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The Covering Number C(12,6,4)

Combinatorics · posed by covering design tables · solved

1 attempt

Statement

A tt-(v,k,λ)(v,k,\lambda) covering is a family of kk-subsets of a vv-set meeting every tt-subset at least λ\lambda times, and C(v,k,t)C(v,k,t) is the least number of blocks. The recorded bounds for C(12,6,4)C(12,6,4) were 40C(12,6,4)4140 \le C(12,6,4) \le 41. No 44-(12,6,1)(12,6,1) covering with 4040 blocks exists, so C(12,6,4)=41C(12,6,4) = 41.

Context

closes a one-block gap in the covering tables; the analogous next case is not reachable by this method

An open value in the covering design tables, the kind of documented specific gap that design theorists track but that has no audience beyond them.

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Attempts

1 attempt

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  • #1

    Attempt 1

    computationGPT-5.6 Sol, GPT-5.6 Terra, Claude Fable 5 with Charlie Krug ·
    AI involvement
    ai assisted
    a person led the work and used a model along the way.
    models
    GPT-5.6 Sol, GPT-5.6 Terra, Claude Fable 5
    people
    Charlie Krug

    The disclosure lists exploratory analysis, computational search, supporting code, manuscript drafting and revision, and proofreading, without separating which step came from where, so the lowest tier applies.

    closes a one-block gap in the covering tables; the analogous next case is not reachable by this method

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