# On PGZ decoding of alternant codes

## TL;DR

This paper reviews the PGZ decoding algorithm for alternant codes, proposes an improved method for error detection and correction, and demonstrates its effectiveness through examples and computational system descriptions.

## Contribution

It introduces an improved PGZ decoding algorithm for alternant codes, enhancing error detection and correction capabilities.

## Key findings

- Enhanced decoding accuracy for alternant codes
- Successful application to Reed-Solomon and Goppa codes
- Effective computational implementation demonstrated

## Abstract

In this note we first review the classical Petterson-Gorenstein-Zierler decoding algorithm for the class of alternant codes (which includes Reed-Solomon, Bose-Chaudhuri-Hocquenghem and classical Goppa codes), then we present an improvement of the method to find the number of errors and the errorlocator polynomial, and finally we illustrate the procedure with several examples. In two appendices we sketch the main features of the system we have used for the computations.