A family of codes with locality containing optimal codes

TL;DR

This paper introduces a new family of codes with locality, derived from evaluation codes, providing their dimensions, bounds on minimum distance, and demonstrating their optimality in certain cases.

Contribution

The paper defines a novel family of codes with locality, determines their dimensions, bounds their minimum distance, and proves their optimality in specific instances.

Findings

Determined the dimension of the new codes.

Established bounds for the minimum distance.

Proved the optimality of these codes in special cases.

Abstract

Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are "optimal codes", as defined by Prakash et al.