P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 16

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 = 16

ObjectiveMinimize the largest coherence μ

Best known, unproven0.388092835the 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 16 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

0.998014025

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"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.003906250",
        "0"
      ],
      [
        "0.000244141",
        "0"
      ],
      [
        "0.000015259",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.015625000",
        "0"
      ],
      [
        "0.001953125",
        "0"
      ],
      [
        "0.000244141",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.187500000",
        "0"
      ],
      [
        "0.035156250",
        "0"
      ],
      [
        "0.006591797",
        "0"
      ],
      [
        "0.001235962",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.015625000",
        "0"
      ],
      [
        "0.003906250",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.312500000",
        "0"
      ],
      [
        "0.097656250",
        "0"
      ],
      [
        "0.030517578",
        "0"
      ],
      [
        "0.009536743",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.375000000",
        "0"
      ],
      [
        "0.140625000",
        "0"
      ],
      [
        "0.052734375",
        "0"
      ],
      [
        "0.019775391",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.437500000",
        "0"
      ],
      [
        "0.191406250",
        "0"
      ],
      [
        "0.083740234",
        "0"
      ],
      [
        "0.036636353",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.562500000",
        "0"
      ],
      [
        "0.316406250",
        "0"
      ],
      [
        "0.177978516",
        "0"
      ],
      [
        "0.100112915",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.625000000",
        "0"
      ],
      [
        "0.390625000",
        "0"
      ],
      [
        "0.244140625",
        "0"
      ],
      [
        "0.152587891",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.687500000",
        "0"
      ],
      [
        "0.472656250",
        "0"
      ],
      [
        "0.324951172",
        "0"
      ],
      [
        "0.223403931",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.750000000",
        "0"
      ],
      [
        "0.562500000",
        "0"
      ],
      [
        "0.421875000",
        "0"
      ],
      [
        "0.316406250",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.812500000",
        "0"
      ],
      [
        "0.660156250",
        "0"
      ],
      [
        "0.536376953",
        "0"
      ],
      [
        "0.435806274",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.875000000",
        "0"
      ],
      [
        "0.765625000",
        "0"
      ],
      [
        "0.669921875",
        "0"
      ],
      [
        "0.586181641",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.937500000",
        "0"
      ],
      [
        "0.878906250",
        "0"
      ],
      [
        "0.823974609",
        "0"
      ],
      [
        "0.772476196",
        "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": 16,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.003906250",
        "0"
      ],
      [
        "0.000244141",
        "0"
      ],
      [
        "0.000015259",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.015625000",
        "0"
      ],
      [
        "0.001953125",
        "0"
      ],
      [
        "0.000244141",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.187500000",
        "0"
      ],
      [
        "0.035156250",
        "0"
      ],
      [
        "0.006591797",
        "0"
      ],
      [
        "0.001235962",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ],
      [
        "0.015625000",
        "0"
      ],
      [
        "0.003906250",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.312500000",
        "0"
      ],
      [
        "0.097656250",
        "0"
      ],
      [
        "0.030517578",
        "0"
      ],
      [
        "0.009536743",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.375000000",
        "0"
      ],
      [
        "0.140625000",
        "0"
      ],
      [
        "0.052734375",
        "0"
      ],
      [
        "0.019775391",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.437500000",
        "0"
      ],
      [
        "0.191406250",
        "0"
      ],
      [
        "0.083740234",
        "0"
      ],
      [
        "0.036636353",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.500000000",
        "0"
      ],
      [
        "0.250000000",
        "0"
      ],
      [
        "0.125000000",
        "0"
      ],
      [
        "0.062500000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.562500000",
        "0"
      ],
      [
        "0.316406250",
        "0"
      ],
      [
        "0.177978516",
        "0"
      ],
      [
        "0.100112915",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
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      ],
      [
        "0.625000000",
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      ],
      [
        "0.390625000",
        "0"
      ],
      [
        "0.244140625",
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      ],
      [
        "0.152587891",
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      ]
    ],
    [
      [
        "1.000000000",
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      [
        "0.687500000",
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      [
        "0.472656250",
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        "0.324951172",
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        "0.223403931",
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      ]
    ],
    [
      [
        "1.000000000",
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      [
        "0.750000000",
        "0"
      ],
      [
        "0.562500000",
        "0"
      ],
      [
        "0.421875000",
        "0"
      ],
      [
        "0.316406250",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
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      [
        "0.812500000",
        "0"
      ],
      [
        "0.660156250",
        "0"
      ],
      [
        "0.536376953",
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      ],
      [
        "0.435806274",
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    ],
    [
      [
        "1.000000000",
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      ],
      [
        "0.875000000",
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      [
        "0.765625000",
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      [
        "0.669921875",
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      [
        "0.586181641",
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    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.937500000",
        "0"
      ],
      [
        "0.878906250",
        "0"
      ],
      [
        "0.823974609",
        "0"
      ],
      [
        "0.772476196",
        "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

DISCUSSION

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