LAUSR

Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic. In this paper, based on the augmentation technique, we present two families of linear codes from some functions over finite fields. The first family of linear codes is constructed from monomial functions over finite fields. The weight distribution of the codes is determined in some cases. The codes are proved to be both optimally or almost optimally extendable and self-orthogonal under certain conditions. The localities of the codes and their duals are also studied and we obtain an infinite family of optimal or almost optimal locally recoverable codes. The second family of linear codes is constructed from weakly regular bent functions over finite fields and its weight distribution is explicitly determined. This family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal. Besides, this family of codes has been proven to have locality 2 or 3 under certain conditions. Particularly, we derive two infinite families of optimal locally recoverable codes. Some infinite families of 2-designs are obtained from the codes in this paper as byproducts.