P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 4, n = 12

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 = 12

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.427742315, 0.427742325]	See source attribution	github.com ↗
Site record	0.42774232	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 12 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 (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.427742321→0.42774232
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.42774232008→0.42774232007
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.42774232009→0.42774232008
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4277423201→0.4277423200
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.4277423209→0.4277423201
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": [
    [
      [
        "-0.032837873",
        "0.784887557"
      ],
      [
        "0.582157750",
        "0.066080348"
      ],
      [
        "-0.320398208",
        "-0.999499847"
      ],
      [
        "-0.331258438",
        "0.659381664"
      ]
    ],
    [
      [
        "0.746801166",
        "0.478786614"
      ],
      [
        "0.472334893",
        "0.040560867"
      ],
      [
        "0.999500375",
        "-0.657939412"
      ],
      [
        "0.438981538",
        "0.533369702"
      ]
    ],
    [
      [
        "0.498557023",
        "0.836427832"
      ],
      [
        "-0.474647632",
        "-0.483013231"
      ],
      [
        "-0.214207151",
        "0.999499872"
      ],
      [
        "-0.334233395",
        "-0.592593156"
      ]
    ],
    [
      [
        "0.132333198",
        "0.543472384"
      ],
      [
        "0.999500662",
        "0.237114002"
      ],
      [
        "-0.059159601",
        "0.010947934"
      ],
      [
        "0.501227738",
        "-0.115830550"
      ]
    ],
    [
      [
        "0.562979727",
        "-0.999501670"
      ],
      [
        "0.119392330",
        "-0.932135746"
      ],
      [
        "0.866452393",
        "0.270544990"
      ],
      [
        "0.614777104",
        "0.693772513"
      ]
    ],
    [
      [
        "0.059863600",
        "0.933253129"
      ],
      [
        "0.262510032",
        "-0.999500029"
      ],
      [
        "0.322366005",
        "0.038052137"
      ],
      [
        "-0.786442583",
        "0.321322665"
      ]
    ],
    [
      [
        "0.999499635",
        "-0.201165798"
      ],
      [
        "0.384169520",
        "0.745392330"
      ],
      [
        "0.635860709",
        "0.629696306"
      ],
      [
        "0.810838205",
        "-0.253357489"
      ]
    ],
    [
      [
        "-0.813017982",
        "0.293079724"
      ],
      [
        "-0.462989650",
        "0.287022242"
      ],
      [
        "-0.112450146",
        "0.677587380"
      ],
      [
        "0.999500333",
        "-0.202785608"
      ]
    ],
    [
      [
        "0.875222992",
        "0.049611226"
      ],
      [
        "0.193958766",
        "0.999501538"
      ],
      [
        "-0.268541780",
        "-0.527730952"
      ],
      [
        "-0.085353065",
        "0.257714886"
      ]
    ],
    [
      [
        "-0.702215088",
        "-0.283013799"
      ],
      [
        "-0.055450613",
        "-0.481679665"
      ],
      [
        "0.785968780",
        "0.002079711"
      ],
      [
        "-0.537117243",
        "0.999499979"
      ]
    ],
    [
      [
        "-0.790650571",
        "0.116019669"
      ],
      [
        "0.803878836",
        "0.058792072"
      ],
      [
        "-0.568299963",
        "0.388338285"
      ],
      [
        "-0.330813539",
        "0.999499097"
      ]
    ],
    [
      [
        "0.183911658",
        "0.815032516"
      ],
      [
        "-0.179584253",
        "-0.621831630"
      ],
      [
        "-0.170893456",
        "-0.999499280"
      ],
      [
        "-0.158086262",
        "-0.707025025"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 12,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      [
        "-0.032837873",
        "0.784887557"
      ],
      [
        "0.582157750",
        "0.066080348"
      ],
      [
        "-0.320398208",
        "-0.999499847"
      ],
      [
        "-0.331258438",
        "0.659381664"
      ]
    ],
    [
      [
        "0.746801166",
        "0.478786614"
      ],
      [
        "0.472334893",
        "0.040560867"
      ],
      [
        "0.999500375",
        "-0.657939412"
      ],
      [
        "0.438981538",
        "0.533369702"
      ]
    ],
    [
      [
        "0.498557023",
        "0.836427832"
      ],
      [
        "-0.474647632",
        "-0.483013231"
      ],
      [
        "-0.214207151",
        "0.999499872"
      ],
      [
        "-0.334233395",
        "-0.592593156"
      ]
    ],
    [
      [
        "0.132333198",
        "0.543472384"
      ],
      [
        "0.999500662",
        "0.237114002"
      ],
      [
        "-0.059159601",
        "0.010947934"
      ],
      [
        "0.501227738",
        "-0.115830550"
      ]
    ],
    [
      [
        "0.562979727",
        "-0.999501670"
      ],
      [
        "0.119392330",
        "-0.932135746"
      ],
      [
        "0.866452393",
        "0.270544990"
      ],
      [
        "0.614777104",
        "0.693772513"
      ]
    ],
    [
      [
        "0.059863600",
        "0.933253129"
      ],
      [
        "0.262510032",
        "-0.999500029"
      ],
      [
        "0.322366005",
        "0.038052137"
      ],
      [
        "-0.786442583",
        "0.321322665"
      ]
    ],
    [
      [
        "0.999499635",
        "-0.201165798"
      ],
      [
        "0.384169520",
        "0.745392330"
      ],
      [
        "0.635860709",
        "0.629696306"
      ],
      [
        "0.810838205",
        "-0.253357489"
      ]
    ],
    [
      [
        "-0.813017982",
        "0.293079724"
      ],
      [
        "-0.462989650",
        "0.287022242"
      ],
      [
        "-0.112450146",
        "0.677587380"
      ],
      [
        "0.999500333",
        "-0.202785608"
      ]
    ],
    [
      [
        "0.875222992",
        "0.049611226"
      ],
      [
        "0.193958766",
        "0.999501538"
      ],
      [
        "-0.268541780",
        "-0.527730952"
      ],
      [
        "-0.085353065",
        "0.257714886"
      ]
    ],
    [
      [
        "-0.702215088",
        "-0.283013799"
      ],
      [
        "-0.055450613",
        "-0.481679665"
      ],
      [
        "0.785968780",
        "0.002079711"
      ],
      [
        "-0.537117243",
        "0.999499979"
      ]
    ],
    [
      [
        "-0.790650571",
        "0.116019669"
      ],
      [
        "0.803878836",
        "0.058792072"
      ],
      [
        "-0.568299963",
        "0.388338285"
      ],
      [
        "-0.330813539",
        "0.999499097"
      ]
    ],
    [
      [
        "0.183911658",
        "0.815032516"
      ],
      [
        "-0.179584253",
        "-0.621831630"
      ],
      [
        "-0.170893456",
        "-0.999499280"
      ],
      [
        "-0.158086262",
        "-0.707025025"
      ]
    ]
  ]
}

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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