Constructions of cyclic codes and extended primitive cyclic codes with their applications

TL;DR

This paper constructs two new families of linear codes with few weights using special polynomials over finite fields, demonstrating their combinatorial design properties and optimal local recoverability for applications in storage and secret sharing.

Contribution

It introduces extended primitive cyclic codes and reducible cyclic codes with specific parameters, and proves their design and locality properties, advancing coding theory and applications.

Findings

Codes hold 2- and 3-designs

Minimum localities are determined

Optimal locally recoverable codes are derived

Abstract

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold $t$-designs, where $t=2,3$. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.