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Abstract

This monograph explores error-correcting codes that play a crucial role in technologies, particularly in information and communication systems. The core of the investigation aimed to answer the following guiding question: How to determine a relationship between parameters of a linear code to ascertain whether it is perfect or not? The main objective of this study is to establish a strong connection between the parameters of a linear code and its quality as a perfect code. Within this scope, the specific goals of this work encompass a deep exploration of fundamental concepts in Linear and Abstract Algebra, addressing elements such as residue classes of integers, vector spaces, fields, linear transformations, and other relevant notions.Furthermore, essential terms such as metric, metric spaces, packing radius of a code, Hamming metric, linear codes, and perfect codes are rigorously defined and examined in depth. Through a comprehensive analysis of these concepts, the focus is not only limited to the introductory presentation of error-correcting codes but also aims to build a solid theoretical foundation for determining perfect codes based on their linear code parameters, as evidenced in the (Pinheiro-Oliveira) Theorem.