Multi-point Codes from Generalized Hermitian Curves

TL;DR

This paper introduces explicit constructions of multi-point algebraic geometric codes from generalized Hermitian curves, demonstrating their favorable properties and achieving record-breaking parameters over finite fields.

Contribution

It provides a basis for Riemann-Roch spaces of divisors on these curves, enabling explicit code construction and parameter calculation, including dual codes.

Findings

Codes have properties similar to Hermitian codes, such as ease of description, encoding, and decoding.

Explicit formulas for dual codes and their parameters are derived.

A new record code with parameters [234,141,≥59] over GF(27) is constructed.

Abstract

We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Alp Bassa, Peter Beelen, Arnaldo Garcia, and Henning Stichtenoth. Our main result is to find a basis of the Riemann-Roch space of a series of divisors, which can be used to construct multi-point codes explicitly. These codes turn out to have nice properties similar to those of Hermitian codes, for example, they are easy to describe, to encode and decode. It is shown that the duals are also such codes and an explicit formula is given. In particular, this formula enables one to calculate the parameters of these codes. Finally, we apply our results to obtain linear codes attaining new records on the parameters. A new record-giving $ [234,141,\geqslant 59] $-code over $ \mathbb{F}_{27} $ is presented as one of the examples.