The covering number C(13,7,4) is 30: A classification-assisted exact proof
We prove that $C(13,7,4)=30$, using the classification of minimum $(12,6,3)$ coverings by Gordon, Patashnik, Petro and Taylor. A hypothetical $29$-block cover has at least five points lying in exactly $15$ blocks. The link at one of these points leaves only $14$ blocks outside it. The other four points force complete families of outside blocks whose overlaps must agree. An exhaustive computation shows that every compatible choice requires $15$ outside blocks, giving a contradiction. The known $30$-block construction is therefore optimal. We give reproducible exact computations and a public…
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