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.
The construction of optimal binary linear complementary pair (LCP) has recently attracted considerable attention in coding theory. Based on the Belov codes, this paper proposes a novel construction of optimal binary LCPs with $θ=4$, extending our previous work on $θ=3$. Our results not only yield abundant new optimal binary LCPs with flexible parameters, but also enable more optimal binary LCPs with smaller minimum distances.
Additive codes generalize linear codes and play an important role in the construc?tion of quantum codes. As generalizations of linear MDS codes, additive MDS codes have received much research attention. As an important subclass of additive codes, ACD codes are widely used for quantum error correction. In this paper, we study single-twist additive generalized twisted Reed-Solomon (GTRS) codes over finite fields. We derive necessary and sufficient conditions for these codes to be MDS for arbitrary hook and twist parameters. Then we employ Schur squares to prove that such additive MDS codes are not monomially equivalent to additive Reed-Solomon codes. We also identify a class of additive GTRS codes whose duals remain additive GTRS codes, and show that, for all sufficiently large prime powers q whose characteristic avoids a finite set, such GTRS codes yield ACD codes.