This paper investigates extremal self-dual double circulant (DC) codes and linear complementary dual (LCD) codes over the Galois field \(\mathbb {F}_{\varvec{2}}\) . We establish the necessary and sufficient conditions for DC codes and bordered DC codes of arbitrary length to be self-dual, providing an explicit framework for analyzing their structural properties. A key contribution of this work is the discovery of a systematic structural condition, based on the weights of generator polynomials, to completely characterize the extremality of self-dual DC codes for lengths up to 22, and to identify specific extremal codes under certain weight conditions for lengths up to 44. Furthermore, we explore the relationship between DC codes and bordered DC codes in terms of self-duality and investigate how the extremality of a DC code relates to that of its bordered counterpart with certain length. Additionally, we study the structural properties of LCD codes within general DC and bordered DC families. Building on recent advances in the characterization of LCD codes, we establish sufficient conditions for bordered DC codes to be LCD under the Euclidean inner product and identify several optimal codes.
