The Weight Enumerator of Three Families of Cyclic Codes

TL;DR

This paper generalizes the weight distribution analysis of certain $p$-ary cyclic codes, extending previous results by connecting exponential sums to graph spectra, and determines their weight distributions for both even and odd parameters.

Contribution

It extends the known weight distribution results of $p$-ary cyclic codes by generalizing the class studied and establishing new formulas for both even and odd cases.

Findings

Weight distributions are explicitly determined for the generalized cyclic codes.

The approach connects exponential sums with the spectrum of Hermitian forms graphs.

Results include the weight distributions of two related families of cyclic codes.

Abstract

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.