Four classes of few-weight self-orthogonal codes and their applications for LCD codes and quantum codes

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.