P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, n = 15

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 = 15

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.379936775, 0.379936785]	See source attribution	github.com ↗
Site record	0.379936783	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 15 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 (11 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.3799367829→0.3799367828
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678296→0.37993678295
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782967→0.379936782962
2026-09-20
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678297→0.37993678296
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782974→0.379936782971
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782977→0.379936782974
2026-09-19
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678298→0.37993678297
2026-09-16
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936782989→0.379936782982
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.37993678299→0.37993678298
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.379936784→0.379936783
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.3799367837→0.3799367830
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": [
    [
      [
        "-0.204043955",
        "0.265211185"
      ],
      [
        "-0.349458825",
        "-0.687806364"
      ],
      [
        "0.878711900",
        "-0.276677894"
      ],
      [
        "0.999504784",
        "0.369551844"
      ],
      [
        "0.011669240",
        "0.076768676"
      ]
    ],
    [
      [
        "-0.406442576",
        "-0.958033587"
      ],
      [
        "-0.597560911",
        "0.297539874"
      ],
      [
        "-0.341158735",
        "0.736565202"
      ],
      [
        "0.023210154",
        "-0.999500588"
      ],
      [
        "0.989411049",
        "-0.382159108"
      ]
    ],
    [
      [
        "-0.999504302",
        "0.092225783"
      ],
      [
        "-0.578235353",
        "-0.105490793"
      ],
      [
        "-0.302026534",
        "-0.914529465"
      ],
      [
        "-0.478136878",
        "-0.210629532"
      ],
      [
        "0.101724900",
        "-0.542821624"
      ]
    ],
    [
      [
        "0.458351129",
        "-0.988915083"
      ],
      [
        "0.235010597",
        "0.161323614"
      ],
      [
        "0.999491328",
        "-0.141549286"
      ],
      [
        "-0.482179419",
        "0.604180383"
      ],
      [
        "-0.770605197",
        "-0.291057579"
      ]
    ],
    [
      [
        "-0.999500143",
        "0.106302717"
      ],
      [
        "0.601087046",
        "-0.146232579"
      ],
      [
        "0.129925710",
        "-0.240303697"
      ],
      [
        "-0.875247649",
        "0.060434260"
      ],
      [
        "0.367525355",
        "0.573719100"
      ]
    ],
    [
      [
        "0.512577117",
        "0.296843851"
      ],
      [
        "0.189997009",
        "0.299959821"
      ],
      [
        "-0.536833463",
        "0.177451095"
      ],
      [
        "-0.439284544",
        "0.331157996"
      ],
      [
        "0.999503744",
        "0.957905424"
      ]
    ],
    [
      [
        "-0.360191480",
        "-0.045741592"
      ],
      [
        "-0.289182244",
        "0.999504186"
      ],
      [
        "0.594758470",
        "-0.715859930"
      ],
      [
        "-0.116495659",
        "-0.049026091"
      ],
      [
        "-0.110830645",
        "0.055959098"
      ]
    ],
    [
      [
        "0.314339398",
        "-0.726118541"
      ],
      [
        "0.257712951",
        "-0.308642734"
      ],
      [
        "-0.466122706",
        "-0.999503915"
      ],
      [
        "-0.666975846",
        "-0.059539793"
      ],
      [
        "0.241922838",
        "0.491632287"
      ]
    ],
    [
      [
        "0.155092136",
        "0.222642647"
      ],
      [
        "-0.623753020",
        "0.999504142"
      ],
      [
        "-0.112429876",
        "0.359866178"
      ],
      [
        "0.518388594",
        "-0.125538401"
      ],
      [
        "0.285924129",
        "0.464179558"
      ]
    ],
    [
      [
        "0.999500224",
        "0.373224313"
      ],
      [
        "-0.970870529",
        "0.428582957"
      ],
      [
        "0.068953468",
        "-0.659011838"
      ],
      [
        "-0.614387923",
        "0.315660025"
      ],
      [
        "-0.607273192",
        "-0.039392892"
      ]
    ],
    [
      [
        "-0.249395477",
        "-0.936088436"
      ],
      [
        "-0.999503178",
        "0.327125483"
      ],
      [
        "-0.547115045",
        "0.358458073"
      ],
      [
        "-0.674130707",
        "0.268200643"
      ],
      [
        "-0.966437129",
        "-0.670231035"
      ]
    ],
    [
      [
        "0.062018533",
        "-0.929665412"
      ],
      [
        "-0.999499122",
        "0.127346585"
      ],
      [
        "-0.035262945",
        "-0.723428800"
      ],
      [
        "0.600547790",
        "0.833887960"
      ],
      [
        "0.498729531",
        "-0.420146094"
      ]
    ],
    [
      [
        "0.172710620",
        "0.191906583"
      ],
      [
        "-0.190650569",
        "0.311415927"
      ],
      [
        "0.442106726",
        "-0.398370417"
      ],
      [
        "-0.999497038",
        "-0.373996161"
      ],
      [
        "0.905074029",
        "-0.082320353"
      ]
    ],
    [
      [
        "0.063492261",
        "0.666895811"
      ],
      [
        "0.230370349",
        "0.307550770"
      ],
      [
        "0.017701653",
        "-0.403118797"
      ],
      [
        "0.214038485",
        "-0.312387055"
      ],
      [
        "0.120110096",
        "-0.999501463"
      ]
    ],
    [
      [
        "0.451054173",
        "0.290398719"
      ],
      [
        "0.039832786",
        "0.732963777"
      ],
      [
        "0.734542447",
        "-0.020700213"
      ],
      [
        "0.270639340",
        "0.999499378"
      ],
      [
        "0.284918026",
        "-0.143027197"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 15,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "-0.204043955",
        "0.265211185"
      ],
      [
        "-0.349458825",
        "-0.687806364"
      ],
      [
        "0.878711900",
        "-0.276677894"
      ],
      [
        "0.999504784",
        "0.369551844"
      ],
      [
        "0.011669240",
        "0.076768676"
      ]
    ],
    [
      [
        "-0.406442576",
        "-0.958033587"
      ],
      [
        "-0.597560911",
        "0.297539874"
      ],
      [
        "-0.341158735",
        "0.736565202"
      ],
      [
        "0.023210154",
        "-0.999500588"
      ],
      [
        "0.989411049",
        "-0.382159108"
      ]
    ],
    [
      [
        "-0.999504302",
        "0.092225783"
      ],
      [
        "-0.578235353",
        "-0.105490793"
      ],
      [
        "-0.302026534",
        "-0.914529465"
      ],
      [
        "-0.478136878",
        "-0.210629532"
      ],
      [
        "0.101724900",
        "-0.542821624"
      ]
    ],
    [
      [
        "0.458351129",
        "-0.988915083"
      ],
      [
        "0.235010597",
        "0.161323614"
      ],
      [
        "0.999491328",
        "-0.141549286"
      ],
      [
        "-0.482179419",
        "0.604180383"
      ],
      [
        "-0.770605197",
        "-0.291057579"
      ]
    ],
    [
      [
        "-0.999500143",
        "0.106302717"
      ],
      [
        "0.601087046",
        "-0.146232579"
      ],
      [
        "0.129925710",
        "-0.240303697"
      ],
      [
        "-0.875247649",
        "0.060434260"
      ],
      [
        "0.367525355",
        "0.573719100"
      ]
    ],
    [
      [
        "0.512577117",
        "0.296843851"
      ],
      [
        "0.189997009",
        "0.299959821"
      ],
      [
        "-0.536833463",
        "0.177451095"
      ],
      [
        "-0.439284544",
        "0.331157996"
      ],
      [
        "0.999503744",
        "0.957905424"
      ]
    ],
    [
      [
        "-0.360191480",
        "-0.045741592"
      ],
      [
        "-0.289182244",
        "0.999504186"
      ],
      [
        "0.594758470",
        "-0.715859930"
      ],
      [
        "-0.116495659",
        "-0.049026091"
      ],
      [
        "-0.110830645",
        "0.055959098"
      ]
    ],
    [
      [
        "0.314339398",
        "-0.726118541"
      ],
      [
        "0.257712951",
        "-0.308642734"
      ],
      [
        "-0.466122706",
        "-0.999503915"
      ],
      [
        "-0.666975846",
        "-0.059539793"
      ],
      [
        "0.241922838",
        "0.491632287"
      ]
    ],
    [
      [
        "0.155092136",
        "0.222642647"
      ],
      [
        "-0.623753020",
        "0.999504142"
      ],
      [
        "-0.112429876",
        "0.359866178"
      ],
      [
        "0.518388594",
        "-0.125538401"
      ],
      [
        "0.285924129",
        "0.464179558"
      ]
    ],
    [
      [
        "0.999500224",
        "0.373224313"
      ],
      [
        "-0.970870529",
        "0.428582957"
      ],
      [
        "0.068953468",
        "-0.659011838"
      ],
      [
        "-0.614387923",
        "0.315660025"
      ],
      [
        "-0.607273192",
        "-0.039392892"
      ]
    ],
    [
      [
        "-0.249395477",
        "-0.936088436"
      ],
      [
        "-0.999503178",
        "0.327125483"
      ],
      [
        "-0.547115045",
        "0.358458073"
      ],
      [
        "-0.674130707",
        "0.268200643"
      ],
      [
        "-0.966437129",
        "-0.670231035"
      ]
    ],
    [
      [
        "0.062018533",
        "-0.929665412"
      ],
      [
        "-0.999499122",
        "0.127346585"
      ],
      [
        "-0.035262945",
        "-0.723428800"
      ],
      [
        "0.600547790",
        "0.833887960"
      ],
      [
        "0.498729531",
        "-0.420146094"
      ]
    ],
    [
      [
        "0.172710620",
        "0.191906583"
      ],
      [
        "-0.190650569",
        "0.311415927"
      ],
      [
        "0.442106726",
        "-0.398370417"
      ],
      [
        "-0.999497038",
        "-0.373996161"
      ],
      [
        "0.905074029",
        "-0.082320353"
      ]
    ],
    [
      [
        "0.063492261",
        "0.666895811"
      ],
      [
        "0.230370349",
        "0.307550770"
      ],
      [
        "0.017701653",
        "-0.403118797"
      ],
      [
        "0.214038485",
        "-0.312387055"
      ],
      [
        "0.120110096",
        "-0.999501463"
      ]
    ],
    [
      [
        "0.451054173",
        "0.290398719"
      ],
      [
        "0.039832786",
        "0.732963777"
      ],
      [
        "0.734542447",
        "-0.020700213"
      ],
      [
        "0.270639340",
        "0.999499378"
      ],
      [
        "0.284918026",
        "-0.143027197"
      ]
    ]
  ]
}

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

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.