The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

TL;DR

This paper explicitly computes the spectra of certain irreducible cyclic codes linked to decomposable generalized Paley graphs, with applications to counting rational points on curves and Gaussian periods.

Contribution

It provides explicit spectral formulas for irreducible cyclic codes associated with decomposable generalized Paley graphs, advancing understanding of their algebraic and combinatorial properties.

Findings

Spectral formulas for associated cyclic codes

Reduction formulas for rational points on Artin-Schreier curves

Methods for computing Gaussian periods

Abstract

We use known characterizations of generalized Paley graphs which are cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.