On Parameters of Subfield Subcodes of Extended Norm-Trace Codes

TL;DR

This paper presents a method to determine the parameters of subfield subcodes of extended Norm-Trace codes using Gröbner bases, providing insights into their dimensions and minimum distances.

Contribution

It introduces a Gröbner basis approach to compute parameters of subfield subcodes of extended Norm-Trace codes, including their dimensions and bounds on minimum distance.

Findings

Dimension formulas for subfield subcodes

BCH-like bound on minimum distance

Method applicable to Norm-Trace code variants

Abstract

In this article we describe how to find the parameters of subfield subcodes of extended Norm--Trace codes. With a Gr\"obner basis of the ideal of the $\mathbb{F}_{q^r}$ rational points of the Norm--Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH--like bound from the minimum distance of the original supercode.