A Family of Five-Weight Cyclic Codes and Their Weight Enumerators

TL;DR

This paper introduces a new family of p-ary cyclic codes with duals having three zeros, determines their weight distribution, and reveals they possess five nonzero weights, enhancing understanding of their structure and applications.

Contribution

The paper proposes a novel family of cyclic codes with specific dual properties and explicitly determines their weight enumerators, which was previously uncharacterized.

Findings

The cyclic codes have exactly five nonzero weights.

The weight distribution of these codes is explicitly determined.

The codes have applications in efficient encoding and decoding.

Abstract

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a family of $p$-ary cyclic codes whose duals have three zeros are proposed. The weight distribution of this family of cyclic codes is determined. It turns out that the proposed cyclic codes have five nonzero weights.