Codes and Stability

TL;DR

This paper introduces new linear codes derived from adelic elements of function fields, generalizing algebraic geometry codes, and explores their properties, including bounds, stability conditions, and minimal distances.

Contribution

It presents a novel class of codes based on adelic elements, extending algebraic geometry codes and analyzing their stability and distance properties.

Findings

Codes coincide with classical AG codes when r=1

Provides improved bounds for code dimensions

Determines minimal distances using stability conditions

Abstract

We introduce new yet easily accessible codes for elements of $GL_r(A)$ with $A$ the adelic ring of a (dimension one) function field over a finite field. They are linear codes, and coincide with classical algebraic geometry codes when $r=1$. Basic properties of these codes are presented. In particular, when offering better bounds for the associated dimensions, naturally introduced is the well-known stability condition. This condition is further used to determine the minimal distances of these codes. To end this paper, for reader's convenience, we add two appendices on some details of the adelic theory of curves and classical AG codes, respectively.