The augmented codes of a family of linear codes with locality 2

In this paper, we first generalize the class of linear codes by Ding and Ding [A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Trans. Inform. Theory 61 (2015) 5835-5842, https://ieeexplore.ieee.org/document/7226827]. Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums to determine the parameters and weight distributions of the augmented codes in some cases. It is shown that the augmented codes are self-orthogonal and have only a few nonzero weights. For another thing, the locality of the augmented codes is proved to be 2, which indicates that the augmented codes are useful in distributed storage. Besides, the augmented codes are projective as the minimum distance of their duals is proved to be 3. In particular, we obtain several (almost) optimal linear codes and locally recoverable codes.