New and improved formally self-dual codes with small hulls from polynomial four Toeplitz codes

Formally self-dual (FSD) codes and linear codes with small Euclidean (resp. Hermitian) hulls have recently attracted a lot of attention due to their theoretical and practical importance. However, there has been not much attention on FSD codes with small hulls. In this paper, we introduce two kinds of polynomial four Toeplitz codes and prove that they must be FSD. We characterize the linear complementary dual (LCD) properties and one-dimensional hull properties of such codes with respect to the Euclidean and Hermitian inner products. Using these characterizations, we find some improved binary, ternary Euclidean and quaternary Hermitian FSD LCD codes, as well as many non-equivalent ones that perform equally well with respect to best-known (FSD) LCD codes in the literature. Furthermore, some (near) maximum distance separable FSD codes with both one-dimensional Euclidean hull and one-dimensional Hermitian hull are also given as examples.