P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 13

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 = 13

ObjectiveMinimize the largest coherence μ

Best known, unproven0.367602315the 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 13 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

0.996965532

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"
      ],
      [
        "0.076923077",
        "0"
      ],
      [
        "0.005917160",
        "0"
      ],
      [
        "0.000455166",
        "0"
      ],
      [
        "0.000035013",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.153846154",
        "0"
      ],
      [
        "0.023668639",
        "0"
      ],
      [
        "0.003641329",
        "0"
      ],
      [
        "0.000560204",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.230769231",
        "0"
      ],
      [
        "0.053254438",
        "0"
      ],
      [
        "0.012289486",
        "0"
      ],
      [
        "0.002836035",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.307692308",
        "0"
      ],
      [
        "0.094674556",
        "0"
      ],
      [
        "0.029130633",
        "0"
      ],
      [
        "0.008963272",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.384615385",
        "0"
      ],
      [
        "0.147928994",
        "0"
      ],
      [
        "0.056895767",
        "0"
      ],
      [
        "0.021882987",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.461538462",
        "0"
      ],
      [
        "0.213017751",
        "0"
      ],
      [
        "0.098315885",
        "0"
      ],
      [
        "0.045376562",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.538461538",
        "0"
      ],
      [
        "0.289940828",
        "0"
      ],
      [
        "0.156121985",
        "0"
      ],
      [
        "0.084065684",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.615384615",
        "0"
      ],
      [
        "0.378698225",
        "0"
      ],
      [
        "0.233045061",
        "0"
      ],
      [
        "0.143412346",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.692307692",
        "0"
      ],
      [
        "0.479289941",
        "0"
      ],
      [
        "0.331816113",
        "0"
      ],
      [
        "0.229718847",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.769230769",
        "0"
      ],
      [
        "0.591715976",
        "0"
      ],
      [
        "0.455166136",
        "0"
      ],
      [
        "0.350127797",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.846153846",
        "0"
      ],
      [
        "0.715976331",
        "0"
      ],
      [
        "0.605826127",
        "0"
      ],
      [
        "0.512622107",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.923076923",
        "0"
      ],
      [
        "0.852071006",
        "0"
      ],
      [
        "0.786527082",
        "0"
      ],
      [
        "0.726024999",
        "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": 13,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.076923077",
        "0"
      ],
      [
        "0.005917160",
        "0"
      ],
      [
        "0.000455166",
        "0"
      ],
      [
        "0.000035013",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.153846154",
        "0"
      ],
      [
        "0.023668639",
        "0"
      ],
      [
        "0.003641329",
        "0"
      ],
      [
        "0.000560204",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.230769231",
        "0"
      ],
      [
        "0.053254438",
        "0"
      ],
      [
        "0.012289486",
        "0"
      ],
      [
        "0.002836035",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.307692308",
        "0"
      ],
      [
        "0.094674556",
        "0"
      ],
      [
        "0.029130633",
        "0"
      ],
      [
        "0.008963272",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.384615385",
        "0"
      ],
      [
        "0.147928994",
        "0"
      ],
      [
        "0.056895767",
        "0"
      ],
      [
        "0.021882987",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.461538462",
        "0"
      ],
      [
        "0.213017751",
        "0"
      ],
      [
        "0.098315885",
        "0"
      ],
      [
        "0.045376562",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.538461538",
        "0"
      ],
      [
        "0.289940828",
        "0"
      ],
      [
        "0.156121985",
        "0"
      ],
      [
        "0.084065684",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.615384615",
        "0"
      ],
      [
        "0.378698225",
        "0"
      ],
      [
        "0.233045061",
        "0"
      ],
      [
        "0.143412346",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.692307692",
        "0"
      ],
      [
        "0.479289941",
        "0"
      ],
      [
        "0.331816113",
        "0"
      ],
      [
        "0.229718847",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.769230769",
        "0"
      ],
      [
        "0.591715976",
        "0"
      ],
      [
        "0.455166136",
        "0"
      ],
      [
        "0.350127797",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.846153846",
        "0"
      ],
      [
        "0.715976331",
        "0"
      ],
      [
        "0.605826127",
        "0"
      ],
      [
        "0.512622107",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.923076923",
        "0"
      ],
      [
        "0.852071006",
        "0"
      ],
      [
        "0.786527082",
        "0"
      ],
      [
        "0.726024999",
        "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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