Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory

We consider a new class of square Fibonacci (p + 1) × (p + 1)-matrices, which are based on the Fibonacci p-numbers (p = 0, 1, 2, 3, …), with a determinant equal to +1 or −1. This unique property leads to a generalization of the “Cassini formula” for Fibonacci numbers. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. In contrast to classical redundant codes a basic peculiarity of the method is that it allows to correct matrix elements that can be theoretically unlimited integers. For the simplest case the correct ability of the method is equal 93.33% which exceeds essentially all well-known correcting codes.