کدگذاری و اعداد فیبوناتچی تعمیم یافته

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 importance characteristic which is the golden number. in this thesis, the golden number is observed in different parts. generally, in this thesis we use the 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 matrix, two solutions are introduced. the first one is to use linear algebra and the techniques within it. the second and 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 class of matrices called mp with determinant ±1 is introduced whose nth power, mpn, has a simple closed-form expression. a similar expression is derived for the inverse matrices mp-n. we define two sequences an and bn that are useful in giving a closed-form expression for mpn and mp-n. the matrices mpn and mp-n are used as the encoding and decoding matrices, respectively. given a message-matrix m, we encode m by e=m× mpn and decode e by m=e×mp-n. due to the structure of mp, 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 classical methods in coding theory are complexity and the code rate. the complexity of fibonacci coding is reasonable and it can be implemented by software without difficulty. the second feature that has a main role in the extension of this method is the rate of error-correction capability. in fact, the rate of error-correction capability for the simplest case of fibonacci coding is about ninety-three percentage and for the second case is about ninety-nine percentage. we have also used another very interesting matrix denoted rm,,t as an encoding matrix. these matrices are well-known for their determinants that are expressed by the fibonacci numbers. we consider two cases of rm,,t matrices for coding process: rm,,0 and rm,2. for rm,0, we introduce a method for computing its nth power that is driven from some topics in linear algebra like eigenvalues and eigenvectors. 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 m 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 investigations on this method. in this thesis the efforts have been on using a family of matrices and applying this coding method to them.