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

TL;DR

This paper generalizes a class of linear codes, studies their augmented versions using Gaussian sums, and demonstrates their usefulness in distributed storage with optimal or near-optimal parameters.

Contribution

It introduces a generalized class of linear codes, analyzes their augmented codes' parameters, weight distributions, and locality, and establishes their optimality and projectiveness.

Findings

Augmented codes have only a few nonzero weights.

They are self-orthogonal with locality 2.

Several (almost) optimal linear and locally recoverable codes are obtained.

Abstract

In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). 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 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.