On the duals of geometric Goppa codes from norm-trace curves

TL;DR

This paper investigates the duals of evaluation codes from norm-trace curves, determining their minimum distance, minimum-weight codeword counts, and dimensions using geometric methods, with applications to Goppa codes.

Contribution

It provides explicit minimum distance and weight bounds for dual codes from norm-trace curves, and computes dimensions of classical two-point Goppa codes using geometric analysis.

Findings

Minimum distance of dual codes explicitly determined

Lower bounds for the number of minimum-weight codewords established

Dimensions of classical two-point Goppa codes computed

Abstract

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppa's construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitly computed.