Multi-point Codes over Kummer Extensions

TL;DR

This paper constructs algebraic geometric codes from Kummer extensions, providing explicit descriptions of Riemann-Roch spaces, Weierstrass semigroups, and pure gaps, leading to the development of multi-point codes with improved parameters.

Contribution

It offers a new explicit characterization of Weierstrass semigroups and pure gaps for Kummer extensions, enabling the construction of better multi-point algebraic geometric codes.

Findings

A new record code with parameters [254,228,≥16] over GF(64)

Explicit bases for Riemann-Roch spaces are provided

Characterization of Weierstrass semigroups and pure gaps for Kummer extensions

Abstract

This paper is concerned with the construction of algebraic geometric codes defined from Kummer extensions. It plays a significant role in the study of such codes to describe bases for the Riemann-Roch spaces associated with totally ramified places. Along this line, we give an explicit characterization of Weierstrass semigroups and pure gaps. Additionally, we determine the floor of a certain type of divisor introduced by Maharaj, Matthews and Pirsic. Finally, we apply these results to find multi-point codes with good parameters. As one of the examples, a presented code with parameters $ [254,228,\geqslant 16] $ over $ \mathbb{F}_{64} $ yields a new record.