "0.121046999",
        "0.878027413"
      ],
      [
        "0.328567244",
        "0.999499996"
      ]
    ]
  ]
}

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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