P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 3, n = 13

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 = 13

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.622143865, 0.622143875]	See source attribution	github.com ↗
Site record	0.622143873	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 13 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
0.6221438729→0.6221438728
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.622143874→0.622143873
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.622143873009→0.622143873005
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.62214387301→0.62214387300
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.62214387302→0.62214387301
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.62214387303→0.62214387302
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.6221438734→0.6221438730
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": [
    [
      [
        "-0.568452811",
        "-0.069406374"
      ],
      [
        "-0.120799874",
        "0.098744024"
      ],
      [
        "-0.370442148",
        "-0.999498877"
      ]
    ],
    [
      [
        "0.999502286",
        "-0.492368232"
      ],
      [
        "-0.324758696",
        "0.285609470"
      ],
      [
        "0.511565744",
        "-0.464530034"
      ]
    ],
    [
      [
        "-0.102650494",
        "-0.388319980"
      ],
      [
        "0.053671284",
        "-0.351700635"
      ],
      [
        "-0.999499039",
        "-0.034027797"
      ]
    ],
    [
      [
        "-0.866150829",
        "0.999498896"
      ],
      [
        "0.983736180",
        "0.471507622"
      ],
      [
        "0.754573474",
        "0.730467238"
      ]
    ],
    [
      [
        "0.542563409",
        "0.042049275"
      ],
      [
        "-0.359046459",
        "0.475805780"
      ],
      [
        "-0.999500009",
        "-0.479747838"
      ]
    ],
    [
      [
        "-0.431201436",
        "-0.179147674"
      ],
      [
        "0.999500408",
        "0.168408186"
      ],
      [
        "-0.024189802",
        "0.148695499"
      ]
    ],
    [
      [
        "0.171713993",
        "0.130048785"
      ],
      [
        "0.999499436",
        "0.556675868"
      ],
      [
        "-0.948375268",
        "-0.396716779"
      ]
    ],
    [
      [
        "-0.444684139",
        "0.446196429"
      ],
      [
        "0.999499650",
        "0.095029131"
      ],
      [
        "0.065519370",
        "-0.904012459"
      ]
    ],
    [
      [
        "0.911358547",
        "0.106468621"
      ],
      [
        "0.999499483",
        "-0.390592419"
      ],
      [
        "0.224173328",
        "0.348272958"
      ]
    ],
    [
      [
        "0.169885126",
        "-0.910184480"
      ],
      [
        "0.999499088",
        "-0.334437347"
      ],
      [
        "-0.214909351",
        "-0.218657964"
      ]
    ],
    [
      [
        "0.624195014",
        "0.999499758"
      ],
      [
        "0.350990387",
        "0.274936719"
      ],
      [
        "-0.167013499",
        "-0.691617278"
      ]
    ],
    [
      [
        "-0.999500563",
        "0.294058632"
      ],
      [
        "0.167670882",
        "-0.455579772"
      ],
      [
        "0.104632505",
        "-0.509325225"
      ]
    ],
    [
      [
        "-0.125179066",
        "-0.254874984"
      ],
      [
        "0.638274899",
        "-0.999501197"
      ],
      [
        "0.435736654",
        "-0.979937609"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 13,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      [
        "-0.568452811",
        "-0.069406374"
      ],
      [
        "-0.120799874",
        "0.098744024"
      ],
      [
        "-0.370442148",
        "-0.999498877"
      ]
    ],
    [
      [
        "0.999502286",
        "-0.492368232"
      ],
      [
        "-0.324758696",
        "0.285609470"
      ],
      [
        "0.511565744",
        "-0.464530034"
      ]
    ],
    [
      [
        "-0.102650494",
        "-0.388319980"
      ],
      [
        "0.053671284",
        "-0.351700635"
      ],
      [
        "-0.999499039",
        "-0.034027797"
      ]
    ],
    [
      [
        "-0.866150829",
        "0.999498896"
      ],
      [
        "0.983736180",
        "0.471507622"
      ],
      [
        "0.754573474",
        "0.730467238"
      ]
    ],
    [
      [
        "0.542563409",
        "0.042049275"
      ],
      [
        "-0.359046459",
        "0.475805780"
      ],
      [
        "-0.999500009",
        "-0.479747838"
      ]
    ],
    [
      [
        "-0.431201436",
        "-0.179147674"
      ],
      [
        "0.999500408",
        "0.168408186"
      ],
      [
        "-0.024189802",
        "0.148695499"
      ]
    ],
    [
      [
        "0.171713993",
        "0.130048785"
      ],
      [
        "0.999499436",
        "0.556675868"
      ],
      [
        "-0.948375268",
        "-0.396716779"
      ]
    ],
    [
      [
        "-0.444684139",
        "0.446196429"
      ],
      [
        "0.999499650",
        "0.095029131"
      ],
      [
        "0.065519370",
        "-0.904012459"
      ]
    ],
    [
      [
        "0.911358547",
        "0.106468621"
      ],
      [
        "0.999499483",
        "-0.390592419"
      ],
      [
        "0.224173328",
        "0.348272958"
      ]
    ],
    [
      [
        "0.169885126",
        "-0.910184480"
      ],
      [
        "0.999499088",
        "-0.334437347"
      ],
      [
        "-0.214909351",
        "-0.218657964"
      ]
    ],
    [
      [
        "0.624195014",
        "0.999499758"
      ],
      [
        "0.350990387",
        "0.274936719"
      ],
      [
        "-0.167013499",
        "-0.691617278"
      ]
    ],
    [
      [
        "-0.999500563",
        "0.294058632"
      ],
      [
        "0.167670882",
        "-0.455579772"
      ],
      [
        "0.104632505",
        "-0.509325225"
      ]
    ],
    [
      [
        "-0.125179066",
        "-0.254874984"
      ],
      [
        "0.638274899",
        "-0.999501197"
      ],
      [
        "0.435736654",
        "-0.979937609"
      ]
    ]
  ]
}

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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