Introducing Three Best Known Goppa Codes

TL;DR

This paper introduces three new binary Goppa codes with higher minimum distances than the current best known codes of similar length and dimension, and explores derived codes with improved parameters.

Contribution

The paper presents three novel binary Goppa codes with superior minimum distances and demonstrates techniques to derive additional codes with better parameters.

Findings

New Goppa codes with higher minimum distances

Derived codes with improved parameters through puncturing, shortening, and extending

Enhanced code performance over existing best known codes

Abstract

The current best known $[239, 21], \, [240, 21], \, \text{and} \, [241, 21]$ binary linear codes have minimum distance 98, 98, and 99 respectively. In this article, we introduce three binary Goppa codes with Goppa polynomials $(x^{17} + 1)^6, (x^{16} + x)^6,\text{ and } (x^{15} + 1)^6$. The Goppa codes are $[239, 21, 103], \, [240, 21, 104], \, \text{and} \, [241, 21, 104]$ binary linear codes respectively. These codes have greater minimum distance than the current best known codes with the respective length and dimension. In addition, with the techniques of puncturing, shortening, and extending, we find more derived codes with a better minimum distance than the current best known codes with the respective length and dimension.