Arithmetic Operators over Finite Field GF($2^m$) for Error Correction Codes Application

TL;DR

This paper reviews the arithmetic operations over GF(2^m) finite fields crucial for error correction codes like BCH and Reed-Solomon, emphasizing their mathematical foundations and implementation in error detection and correction.

Contribution

It provides a comprehensive examination of finite field arithmetic operations essential for implementing BCH and Reed-Solomon codes, highlighting their significance in error correction.

Findings

Detailed analysis of GF(2^m) arithmetic operations

Insights into implementing error correction codes

Clarification of mathematical principles involved

Abstract

Galois field arithmetic circuits find application in a range of domains including error correction codes, communications, signal processing, and security engineering. This paper aims to elucidate the importance of error detection and correction techniques, while also scrutinizing the fundamental principles and wide array of techniques that can be employed. Additionally, a comprehensive understanding of the mathematical intricacies involved in BCH and Reed-Solomon codes requires extensive employment of GF(2m) arithmetic. Consequently, the primary contribution of this research is to critically examine the arithmetic operations performed over a finite field, which are essential for the successful implementation of BCH and Reed-Solomon codes. These operations encompass division, multiplication, exponentiation, multiplication inverses, addition, and subtraction