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 locality of them is determined and the weight distributions of two subfamilies of the codes are also given. An infinite family of locally recoverable codes which are at least almost optimal and some optimal recoverable codes are obtained from the linear codes. In particular, the two subfamilies of the codes are proved to be both optimally or almost optimally extendable and self-orthogonal. The second family of linear codes is constructed from weakly regular bent functions over finite fields and their weight distribution is determined. This family of codes is proved to have locality 3 for some cases and is conjectured to have locality 2 for other cases. Particularly, two families of optimal locally recoverable codes are derived from the linear codes. Besides, this family of codes is also proved to be both optimally or almost optimally extendable and self-orthogonal.

翻译：线性码在编码理论中被广泛研究，因其在分布式存储、组合数学、格密码学等领域的良好应用。构造具有期望性质的线性码是一个有趣的研究课题。本文基于扩充技术，从有限域上的若干函数出发，提出了两类线性码。第一类线性码由有限域上的单项式函数构造，确定了其局部性，并给出了两个子族码的重量分布。从这些线性码中得到了一个至少接近最优的无限族局部可修复码和一些最优可修复码。特别地，证明了这两个子族码均为最优或接近最优可扩码且自正交。第二类线性码由有限域上的弱正则弯曲函数构造，并确定了其重量分布。证明了该族码在某些情况下具有局部性3，并在其他情况下推测具有局部性2。特别地，从这些线性码中推导出两类最优局部可修复码。此外，该族码也被证明为最优或接近最优可扩码且自正交。