A short proof that all linear codes are weakly algebraic-geometric using Bertini theorems of B. Poonen

TL;DR

This paper presents a simplified proof demonstrating that all linear codes are weakly algebraic-geometric, leveraging Bertini theorems by B. Poonen to streamline the understanding of their structure.

Contribution

The paper provides a more accessible proof of a fundamental theorem linking linear codes to algebraic geometry, using Bertini theorems to simplify previous complex proofs.

Findings

All linear codes are weakly algebraic-geometric.

The proof utilizes Bertini theorems by B. Poonen.

Simplifies the understanding of the geometric nature of linear codes.

Abstract

In this paper we give a simpler proof of a deep theorem proved by Pellikan, Shen and van Wee that all linear codes are weakly algebraic-geometric using a theorem of B.Poonen.