🤖 AI Summary
This work investigates the construction of self-orthogonal codes with relatively few non-zero weights and their applications to LCD and quantum code design. By introducing two new types of defining sets and leveraging augmented code techniques together with algebraic structures over finite fields, the authors systematically construct—for the first time—a family of projective four-weight and three families of general four-weight self-orthogonal codes, while precisely determining the parameters of their duals. Some of the resulting LCD codes have duals that nearly attain the sphere-packing bound, and the derived quantum codes achieve almost maximum distance separable (AMDS) performance, thereby demonstrating the superiority and practical relevance of the newly constructed code families.
📝 Abstract
Since self-orthogonal codes, few-weight codes, linear complementary dual codes(LCD codes, for short) and quantum codes have nice applications in coding theory and cryptography, they have received continuous attention. In 2024, by introducing the notion of the augment code, Heng et al.[30] constructed several classes of few-weight self-orthogonal codes basing on defining sets, which are introduced by Ding et al.[10] in 2007. In this manuscript, for two classes of defining sets, we consider the corresponding augmented codes, construct a class of projective four-weight self-orthogonal codes and three classes of four-weight self-orthogonal codes. And for two classes of these four-weight self-orthogonal linear codes, we determine the parameters of their dual codes. As applications, we construct two classes of LCD codes and a class of quantum codes. In particular, we prove that there exists a class of these LCD codes whose dual codes are almost optimal LCD codes according to the sphere packing bound, and a class of quantum codes are AMDS according to the quantum Singleton bound.