Improved Two-Point Codes on Hermitian Curves

TL;DR

This paper combines two approaches to construct improved two-point codes on Hermitian curves, resulting in codes with better parameters and minimum distance for a wide range of designed distances.

Contribution

It introduces an elementary construction that merges Feng-Rao and Matthews methods to enhance two-point Hermitian codes.

Findings

Codes have improved parameters over previous constructions.

Minimum distance and redundancy are optimized.

Improvements are significant for a large range of designed distances.

Abstract

One-point codes on the Hermitian curve produce long codes with excellent parameters. Feng and Rao introduced a modified construction that improves the parameters while still using one-point divisors. A separate improvement of the parameters was introduced by Matthews considering the classical construction but with two-point divisors. Those two approaches are combined to describe an elementary construction of two-point improved codes. Upon analysis of their minimum distance and redundancy, it is observed that they improve on the previous constructions for a large range of designed distances.