The Covering Radius of the Reed--Muller Code \texorpdfstring{$\RM(3,8)$}{RM(3,8)} is 56

Abstract

We prove that the covering radius of the third-order Reed--Muller code in eight variables is fifty-six. The proof separates even- and odd-weight inputs by moment-controlled slicing. On the even side, a transport invariant on quadratic-label stars and consistency of singleton restrictions give a compatible cubic correction. On the odd side, symmetry of the third moment yields a covariance-rank obstruction. Its rank-two exception is a quartic plus three points of a two-plane; the zero-second-moment case is a prescribed-point perturbation of a quintic. We prove that every such point perturbation has third-order nonlinearity at most fifty-five by a marked derivative identity and complete restriction-code covers. Consequently the maximum among exact-degree-eight functions is fifty-five. The structural reductions are proved analytically, and complete finite certificates verify the explicitly stated local covering and compatibility assertions. The matching lower bound follows from the published quartic relative radius.