Minimal codewords in Norm-Trace codes

TL;DR

This paper studies the minimal codewords in Norm-Trace codes, focusing on their weight distribution and algebraic structure using algebraic geometry techniques to analyze the irreducibility of related varieties.

Contribution

It introduces a geometric approach to analyze minimal codewords in Norm-Trace codes, providing new insights into their weight distribution and algebraic properties.

Findings

Characterization of minimal codewords in Norm-Trace codes

Analysis of weight distribution using algebraic geometry

Identification of conditions for absolute irreducibility

Abstract

In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.