The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs

TL;DR

This paper determines the weight distribution of a class of reducible cyclic codes by linking exponential sums to the spectra of Hermitian forms graphs, simplifying a complex computational problem.

Contribution

It introduces a novel connection between exponential sums of cyclic codes and the spectra of Hermitian forms graphs, enabling explicit weight distribution calculations.

Findings

Explicit weight distribution formulas for the studied cyclic codes

Establishment of a link between exponential sums and graph spectra

Simplification of weight distribution computation for complex codes

Abstract

The determination of weight distribution of cyclic codes involves evaluation of Gauss sums and exponential sums. Despite of some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectrums of Hermitian forms graphs.