English One of the most important number sequences in mathematics is Fibonacci sequence. Fibonacci sequence except for mathematics is applied to other branches of science such as Physics and Arts. In fact, between anesthetics and this sequence there exists a wonderful relation. Fibonacci sequence has an important characteristic which is its relation with the golden number. In this thesis, the golden number is observed in different parts. Generally, in this thesis we use some matrices that can be formalized which means that the determinant and the nth power of them have a closed form expression. In order to compute the nth power of the considered matrices, two solutions are introduced. The first one is to use linear algebra and the techniques within it. The second and the most efficient solution is to use sequences to compute the nth power of matrices. In this thesis, the application of Fibonacci-based sequences in coding theory is investigated. A new M p n and decode E by M=E×M p -n . Due to the structure of M p , some relations between the entries of the code-message matrices exist that are used in the error-correction process. The main differences between this new coding theory and other dir=ltr align=left In order to be able to answer the question that whether this coding method is acceptable or not, we should answer some important questions such as what kind of relationship exists between the size of the massage matrix and the power of encoding matrix which means how p and n should be chosen in order to have the least error in the error correction. One other issue with the coding method is the solution of ill-conditioned systems while correcting errors. In fact, to utilize this coding method in a communication system one needs more numerical analysis investigating this method. In this thesis the efforts have been on using a family of matrices and applying this coding method to them.