On the automorphism groups of some AG-codes based on $C_{a, b}$ curves

TL;DR

This paper investigates the automorphism groups of algebraic geometry codes derived from specific $C_{a, b}$ curves, highlighting their structure and applications to decoding algorithms and MDS code construction.

Contribution

It characterizes automorphism groups of certain $C_{a, b}$ curve-based codes and explores their use in decoding and constructing MDS codes.

Findings

Automorphism groups are determined for specific $C_{a, b}$ curves.

Certain $C_{a, b}$ curves yield codes with extra automorphisms.

Results facilitate decoding and code construction using automorphisms.

Abstract

We study $C_{a, b}$ curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how $C_{a, b}$ curves can be used to construct MDS codes and focus on some $C_{a, b}$ curves with extra automorphisms, namely $y^3=x^4+1, y^3=x^4-x, y^3-y=x^4$. The automorphism groups of such codes are determined in most characteristics.