On Coding by (2,q)-Distance Fibonacci Numbers

Abstract

In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2,q)-distance Fibonacci numbers F2(q,n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (Tq(n))n=−∞∞, which is an extension of Włoch’s sequence (F2(q,n))n=0∞.

Generalization of the Distance Fibonacci Sequences

“…This revelation established a profound connection between two fundamental sequences in mathematics: the Fibonacci sequence and the binomial coefficients represented in Pascal's triangle. Lucasʹs insight not only highlighted an elegant combinatorial property but also enriched the mathematical understanding of Fibonacci numbers, demonstrating their pervasive presence in various mathematical structures and sequences (see Figure 2) [18]. In [25][26][27] and in this study, the connection between Pascal's triangle and distance Fibonacci numbers is examined.…”

“…In 2020, I. Matoušová and P. Trojovský [18] investigated the application of (2,q)-distance number sequences in the development of algorithms and mathematical operations. Their research demonstrated that these sequences have practical applications in various fields, including data processing, random number generation, and error correction.…”

“…Their application underscores the importance of matrix methods in achieving precision, speed, and scalability in mathematical computations. In [18,[25][26][27], using matrices provided an alternative way to find distance Fibonacci numbers.…”

Distance Fibonacci Polynomials by Graph Methods

“…It is well known that Fibonacci numbers can be obtained from Pascal's triangle. Many authors studying Fibonacci type sequences also give connections with Pascal's triangle or modified Pascal's triangle, see [23][24][25]. Such interpretations can be used to derive binomial formulas for Fibonacci type sequences.…”

Multidimensional Fibonacci Coding

“…In terms of data compression, Fibonacci codes are optimal under some distributions and can be used as alternatives to Huffman codes [1,8]. Fibonacci codes are naturally suitable for integer data, where fast encoding and decoding algorithms can be implemented [9][10][11]. In addition, compared with other variable-length codes for the integers such as logarithmic ramp [12] and Elias codes [13], Fibonacci codes provide much better resistance against insertion and deletion errors.…”