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 almost optimal recoverable codes 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：线性码由于在分布式存储、组合学、格、密码学等领域具有良好应用，在编码理论中被广泛研究。构造具有理想性质的线性码是一个有趣的研究课题。本文基于增广技术，从有限域上的函数出发，提出了两类线性码。第一类线性码由有限域上的单项函数构造，确定了其局部性，并给出了两个子码族的重量分布。从这些线性码中获得了无限族近似最优可恢复码和若干最优可恢复码。特别地，证明这两个子码族同时具备最优或近乎最优可扩展性与自对偶性。第二类线性码由有限域上的弱正则bent函数构造，并确定了其重量分布。证明该类码在某些情况下具有局部性3，并推测其他情况下具有局部性2。特别地，从这些线性码推导出两类最优局部可恢复码。此外，还证明该类码同时具备最优或近乎最优可扩展性与自对偶性。