The thesis presents a novel approach to key themes in algebraic coding theory, centering on the concept of a partial-inverse polynomial within the quotient ring F[x]/m(x). It highlights that the decoding of Reed-Solomon codes relies on computing a partial-inverse polynomial, with practical computation linked to shift-register synthesis and the Berlekamp-Massey algorithm. A significant contribution is a new algorithm for computing partial-inverse polynomials, closely related to Berlekamp-Massey but applicable to extended Reed-Solomon codes and polynomial remainder codes. This algorithm can be transformed into the Euclidean algorithm, offering a fresh derivation. For Reed-Solomon decoding, it can be applied to the classical key equation or a newly proposed key equation, particularly relevant for generalizations of these codes, along with two new interpolation methods. Additionally, the work delves into polynomial remainder codes, a natural extension of Reed-Solomon codes, establishing a robust theoretical framework that accommodates varying degrees of remainders and introduces two definitions of distance between codewords. The decoding process for these codes leads directly to the new key equation. The thesis also addresses the challenge of decoding errors beyond half the minimum distance, proposing a generalization of the Berlekamp-Massey algorithm for multiple parallel sequences, alongside a corresponding decoding algorithm ut
