A Note on Weight Distributions of Irreducible Cyclic Codes

TL;DR

This paper investigates the weight distributions of irreducible cyclic codes, providing necessary and sufficient conditions for codes with maximal and minimal nonzero weights, and determining their distributions in specific cases.

Contribution

It offers new criteria for identifying irreducible cyclic codes with maximal or single nonzero weight and computes their weight distributions in certain scenarios.

Findings

Characterization of codes with maximal nonzero weights

Conditions for codes with only one nonzero weight

Explicit weight distribution in specific cases

Abstract

Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.