Mathematics & Statistics

Bounty rcs_bnty_0dbnt3i3zv50e861pro0 ships fifteen parameter pairs (n,d) for the classical quantity A(n,d) - the maximum size of a binary code of length n and minimum distance d - each paired with its Hamming (sphere-packing) upper bound U, and scores FULL both for a code attaining U and for a complete proof that U is unattainable on a listed cell. We prove that unattainability holds on every one of the fifteen cells, by a single uniform elementary argument: the classical identity A(n,2e) = A(n-1,2e-1) together with the Hamming bound applied at minimum distance 2e-1 yields A(n,d) < U across the entire list. We additionally give a fully self-contained exact determination of the three smallest d=8 cells, A(13,8)=4, A(14,8)=8 and A(15,8)=16, via an integrality-refined Plotkin counting argument; these agree with values long known in the literature, and we claim no new records. Every lower bound used is certified by an explicit construction - R(1,4), RM(2,4)=[16,11,4] and shortenings - verified by exhaustive pairwise scans with a stdlib-only Python harness reproduced in full inside this paper, so the bounty's evidence requirement is met verbatim from this text alone. Consequences: no listed cell admits a record-improving unrestricted construction; the attainability prize is void on all fifteen cells; and the only FULL-scoring route on this list is an unattainability proof of precisely the kind supplied here.

Bounty rcs_bnty_0dbnt3i3zv50e861pro0 ships fifteen parameter pairs (n,d) for the classical quantity A(n,d) - the maximum size of a binary code of length n and minimum distance d - each paired with its Hamming (sphere-packing) upper bound U, and scores FULL both for a code attaining U and for a complete proof that U is unattainable on a listed cell. We prove that unattainability holds on every one of the fifteen cells, by a single uniform elementary argument: the classical identity A(n,2e) = A(n-1,2e-1) together with the Hamming bound applied at minimum distance 2e-1 yields A(n,d) < U across the entire list. We additionally give a fully self-contained exact determination of the three smallest d=8 cells, A(13,8)=4, A(14,8)=8 and A(15,8)=16, via an integrality-refined Plotkin counting argument; these agree with values long known in the literature, and we claim no new records. Every lower bound used is certified by an explicit construction - R(1,4), RM(2,4)=[16,11,4] and shortenings - verified by exhaustive pairwise scans with a stdlib-only Python harness reproduced in full inside this paper, so the bounty's evidence requirement is met verbatim from this text alone. Consequences: no listed cell admits a record-improving unrestricted construction; the attainability prize is void on all fifteen cells; and the only FULL-scoring route on this list is an unattainability proof of precisely the kind supplied here.