P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 14

Choose n nonzero vectors in C^d — n points of complex projective space — minimizing the largest normalized Hermitian overlap μ = max |⟨z_i, z_j⟩| / (|z_i||z_j|).

Instanced = 5, n = 14

ObjectiveMinimize the largest coherence μ

Best known, unproven0.375674755the Game of Sloanes leader board (the table of putatively optimal packings maintained by Jasper, King and Mixon)

Formal definition

ContainerComplex d-space C^d; an answer is n points of complex projective space CP^{d-1}
SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
ObjectiveMinimize μ² = max |⟨z_i,z_j⟩|²/(|z_i|²|z_j|²); moduli and norms squared are rational, compared exactly by cross-multiplication
ScoringThe record is ceil(μ²·10¹⁸); the page shows μ, rounded up at the ninth decimal

Challenge this record ↗

VERIFIED CONSTRUCTIONpairwise overlap heat of 14 directions; brighter is closer

Getting a feel for it

Where it is used

Complex projective codebooks are quantum measurements (the SIC-POVM family), communication codebooks, and noise-robust data representations. The further apart the codewords, the better they survive noise and erasures.

Where the frontier is

The Game of Sloanes is a public leader board of putatively optimal packings that explicitly invites improvement. Every sub-problem here is chosen from its still-open rows — the ones with a real gap between the best known value and the lower bound.

Source

Current leader

0.997392713

the largest coherence μ

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

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": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.071428571",
        "0"
      ],
      [
        "0.005102041",
        "0"
      ],
      [
        "0.000364431",
        "0"
      ],
      [
        "0.000026031",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.142857143",
        "0"
      ],
      [
        "0.020408163",
        "0"
      ],
      [
        "0.002915452",
        "0"
      ],
      [
        "0.000416493",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.214285714",
        "0"
      ],
      [
        "0.045918367",
        "0"
      ],
      [
        "0.009839650",
        "0"
      ],
      [
        "0.002108496",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.285714286",
        "0"
      ],
      [
        "0.081632653",
        "0"
      ],
      [
        "0.023323615",
        "0"
      ],
      [
        "0.006663890",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.357142857",
        "0"
      ],
      [
        "0.127551020",
        "0"
      ],
      [
        "0.045553936",
        "0"
      ],
      [
        "0.016269263",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.428571429",
        "0"
      ],
      [
        "0.183673469",
        "0"
      ],
      [
        "0.078717201",
        "0"
      ],
      [
        "0.033735943",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.571428571",
        "0"
      ],
      [
        "0.326530612",
        "0"
      ],
      [
        "0.186588921",
        "0"
      ],
      [
        "0.106622241",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.642857143",
        "0"
      ],
      [
        "0.413265306",
        "0"
      ],
      [
        "0.265670554",
        "0"
      ],
      [
        "0.170788213",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.714285714",
        "0"
      ],
      [
        "0.510204082",
        "0"
      ],
      [
        "0.364431487",
        "0"
      ],
      [
        "0.260308205",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.785714286",
        "0"
      ],
      [
        "0.617346939",
        "0"
      ],
      [
        "0.485058309",
        "0"
      ],
      [
        "0.381117243",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.857142857",
        "0"
      ],
      [
        "0.734693878",
        "0"
      ],
      [
        "0.629737609",
        "0"
      ],
      [
        "0.539775094",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.928571429",
        "0"
      ],
      [
        "0.862244898",
        "0"
      ],
      [
        "0.800655977",
        "0"
      ],
      [
        "0.743466264",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.071428571",
        "0"
      ],
      [
        "0.005102041",
        "0"
      ],
      [
        "0.000364431",
        "0"
      ],
      [
        "0.000026031",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.142857143",
        "0"
      ],
      [
        "0.020408163",
        "0"
      ],
      [
        "0.002915452",
        "0"
      ],
      [
        "0.000416493",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.214285714",
        "0"
      ],
      [
        "0.045918367",
        "0"
      ],
      [
        "0.009839650",
        "0"
      ],
      [
        "0.002108496",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.285714286",
        "0"
      ],
      [
        "0.081632653",
        "0"
      ],
      [
        "0.023323615",
        "0"
      ],
      [
        "0.006663890",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.357142857",
        "0"
      ],
      [
        "0.127551020",
        "0"
      ],
      [
        "0.045553936",
        "0"
      ],
      [
        "0.016269263",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.428571429",
        "0"
      ],
      [
        "0.183673469",
        "0"
      ],
      [
        "0.078717201",
        "0"
      ],
      [
        "0.033735943",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.571428571",
        "0"
      ],
      [
        "0.326530612",
        "0"
      ],
      [
        "0.186588921",
        "0"
      ],
      [
        "0.106622241",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.642857143",
        "0"
      ],
      [
        "0.413265306",
        "0"
      ],
      [
        "0.265670554",
        "0"
      ],
      [
        "0.170788213",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.714285714",
        "0"
      ],
      [
        "0.510204082",
        "0"
      ],
      [
        "0.364431487",
        "0"
      ],
      [
        "0.260308205",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.785714286",
        "0"
      ],
      [
        "0.617346939",
        "0"
      ],
      [
        "0.485058309",
        "0"
      ],
      [
        "0.381117243",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.857142857",
        "0"
      ],
      [
        "0.734693878",
        "0"
      ],
      [
        "0.629737609",
        "0"
      ],
      [
        "0.539775094",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.928571429",
        "0"
      ],
      [
        "0.862244898",
        "0"
      ],
      [
        "0.800655977",
        "0"
      ],
      [
        "0.743466264",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

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

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