P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · d = 5, n = 8

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 = 8

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.295208075, 0.295208085]	See source attribution	github.com ↗
Site record	0.295208077	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 8 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
0.2952080765→0.2952080764
2026-10-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.29520807653→0.29520807652
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.29520807655→0.29520807653
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.295208078→0.295208077
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": [
    [
      [
        "0.520073693",
        "0.095331575"
      ],
      [
        "0.999999622",
        "-0.214529530"
      ],
      [
        "-0.396313189",
        "-0.783004109"
      ],
      [
        "0.732394977",
        "-0.222564354"
      ],
      [
        "0.206573379",
        "-0.215478139"
      ]
    ],
    [
      [
        "0.017964729",
        "-0.999498882"
      ],
      [
        "0.085568582",
        "0.385459261"
      ],
      [
        "0.162711568",
        "-0.473916161"
      ],
      [
        "0.881173036",
        "0.025619976"
      ],
      [
        "-0.084691218",
        "0.975360943"
      ]
    ],
    [
      [
        "-0.139686456",
        "-0.814723458"
      ],
      [
        "-0.729135341",
        "0.239419578"
      ],
      [
        "0.296074732",
        "0.258883667"
      ],
      [
        "0.453349228",
        "0.408456097"
      ],
      [
        "0.999999695",
        "-0.256974900"
      ]
    ],
    [
      [
        "-0.189184748",
        "0.183331227"
      ],
      [
        "-0.999499797",
        "0.051946483"
      ],
      [
        "0.207113108",
        "-0.929221990"
      ],
      [
        "-0.531073267",
        "0.143792192"
      ],
      [
        "0.168760948",
        "-0.592908934"
      ]
    ],
    [
      [
        "0.646919352",
        "0.076279876"
      ],
      [
        "0.123629209",
        "0.976555316"
      ],
      [
        "-0.436395494",
        "0.915044175"
      ],
      [
        "-0.999500091",
        "0.155286765"
      ],
      [
        "0.729121319",
        "0.080931430"
      ]
    ],
    [
      [
        "0.583365610",
        "-0.197198767"
      ],
      [
        "-0.869528548",
        "0.226762657"
      ],
      [
        "-0.495484025",
        "0.393106564"
      ],
      [
        "0.999999699",
        "-0.363047818"
      ],
      [
        "-0.280925335",
        "-0.330315204"
      ]
    ],
    [
      [
        "0.444702769",
        "-0.823981506"
      ],
      [
        "0.248490532",
        "0.198601473"
      ],
      [
        "-0.556269703",
        "-0.999999592"
      ],
      [
        "-0.752274833",
        "0.141696631"
      ],
      [
        "0.688929598",
        "0.145665771"
      ]
    ],
    [
      [
        "-0.978596002",
        "-0.058048427"
      ],
      [
        "-0.031269830",
        "-0.057332046"
      ],
      [
        "-0.484434949",
        "-0.119847714"
      ],
      [
        "-0.137114575",
        "0.657604767"
      ],
      [
        "-0.673064827",
        "0.999499967"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 8,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "0.520073693",
        "0.095331575"
      ],
      [
        "0.999999622",
        "-0.214529530"
      ],
      [
        "-0.396313189",
        "-0.783004109"
      ],
      [
        "0.732394977",
        "-0.222564354"
      ],
      [
        "0.206573379",
        "-0.215478139"
      ]
    ],
    [
      [
        "0.017964729",
        "-0.999498882"
      ],
      [
        "0.085568582",
        "0.385459261"
      ],
      [
        "0.162711568",
        "-0.473916161"
      ],
      [
        "0.881173036",
        "0.025619976"
      ],
      [
        "-0.084691218",
        "0.975360943"
      ]
    ],
    [
      [
        "-0.139686456",
        "-0.814723458"
      ],
      [
        "-0.729135341",
        "0.239419578"
      ],
      [
        "0.296074732",
        "0.258883667"
      ],
      [
        "0.453349228",
        "0.408456097"
      ],
      [
        "0.999999695",
        "-0.256974900"
      ]
    ],
    [
      [
        "-0.189184748",
        "0.183331227"
      ],
      [
        "-0.999499797",
        "0.051946483"
      ],
      [
        "0.207113108",
        "-0.929221990"
      ],
      [
        "-0.531073267",
        "0.143792192"
      ],
      [
        "0.168760948",
        "-0.592908934"
      ]
    ],
    [
      [
        "0.646919352",
        "0.076279876"
      ],
      [
        "0.123629209",
        "0.976555316"
      ],
      [
        "-0.436395494",
        "0.915044175"
      ],
      [
        "-0.999500091",
        "0.155286765"
      ],
      [
        "0.729121319",
        "0.080931430"
      ]
    ],
    [
      [
        "0.583365610",
        "-0.197198767"
      ],
      [
        "-0.869528548",
        "0.226762657"
      ],
      [
        "-0.495484025",
        "0.393106564"
      ],
      [
        "0.999999699",
        "-0.363047818"
      ],
      [
        "-0.280925335",
        "-0.330315204"
      ]
    ],
    [
      [
        "0.444702769",
        "-0.823981506"
      ],
      [
        "0.248490532",
        "0.198601473"
      ],
      [
        "-0.556269703",
        "-0.999999592"
      ],
      [
        "-0.752274833",
        "0.141696631"
      ],
      [
        "0.688929598",
        "0.145665771"
      ]
    ],
    [
      [
        "-0.978596002",
        "-0.058048427"
      ],
      [
        "-0.031269830",
        "-0.057332046"
      ],
      [
        "-0.484434949",
        "-0.119847714"
      ],
      [
        "-0.137114575",
        "0.657604767"
      ],
      [
        "-0.673064827",
        "0.999499967"
      ]
    ]
  ]
}

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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