Some New Fibonacci Matrices and Error Detecting-Correcting Codes

Abstract

The rapid and accurate transfer of information is essential in today’s digital world, where error detecting and correcting codes play a crucial role. In this context, Fibonacci numbers have been extensively applied in coding theory, particularly in coding structures based on Fibonacci polynomials and symmetric matrices. This study proposes a novel coding framework by defining and using some new Fibonacci matrices with their determinants. The properties of these matrices are analyzed to construct efficient coding and error correction mechanisms. The proposed structure enables the transmission of larger data volumes in a single iteration without compromising the error correction performance, demonstrating the potential of Fibonacci matrices for reliable and efficient information transfer.

1. Introduction

In recent years, advancements in the field of information transfer have gained significant momentum. This acceleration has primarily been driven by the increasing reliance on the digital world for a substantial portion of our activities, including those in social life. Consequently, the rapid and error-free transfer of information has become critically important. A mathematical solution framework designed to address this challenge, known as error detecting and correcting codes, has been extensively studied by researchers. In this context, various methods and structures have been developed in this domain from algebraic, combinatorial, and number-theoretical perspectives. For instance, studies utilizing Fibonacci numbers also form a subset of this research. The literature includes algebraic coding studies based on Fibonacci polynomials, such as those by Esmaeili et al. [1] and Köroğlu et al. [2], as well as works based on symmetric Fibonacci matrices, such as those by Basu and Prasad [3], Stakhov [4], Kuloğlu and Özkan [5], Akbiyik [6], and Yamaç Akbiyik [7]. Our study belongs to the category of research that involves Fibonacci matrices. These studies focus on the construction of coding and decoding frameworks using matrices that have specific determinants, using their inherent properties to prevent potential errors. In addition, there are various studies that explore generalizations of Fibonacci numbers and their other properties. Some of the most significant and innovative ones can be presented as follows. Caldarola et al. introduced new algebraic constructs for Fibonacci numbers [8], Sirivoravit et al. and Erkan et al. gave new generalizations of Fibonacci numbers [9,10], Biban et al. and Xu worked on encryption by using a Fibonacci matrix and Fibonacci transformations [11,12]. For further related articles, see [13,14].

Key challenges in coding structures include speed and the error correction capability. The proposed framework is designed to transmit more data in a single iteration compared to existing approaches, without any compromise in the error correction efficiency. To reach this goal, we first define some matrices using Fibonacci numbers and analyze their properties. Subsequently, we build a symmetric coding structure based on these matrices, followed by an examination of error correction scenarios.

The outline of the paper is as follows: We will first discuss some fundamental concepts in Section 2. Subsequently, in Section 3, we will introduce newly defined matrices, along with their properties. Section 4 will focus on the construction of coding and decoding schemes, and Section 5 will conclude the study with a brief summary.

2. Preliminaries

The Fibonacci sequence is an integer sequence defined by the recurrence relation $F_n=F_{n-1}+F_{n-2}$ with the initial values $F_0=0$ and $F_1=1$. The characteristic equation of this recurrence relation is $x^2-x-1=0$, with the roots $\alpha ={\displaystyle \frac{1+\sqrt{5}}{2}}$ and $\beta ={\displaystyle \frac{1-\sqrt{5}}{2}}$. The Binet formula, which gives the nth term of this sequence, is expressed as $F_n={\alpha }^n+{\beta }^n$. Moreover, negative-indexed Fibonacci numbers can be obtained with the formula $F_{-n}={(-1)}^{n+1}{F}_n$. We can give some negative-indexed and positive-indexed Fibonacci numbers, as in the table below.

$$\begin{array}{|ccccccccccccccccc|}\hline \cdots & F_{-7}& F_{-6}& F_{-5}& F_{-4}& F_{-3}& F_{-2}& F_{-1}& F_0& F_1& F_2& F_3& F_4& F_5& F_6& F_7& \cdots \\ \cdots & 13& -8& 5& -3& 2& -1& 1& 0& 1& 1& 2& 3& 5& 8& 13& \cdots \\ \hline\end{array}$$

In 1977, Stakhov introduced Fibonacci p-numbers [15]. They are defined by the recurrence relation $F_p\left(n\right)=F_p(n-1)+F_p(n-p-1)$ with the initial values $F_p\left(1\right)=F_p\left(2\right)=\cdots =F_p(p+1)=1$ for $n>p+1$. It was observed that when computing elements with negative indices, $F_{-p-1}=F_{-p-2}=\cdots =F_{-2p+1}=0$. Note that when $p=1$, the classical Fibonacci sequence is obtained.

Matrices constructed from these integer sequences are the subject of study by many researchers and find applications in various fields. The Fibonacci Q-matrix, as defined by Hoggat [16] and Gould [17], is a $2\times 2$ square matrix given by

$$Q=\left[\begin{array}{cc}{F}_2& F_1\\ F_1& F_0\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right].$$

The most important properties of these matrices are $Q^n=\left[\begin{array}{cc}{F}_{n+1}& F_n\\ F_n& F_{n-1}\end{array}\right]$ and $det\left(Q^n\right)={(-1)}^n$ [16]. The determinant property is also known as Cassini’s identity.

In [18], Stakhov provided a more general form of the Fibonacci Q-matrices. The generalized Fibonacci matrices $Q_p$ are defined as

$$Q_p={\left[\begin{array}{ccc}1& & \\ 0& & I_{p\times p}\\ ⋮& & \\ 1& 0& \cdots & 0\end{array}\right]}_{(p+1)\times (p+1)}$$

where I is an identity matrix with p dimensions. The most important properties of these matrices are

$$Q_p^n=\left[\begin{array}{cccc}{F}_p(n+1)& F_p\left(n\right)& \cdots & F_p(n-p+1)\\ F_p(n-p+1)& F_p(n-p)& \cdots & F_p(n-2p+1)\\ ⋮& ⋮& \ddots & ⋮\\ F_p\left(n\right)& F_p(n-1)& \cdots & F_p(n-p)\end{array}\right]$$

and

$$det\left(Q_p\right)={(-1)}^p.$$

Additionally, in the same work, Stakhov also presented the most general form of the Cassini formula as $det\left(Q_p^n\right)$.

3. Some New Special Matrices of Fibonacci Numbers

In this section, we will define some special matrices that we will use in the next section and examine their basic properties. First, let us define the matrix $X_m^n$, which is inspired by the matrix $Q_p^n$.

Definition 1.

For Fibonacci numbers, $F_n$, the matrix $X_m^n$ is defined as

$$X_m^n=\left[\begin{array}{cccc}{F}_n& F_{n-1}& \cdots & F_{n-m+1}\\ F_{n-m+1}& F_{n-m}& \cdots & F_{n-2m+2}\\ ⋮& ⋮& \ddots & ⋮\\ F_{n-{(m-1)}^2}& F_{n-{(m-1)}^2-1}& \cdots & F_{n-m^2+m}\end{array}\right]$$

where the square matrix $X_m^n$ is called the Fibonacci X matrix. Here, m and n stand for the number of rows and the index of the starting Fibonacci number, respectively.

Proposition 1.

For $m\ge 1$, the determinant of the Fibonacci X matrix is 0.

Proof. 

In Fibonacci X matrices, if we multiply the second and third columns by $-1$ and add them to the first column, then the first column becomes 0. Since the determinant of a matrix with any row or column equal to zero is zero, then it is seen that the determinant is zero. □

Example 1.

Let $m=4$ and $n=11$; then, the matrix $X_m^n$ and $det\left(X_4^{11}\right)$ can be obtained as follows, respectively:

$$X_4^{11}=\left[\begin{array}{cccc}{F}_{11}& F_{10}& F_9& F_8\\ F_8& F_7& F_6& F_5\\ F_5& F_4& F_3& F_2\\ F_2& F_1& F_0& F_{-1}\end{array}\right]$$

$$det\left(X_4^{11}\right)=\left|\begin{array}{cccc}89& 55& 34& 21\\ 21& 13& 8& 5\\ 5& 3& 2& 1\\ 1& 1& 0& 1\end{array}\right|=0$$

The matrix structure, whose definition is provided below, will be utilized for Method 1 in the next section. This matrix is inspired by the Fibonacci X matrix.

Definition 2.

For $n\in \mathbb{Z}$, the matrix that can be defined as the augmented matrix of the Fibonacci X matrix is

$$K_n=\left[\begin{array}{cccc}& & & 1\\ & X_3^n& & 1\\ & & & 1\\ 1& 1& 1& 1\end{array}\right]=\left[\begin{array}{cccc}{F}_n& F_{n-1}& F_{n-2}& 1\\ F_{n-2}& F_{n-3}& F_{n-4}& 1\\ F_{n-4}& F_{n-5}& F_{n-6}& 1\\ 1& 1& 1& 1\end{array}\right]$$

This $4\times 4$ matrix is called the Fibonacci K matrix.

Proposition 2.

For $n\in \mathbb{Z}$, the determinant of the matrix $K_n$ equals ${(-1)}^n$.

Proof. 

If we multiply the second and third columns of the matrix $K_n$ by $-1$ and add them to the first column, using the identities $F_n{F}_{n-1}=F_n^2-F_{n-1}^2+{(-1)}^n$ [19] and $F_{n-1}{F}_{n+3}-F_n{F}_{n+2}=2{(-1)}^n$ [20], we obtain the following:

$$\begin{array}{cc}\hfill det\left(K_n\right)& =\left|\begin{array}{cccc}0& F_{n-1}& F_{n-2}& 1\\ 0& F_{n-3}& F_{n-4}& 1\\ 0& F_{n-5}& F_{n-6}& 1\\ -1& 1& 1& 1\end{array}\right|=\left|\begin{array}{ccc}{F}_{n-1}& F_{n-2}& 1\\ F_{n-3}& F_{n-4}& 1\\ F_{n-5}& F_{n-6}& 1\end{array}\right|\hfill \\ \hfill & =\left|\begin{array}{cc}{F}_{n-3}& F_{n-4}\\ F_{n-5}& F_{n-6}\end{array}\right|-\left|\begin{array}{cc}{F}_{n-1}& F_{n-2}\\ F_{n-5}& F_{n-6}\end{array}\right|+\left|\begin{array}{cc}{F}_{n-1}& F_{n-2}\\ F_{n-3}& F_{n-4}\end{array}\right|\hfill \\ \hfill & =F_{n-3}{F}_{n-6}-F_{n-5}{F}_{n-4}-(F_{n-1}{F}_{n-6}-F_{n-2}{F}_{n-5})+F_{n-4}{F}_{n-1}-F_{n-3}{F}_{n-2}\hfill \\ \hfill & ={(-1)}^n-\left[(F_{n-2}+F_{n-3})(F_{n-4}-F_{n-5})-F_{n-2}{F}_{n-5}\right]+{(-1)}^{n+2}\hfill \\ \hfill & ={(-1)}^n-(F_{n-2}{F}_{n-4}-F_{n-2}{F}_{n-5}+F_{n-3}{F}_{n-4}-F_{n-3}{F}_{n-5}-F_{n-2}{F}_{n-5})+{(-1)}^n\hfill \\ \hfill & =2{(-1)}^n-\left[F_{n-2}{F}_{n-4}-(F_{n-2}{F}_{n-5}+F_{n-3}{F}_{n-5})+(F_{n-3}{F}_{n-4}-F_{n-2}{F}_{n-5})\right]\hfill \\ \hfill & =2{(-1)}^n-\left[F_{n-2}{F}_{n-4}-F_{n-1}{F}_{n-5}+{(-1)}^n\right]\hfill \\ \hfill & =2{(-1)}^n-\left[2{(-1)}^n+{(-1)}^n\right]\hfill \\ \hfill & =-{(-1)}^n={(-1)}^{n+1}.\hfill \end{array}$$

□

Example 2.

For $n=9$,

$$det\left(K_9\right)=\left|\begin{array}{cccc}34& 21& 13& 1\\ 13& 8& 5& 1\\ 5& 3& 2& 1\\ 1& 1& 1& 1\end{array}\right|=-1$$

Example 3.

For $n=8$,

$$det\left(K_8\right)=\left|\begin{array}{cccc}21& 13& 8& 1\\ 8& 5& 3& 1\\ 3& 2& 1& 1\\ 1& 1& 1& 1\end{array}\right|=1$$

Next, we present a property of the matrix $K_n$ that will be useful in the coding processes that involve this matrix.

Proposition 3.

Let the pth power of the matrix $K_n$ be denoted as

$${\left(K_n\right)}^p=\left[\begin{array}{cccc}{k}_{11}& k_{12}& k_{13}& k_{14}\\ k_{21}& k_{22}& k_{23}& k_{24}\\ k_{31}& k_{32}& k_{33}& k_{34}\\ k_{41}& k_{42}& k_{43}& k_{44}\end{array}\right].$$

In this case,

(i) 

${lim}_{p\to \infty }\left({\displaystyle \frac{k_{1j}}{k_{2j}}}\right)\approx {lim}_{p\to \infty }\left({\displaystyle \frac{k_{2j}}{k_{3j}}}\right)\approx \alpha +1$

(ii) 

${lim}_{p\to \infty }\left({\displaystyle \frac{k_{3j}}{k_{4j}}}\right)\approx {\alpha }^{n-6}.(\alpha -0.7)$

(iii) 

${lim}_{p\to \infty }\left({\displaystyle \frac{k_{i1}}{k_{i2}}}\right)\approx {lim}_{p\to \infty }\left({\displaystyle \frac{k_{i2}}{k_{i3}}}\right)\approx \alpha$

(iv) 

${lim}_{p\to \infty }\left({\displaystyle \frac{k_{i3}}{k_{i4}}}\right)\approx {\alpha }^{n-5}.(\alpha -0.4)$.

where α is the root of the characteristic equation $x^2-x-1$.

Proof .

(i)   The matrix $K_n$ includes blocks derived from Fibonacci numbers, as well as a constant column of ones comprising the fourth column. In the absence of this column, the dominant eigenvalue of the matrix would correspond to the growth rate of Fibonacci numbers, which is asymptotically $\alpha$. However, the addition of the column ${[1,1,1,1]}^T$ modifies the spectral properties of $K_n$, leading to an adjustment in the dominant eigenvalue.

• For a large p, the behavior of ${\left(K_n\right)}^p$ is dominated by the largest eigenvalue ${\lambda }_1$ and the corresponding eigenvector ${\mathbf{v}}_1$. Specifically, we have

  $${\left(K_n\right)}^p\sim {\lambda }_1^p{\mathbf{v}}_1{\mathbf{w}}_1^T,$$

  where ${\lambda }_1$ is the dominant eigenvalue, ${\mathbf{v}}_1$ is the corresponding right eigenvector, and ${\mathbf{w}}_1$ is the corresponding left eigenvector.
  The fourth column of ones in $K_n$ plays a significant role in altering the dominant eigenvector. Without this column, the dominant eigenvector would be associated with Fibonacci numbers, and the ratios ${\displaystyle \frac{k_{1j}}{k_{2j}}}$ would converge to $\alpha$. However, the presence of the column introduces a constant additive term that shifts these ratios.
  For the dominant eigenvector ${\mathbf{v}}_1={[v_1,v_2,v_3,v_4]}^T$, the components $v_1,v_2,\cdots ,v_4$ reflect a combination of Fibonacci growth and the contribution of the constant column. This results in

  $${\displaystyle \frac{v_1}{v_2}}\to \alpha +1,$$

  where the additional $+1$ arises due to the structural contribution of the constant column.
  The ratios ${\displaystyle \frac{k_{1j}}{k_{2j}}}$ converge to $\alpha +1$ as $p\to \infty$. This convergence highlights how the structure of $K_n$, particularly the constant column, modifies the asymptotic behavior of the ratios from the expected $\alpha$ to $\alpha +1$.
  The others can be proven in a similar manner.

□

Proposition 4.

Let M be any $4\times 4$ matrix, and let $K_n$ be the Fibonacci K matrix. Define the product:

$${\left(K_n\right)}^p\times M=\left[\begin{array}{cccc}{e}_{11}& e_{12}& e_{13}& e_{14}\\ e_{21}& e_{22}& e_{23}& e_{24}\\ e_{31}& e_{32}& e_{33}& e_{34}\\ e_{41}& e_{42}& e_{43}& e_{44}\end{array}\right].$$

In this case,

(i) ${lim}_{p\to \infty }\left({\displaystyle \frac{e_{1j}}{e_{2j}}}\right)\approx {lim}_{p\to \infty }\left({\displaystyle \frac{e_{2j}}{e_{3j}}}\right)\approx \alpha +1$

(ii) ${lim}_{p\to \infty }\approx {\alpha }^{n-6}$.

Proof. 

The proof of this proposition can be constructed in a manner similar to that of the proof of Proposition 3. □

When we increase the size of the Fibonacci K matrix, we obtain matrices with a determinant of zero, similarly to the Fibonacci X matrix.

Proposition 5.

For $n\in \mathbb{Z}$, the determinant of the augmented matrix $K_n$ is as follows:

$$det\left(K_n\right)=\left|\begin{array}{ccccc}{F}_n& F_{n-1}& F_{n-2}& F_{n-3}& 1\\ F_{n-3}& F_{n-4}& F_{n-5}& F_{n-6}& 1\\ F_{n-6}& F_{n-7}& F_{n-8}& F_{n-9}& 1\\ F_{n-9}& F_{n-10}& F_{n-11}& F_{n-12}& 1\\ 1& 1& 1& 1& 1\end{array}\right|=0.$$

Proof. 

By multiplying the second and third columns by $-1$ and adding them to the first column, and then multiplying the third and fourth columns by $-1$ and adding them to the second column, we obtain

$$\left|\begin{array}{ccccc}0& 0& F_{n-2}& F_{n-3}& 1\\ 0& 0& F_{n-5}& F_{n-6}& 1\\ 0& 0& F_{n-8}& F_{n-9}& 1\\ 0& 0& F_{n-11}& F_{n-12}& 1\\ -1& -1& 1& 1& 1\end{array}\right|=-1·\left|\begin{array}{cccc}0& F_{n-2}& F_{n-3}& 1\\ 0& F_{n-5}& F_{n-6}& 1\\ 0& F_{n-8}& F_{n-9}& 1\\ 0& F_{n-11}& F_{n-12}& 1\end{array}\right|=0.$$

□

Example 4.

$$det\left(K_{12}\right)=\left|\begin{array}{ccccc}144& 89& 55& 34& 1\\ 34& 21& 13& 8& 1\\ 8& 5& 3& 2& 1\\ 2& 1& 1& 0& 1\\ 1& 1& 1& 1& 1\end{array}\right|=0.$$

This property holds even as the matrix size increases further.

Now, we will define the new matrix used in the second coding structure.

Definition 3.

For $n\in \mathbb{Z}$, the matrix defined as

$$S_n=\left[\begin{array}{ccc}{F}_n^2& F_{n-1}^2& F_{n-2}^2\\ F_{n-2}^2& F_{n-3}^2& F_{n-4}^2\\ F_{n-4}^2& F_{n-5}^2& F_{n-6}^2\end{array}\right]$$

is referred to as the Fibonacci S matrix.

Having knowledge about the determinants of matrices is crucial for using them in coding structures. The next proposition presents a property that allows Fibonacci S matrices to be utilized in coding structures.

Proposition 6.

For $n\in \mathbb{Z}$, the determinant of the Fibonacci S matrix is given by

$$det\left(S_n\right)=6{(-1)}^n.$$

Proof. 

When a row of a matrix is multiplied by a scalar and added to another row, the determinant of the matrix remains unchanged. Therefore, we can multiply the second row by $-1$ and add it to the first row and similarly multiply the third row by $-1$ and add it to the second row.

This results in the following transformation:

$$\left|\begin{array}{ccc}{\left(F_n\right)}^2& {\left(F_{n-1}\right)}^2& {\left(F_{n-2}\right)}^2\\ {\left(F_{n-2}\right)}^2& {\left(F_{n-3}\right)}^2& {\left(F_{n-4}\right)}^2\\ {\left(F_{n-4}\right)}^2& {\left(F_{n-5}\right)}^2& {\left(F_{n-6}\right)}^2\end{array}\right|=\left|\begin{array}{ccc}{\left(F_n\right)}^2-{\left(F_{n-2}\right)}^2& {\left(F_{n-1}\right)}^2-{\left(F_{n-3}\right)}^2& {\left(F_{n-2}\right)}^2-{\left(F_{n-4}\right)}^2\\ {\left(F_{n-2}\right)}^2-{\left(F_{n-4}\right)}^2& {\left(F_{n-3}\right)}^2-{\left(F_{n-5}\right)}^2& {\left(F_{n-4}\right)}^2-{\left(F_{n-6}\right)}^2\\ {\left(F_{n-4}\right)}^2& {\left(F_{n-5}\right)}^2& {\left(F_{n-6}\right)}^2\end{array}\right|.$$

Using the property of Fibonacci numbers ${\left(F_{n+1}\right)}^2-{\left(F_{n-1}\right)}^2=F_{2n}$ [20], it can be observed that the given determinant is equivalent to

$$\left|\begin{array}{ccc}{F}_{2n-2}& F_{2n-4}& F_{2n-6}\\ F_{2n-6}& F_{2n-8}& F_{2n-10}\\ {\left(F_{n-4}\right)}^2& {\left(F_{n-5}\right)}^2& {\left(F_{n-6}\right)}^2\end{array}\right|.$$

Performing a Laplace expansion along the last row of this determinant yields

$$\begin{array}{cc}\hfill \left|\begin{array}{ccc}{F}_{2n-2}& F_{2n-4}& F_{2n-6}\\ F_{2n-6}& F_{2n-8}& F_{2n-10}\\ {\left(F_{n-4}\right)}^2& {\left(F_{n-5}\right)}^2& {\left(F_{n-6}\right)}^2\end{array}\right|& =F_{n-4}^2\left|\begin{array}{cc}{F}_{2n-4}& F_{2n-6}\\ F_{2n-8}& F_{2n-10}\end{array}\right|-F_{n-5}^2\left|\begin{array}{cc}{F}_{2n-2}& F_{2n-6}\\ F_{2n-6}& F_{2n-10}\end{array}\right|\hfill \\ \hfill & +F_{n-6}^2\left|\begin{array}{cc}{F}_{2n-2}& F_{2n-4}\\ F_{2n-6}& F_{2n-8}\end{array}\right|.\hfill \end{array}$$

Simplifying it further,

$$\begin{array}{cc}\hfill \left|\begin{array}{ccc}{F}_{2n-2}& F_{2n-4}& F_{2n-6}\\ F_{2n-6}& F_{2n-8}& F_{2n-10}\\ {\left(F_{n-4}\right)}^2& {\left(F_{n-5}\right)}^2& {\left(F_{n-6}\right)}^2\end{array}\right|& =F_{n-4}^2(F_{2n-4}{F}_{2n-10}-F_{2n-6}{F}_{2n-8})\hfill \\ \hfill & -F_{n-5}^2(F_{2n-2}{F}_{2n-10}-F_{2n-6}^2)\hfill \\ \hfill & +F_{n-6}^2(F_{2n-2}{F}_{2n-8}-F_{2n-4}{F}_{2n-6}).\hfill \end{array}$$

Using the property of Fibonacci numbers $F_n{F}_{m+r}-F_{n+r}{F}_m={(-1)}^m{F}_{n-m}{F}_r$ [16], we simplify the terms within parentheses to obtain the determinant as

$$-3F_{n-4}^2+9F_{n-5}^2-3F_{n-6}^2.$$

Next, using the property $F_n{F}_{n-1}=F_n^2-F_{n-1}^2+{(-1)}^n$ [19], the determinant becomes

$$-3(F_{n-4}{F}_{n-5}-{(-1)}^{n-4})+3F_{n-5}^2+3(F_{n-5}{F}_{n-6}-{(-1)}^{n-5}).$$

Simplifying it further, this expression is equivalent to

$$-3F_{n-5}(F_{n-4}-F_{n-5}-F_{n-6})+6{(-1)}^{n-4}.$$

Thus, it is proven that the determinant equals $6·{(-1)}^n$.

□

When we increase the size of the Fibonacci S matrix, we obtain matrices with a determinant of 0. To illustrate this, we first define the most general form of the Fibonacci S matrices.

Definition 4.

For $m\ge 3$, the $m\times m$ matrix is defined as

$$S_m^n=\left[\begin{array}{cccc}{F}_n^2& F_{n-1}^2& \cdots & F_{n-m+1}^2\\ F_{n-m+1}^2& F_{n-m}^2& \cdots & F_{n-2m+2}^2\\ ⋮& ⋮& \ddots & ⋮\\ F_{n-{(m-1)}^2}^2& F_{n-{(m-1)}^2-1}^2& \cdots & F_{n-m^2+m}^2\end{array}\right].$$

This matrix is called the generalized Fibonacci S matrix. For $m=3$, we obtain the Fibonacci S matrix given in the previous definition.

Proposition 7.

For $m\ge 4$, $det\left(S_m^n\right)=0$.

Proof. 

It is straightforward. □

Example 5.

$$S_4^{10}=\left|\begin{array}{cccc}{55}^2& {34}^2& {21}^2& {13}^2\\ {13}^3& 8^2& 5^2& 3^2\\ 3^2& 2^2& 1^2& 1^2\\ 1^2& 0^2& 1^2& {(-1)}^2\end{array}\right|=0.$$

4. Coding and Decoding Methods

In the literature, coding and decoding methods using Fibonacci-type numbers have employed $2\times 2$ and $3\times 3$ matrices [3,4,21,22]. In this study, we developed coding and decoding methods using $3\times 3$ and $4\times 4$ matrices based on some matrices obtained in Section 3. These methods aim to send more messages simultaneously compared to previous methods. Additionally, the goal is to transmit messages faster without any loss, while maintaining the error detection and correction capabilities of earlier methods.

Similarly to previous approaches, our method also utilizes a coding matrix and its inverse decoding matrix. It is necessary for the determinant of the coding matrix to be known; therefore, some matrices mentioned in the previous section can be used. Furthermore, since we use the inverse of the coding matrix in the decoding process, the determinant of the coding matrix must be non-zero. Consequently, some matrices defined in the previous section cannot be used.

4.1. Coding Method 1

In this section, coding is performed using Fibonacci $K_n$ matrices. The message matrix M must have four dimensions. Additionally, to facilitate the decoding process, we will assume that the entries of M are positive. The encoded matrix E is obtained as ${\left(K_n\right)}^p\times M=E$. The parameters n, p, and $det\left(M\right)$ are sent to the receiver along with E. The receiving side retrieves the message as $M={\left(K_n^{-1}\right)}^p\times E$.

Error detection and correction: If the determinant of the received matrix E differs from $det\left(M\right)·{(-1)}^n$, it is understood that there is at least an error in the matrix E. The entries of the matrix E are checked according to the ratios given in Proposition 3 for the $K_n$ matrix, and the erroneous entries are identified. The erroneous entries are corrected by testing values that satisfy these ratios and the determinant, and subsequently, the message matrix is obtained.

Example 6.

Let us send the message “MATHISINCREDIBLE” using this method. First, we present the binary and decimal ASCII code equivalents of this sentence in a table (Table 1), as shown below [23].

Table 1. ASCII codes of letters in “MATH IS INCREDIBLE” [23].

Letter	Binary	Decimal
M	01001101	77
A	01000001	65
T	01010100	84
H	01001000	72
I	01001001	73
S	01010011	83
I	01001001	73
N	01001110	78
C	01000011	67
R	01010010	82
E	01000101	69
D	01000100	68
I	01001001	73
B	01000010	66
L	01001100	76
E	01000101	69

Then, using the decimal values of these code equivalents, we construct our message matrix:

$$M=\left[\begin{array}{cccc}77& 65& 84& 72\\ 73& 83& 73& 78\\ 67& 82& 69& 68\\ 73& 66& 76& 69\end{array}\right].$$

If we choose the encoding matrix as the fifth power of the augmented matrix $K_{11}$, then we obtain the encoded matrix E as follows:

$$\begin{array}{cc}\hfill E=& {\left(K_{11}\right)}^5\times M\hfill \\ \hfill =& \left[\begin{array}{cccc}2,318,755,968,828& 2,315,490,214,089& 2,440,409,852,317& 2,293,878,416,005\\ 885,930,470,818& 884,682,719,152& 932,410,947,295& 876,425,467,969\\ 338,765,131,936& 338,288,012,379& 356,538,495,942& 335,130,576,345\\ 31,084,364,835& 31,040,585,374& 32,715,210,738& 30,750,865,774\end{array}\right].\hfill \end{array}$$

The encoded matrix E, along with the parameters $n=11$, $p=5$, and $det\left(M\right)=-31,259$, is sent to the receiver. The receiver retrieves the message matrix by calculating ${\left(K_{11}^{-1}\right)}^5\times E$. If the determinant of E, $det\left(E\right)$, equals $det\left(M\right)·{(-1)}^{11}=31,259$, it indicates that the message has been received without any issues.

We examine the error cases here.

Case of one error: Suppose the entry $e_{33}$ in E is incorrectly received as 856714305912. In this case, since $det\left(E\right)\ne 31,259$, it is clear that at least one error has occurred. Checking the ratios, ${\displaystyle \frac{e_{23}}{e_{33}}}\approx 1.08\not\approx \alpha +1$ and ${\displaystyle \frac{e_{33}}{e_{43}}}\approx 26.187\not\approx {\alpha }^5·(\alpha -0.7)$. Since other ratios from Proposition 3 are satisfied, it is determined that only $e_{33}$ is incorrect.

Single error correction: Taking $e_{33}=x$ and expanding the determinant along the third column using Laplace expansion, we equate the determinant to 31,259 to find x.

Case of two errors: We will analyze three different scenarios based on whether the errors are in the same column or not and whether they are adjacent.

Case 1 (two errors in different columns): The positions of the errors are determined using the ratios specified in the relevant proposition.

Error correction: For example, if $e_{12}$ and $e_{23}$ are erroneous, let $e_{12}=x$ and $e_{23}=y$. By performing Laplace expansion along the second and third columns and equating $det\left(E\right)=-det\left(M\right)$, we obtain a Diophantine equation with two unknowns. Additionally, we solve the equation by finding integer values satisfying ${\displaystyle \frac{e_{12}}{e_{22}}}\approx \alpha +1$ and ${\displaystyle \frac{e_{23}}{e_{33}}}\approx \alpha +1$, thereby correcting the erroneous values.

Case 2 (two errors in the same column and adjacent): The error positions are identified using the ratios from the relevant proposition. For example, if $e_{12}$ and $e_{22}$ are erroneous, then ${\displaystyle \frac{e_{12}}{e_{22}}}\not\approx \alpha +1$, ${\displaystyle \frac{e_{22}}{e_{32}}}\not\approx \alpha +1$, ${\displaystyle \frac{e_{12}}{e_{32}}}\not\approx {(\alpha +1)}^2$, and ${\displaystyle \frac{e_{32}}{e_{42}}}\not\approx {\alpha }^5·(\alpha -0.7)$. Even if ${\displaystyle \frac{e_{12}}{e_{22}}}\approx \alpha +1$, if ${\displaystyle \frac{e_{22}}{e_{32}}}\not\approx \alpha +1$, ${\displaystyle \frac{e_{12}}{e_{32}}}\not\approx {(\alpha +1)}^2$, and ${\displaystyle \frac{e_{32}}{e_{42}}}\not\approx {\alpha }^5·(\alpha -0.7)$, it is understood that $e_{12}$ and $e_{22}$ are erroneous.

Error correction: The errors are corrected as in the previous scenario.

Case 3 (two errors in the same column and not adjacent): If, for instance, $e_{13}$ and $e_{33}$ are erroneous, then the conditions ${\displaystyle \frac{e_{13}}{e_{33}}}\not\approx \alpha +1$, ${\displaystyle \frac{e_{23}}{e_{33}}}\not\approx \alpha +1$, ${\displaystyle \frac{e_{33}}{e_{43}}}\not\approx {\alpha }^5·(\alpha -0.7)$, and ${\displaystyle \frac{e_{23}}{e_{43}}}\approx (\alpha +1)·{\alpha }^5·(\alpha -0.7)$ indicate that $e_{13}$ and $e_{33}$ are incorrect.

When there are three or four errors in the same column, it is evident that there is an error because $det\left(E\right)\ne 31,259$, but the errors cannot be corrected.

In cases where there are not three or four errors in the same column (for example, two errors in the first column, one error in the third, and two errors in the fourth), the errors can be corrected using the methods described in the three cases we provided.

Example 7.

If we encode the message "MATHISINCREDIBLE" using ASCII and then apply the encoding matrix ${\left(K_6\right)}^3$,

$$M=\left|\begin{array}{cccc}77& 65& 84& 72\\ 73& 83& 73& 78\\ 67& 82& 69& 68\\ 73& 66& 76& 69\end{array}\right|,K_6=\left|\begin{array}{cccc}8& 5& 3& 1\\ 3& 2& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 1\end{array}\right|$$

Then,

$$E={\left(K_6\right)}^3\times M=\left|\begin{array}{cccc}143,703& 142,439& 151,006& 141,772\\ 55,723& 55,233& 58,553& 54,974\\ 21,181& 20,994& 22,255& 20,896\\ 23,243& 23,046& 24,420& 22,931\end{array}\right|$$

During communication, let $e_{21}$, $e_{31}$, $e_{32}$, and $e_{43}$ be corrupted and transmitted incorrectly. Using the ratios mentioned in the relevant proposition and the methods described in the three cases, the error locations can be identified. Specifically, the conditions $e_{11}\approx e_{21}·(\alpha +1)$, $e_{31}\approx e_{41}·(\alpha -0.7)$, $e_{22}\approx e_{32}·(\alpha +1)$, $e_{33}\approx e_{43}·(\alpha -0.7)$, and $det\left(E\right)=-det\left(M\right)=31,259$ help determine the erroneous values $e_{21}$, $e_{31}$, $e_{32}$, and $e_{43}$.

4.2. Coding Method 2

In this method, $S_n$, defined in the previous section, will be used. Since this matrix has three dimensions, the message matrix M must also have three dimensions. Let the message matrix be

$$M=\left[\begin{array}{ccc}{m}_{11}& m_{12}& m_{13}\\ m_{21}& m_{22}& m_{23}\\ m_{31}& m_{32}& m_{33}\end{array}\right].$$

The encoded matrix E is obtained by encoding the message matrix as $S_n\times M\times S_n=E$. Since $det\left(S_n\right)·det\left(M\right)·det\left(S_n\right)=det\left(E\right)$ and $det\left(S_n\right)=6·{(-1)}^n$, it follows that $det\left(E\right)=36·det\left(M\right)$. The parameters E, n, and $det\left(M\right)$ are sent to the recipient. The recipient recovers the message as ${\left(S_n\right)}^{-1}\times E\times {\left(S_n\right)}^{-1}=M$.

Error detection and correction: If the determinant of the received matrix E differs from $36·det\left(M\right)$, an error in E is detected. Using the ratios ${\displaystyle \frac{e_{ij}}{e_{i+1,j}}}\approx {\alpha }^4$ and ${\displaystyle \frac{e_{ij}}{e_{i,j+1}}}\approx {\alpha }^2$, erroneous entries are identified and corrected by finding integer values that satisfy these ratios as well as the equation $det\left(E\right)=36·det\left(M\right)$.

We will use a two-step error detection system to locate erroneous entries. In the first step, for each row and column, entries that satisfy the conditions ${\displaystyle \frac{e_{ij}}{e_{i+1,j}}}\approx {\alpha }^4$, ${\displaystyle \frac{e_{ij}}{e_{i+2,j}}}\approx {\alpha }^8$, ${\displaystyle \frac{e_{ij}}{e_{i,j+1}}}\approx {\alpha }^2$, and ${\displaystyle \frac{e_{ij}}{e_{i,j+2}}}\approx {\alpha }^4$ are identified. If there are two or more erroneous entries in any row or column, their locations cannot be determined. In this case, the second step is applied. In the second step, the correct entries identified in the first step are used to locate all erroneous entries using the ratios given in the table below.

$${\displaystyle \frac{e_{13}}{e_{21}}}\approx {\displaystyle \frac{e_{23}}{e_{31}}}\approx 1$$

$${\displaystyle \frac{e_{11}}{e_{12}}}\approx {\displaystyle \frac{e_{12}}{e_{13}}}\approx {\displaystyle \frac{e_{12}}{e_{21}}}\approx {\displaystyle \frac{e_{13}}{e_{22}}}\approx {\displaystyle \frac{e_{21}}{e_{22}}}\approx {\displaystyle \frac{e_{22}}{e_{23}}}\approx {\displaystyle \frac{e_{22}}{e_{31}}}\approx {\displaystyle \frac{e_{23}}{e_{32}}}\approx {\displaystyle \frac{e_{31}}{e_{32}}}\approx {\displaystyle \frac{e_{32}}{e_{33}}}\approx {\alpha }^2$$

$${\displaystyle \frac{e_{11}}{e_{13}}}\approx {\displaystyle \frac{e_{11}}{e_{21}}}\approx {\displaystyle \frac{e_{12}}{e_{22}}}\approx {\displaystyle \frac{e_{13}}{e_{23}}}\approx {\displaystyle \frac{e_{13}}{e_{31}}}\approx {\displaystyle \frac{e_{21}}{e_{23}}}\approx {\displaystyle \frac{e_{21}}{e_{31}}}\approx {\displaystyle \frac{e_{22}}{e_{32}}}\approx {\displaystyle \frac{e_{23}}{e_{33}}}\approx {\displaystyle \frac{e_{32}}{e_{33}}}\approx {\alpha }^4$$

$${\displaystyle \frac{e_{11}}{e_{22}}}\approx {\displaystyle \frac{e_{12}}{e_{23}}}\approx {\displaystyle \frac{e_{12}}{e_{31}}}\approx {\displaystyle \frac{e_{13}}{e_{32}}}\approx {\displaystyle \frac{e_{21}}{e_{32}}}\approx {\displaystyle \frac{e_{22}}{e_{33}}}\approx {\alpha }^6$$

$${\displaystyle \frac{e_{11}}{e_{23}}}\approx {\displaystyle \frac{e_{11}}{e_{31}}}\approx {\displaystyle \frac{e_{12}}{e_{32}}}\approx {\displaystyle \frac{e_{13}}{e_{33}}}\approx {\displaystyle \frac{e_{21}}{e_{32}}}\approx {\alpha }^8$$

$${\displaystyle \frac{e_{11}}{e_{32}}}\approx {\displaystyle \frac{e_{12}}{e_{33}}}\approx {\alpha }^{10}$$

$${\displaystyle \frac{e_{11}}{e_{33}}}\approx {\alpha }^{12}$$

If there are two or more errors in every row and column, no correct entries can be identified in the first step. In this case, the correct entries are determined using the ratios in the second step. Once correct and erroneous entries are identified, the approximate integer values of the erroneous ones are determined using the correct entries, and integers that satisfy $det\left(E\right)=36·det\left(M\right)$ are found by testing. The range of these integers can be determined using the bounds $0.9<{\alpha }^0<1.1$, $2.5<{\alpha }^2<2.7$, $6.7<{\alpha }^4<7.1$, $17.7<{\alpha }^6<18$, $45<{\alpha }^8<49$, $120<{\alpha }^{10}<126$, and $300<{\alpha }^{12}<340$.

Example 8.

Let us encode the message “MATHISFUN” using its ASCII code and the coding matrix $S_8$ and send the encoded message.

$$M=\left[\begin{array}{ccc}77& 65& 84\\ 72& 73& 83\\ 70& 85& 78\end{array}\right],S_8=\left[\begin{array}{ccc}{21}^2& {13}^2& 8^2\\ 8^2& 5^2& 3^2\\ 3^2& 2^2& 1^2\end{array}\right]$$

The encoded matrix is computed as

$$E=S_8\times M\times S_8=\left[\begin{array}{ccc}25,793,660& 9,937,547& 3,712,761\\ 3,7502,55& 1,444,864& 539,815\\ 535,553& 206,333& 77,088\end{array}\right]$$

The values $det\left(M\right)=-7347$ and $n=8$, and the matrix E are sent to the receiver. The receiver obtains the message matrix M by computing $S_8^{-1}\times E\times S_8^{-1}$.

Now, let us analyze the error detection and correction process using this example.

Let $e_{11}$, $e_{21}$, $e_{32}$, $e_{13}$, $e_{23}$, and $e_{33}$ be received incorrectly. For example, suppose $e_{11}=16,786,425$, $e_{21}=2,073,710$, $e_{32}=415,255$, $e_{13}=4,786,501$, $e_{23}=332,456$, and $e_{33}=14,056$. Since $det\left(E\right)\ne -264,492$, it is understood that there is an error or multiple errors in matrix E.

Let us first examine the columns of E:

$${\displaystyle \frac{e_{11}}{e_{21}}}=3.308\ne {\alpha }^4,{\displaystyle \frac{e_{21}}{e_{31}}}=9.473\ne {\alpha }^4,{\displaystyle \frac{e_{11}}{e_{31}}}=31.344\ne {\alpha }^8$$

indicating that there are two or more errors in the first column. For the second column,

$${\displaystyle \frac{e_{12}}{e_{22}}}=6.877\approx {\alpha }^4,{\displaystyle \frac{e_{22}}{e_{32}}}=3.479\ne {\alpha }^4,{\displaystyle \frac{e_{12}}{e_{32}}}=23.931\ne {\alpha }^8$$

suggesting that $e_{12}$ and $e_{22}$ are correct, while $e_{32}$ is erroneous. For the third column,

$${\displaystyle \frac{e_{13}}{e_{23}}}=14.397\ne {\alpha }^4,{\displaystyle \frac{e_{23}}{e_{33}}}=23.652\ne {\alpha }^4,{\displaystyle \frac{e_{13}}{e_{33}}}=340.530\ne {\alpha }^8$$

indicating that there are two or more errors in the third column.

Now, let us examine the rows of E:

$${\displaystyle \frac{e_{11}}{e_{12}}}=1.689\ne {\alpha }^2,{\displaystyle \frac{e_{12}}{e_{13}}}=2.076\ne {\alpha }^2,{\displaystyle \frac{e_{11}}{e_{13}}}=3.507\ne {\alpha }^4$$

which implies that there are two or more errors in the first row.

$${\displaystyle \frac{e_{21}}{e_{22}}}=3.511\ne {\alpha }^2,{\displaystyle \frac{e_{22}}{e_{23}}}=4.346\ne {\alpha }^2,{\displaystyle \frac{e_{21}}{e_{23}}}=15.261\ne {\alpha }^4$$

suggests two or more errors in the second row.

$${\displaystyle \frac{e_{31}}{e_{32}}}=1.289\ne {\alpha }^2,{\displaystyle \frac{e_{32}}{e_{33}}}=29.542\ne {\alpha }^2,{\displaystyle \frac{e_{31}}{e_{33}}}=264.140\ne {\alpha }^4$$

indicating two or more errors in the third row.

Additionally, since $e_{12}$ and $e_{22}$ are known to be correct, we can conclude that $e_{11}$, $e_{13}$, $e_{21}$, and $e_{23}$ are erroneous. Proceeding to the second stage, we examine $e_{31}$ and $e_{33}$:

$${\displaystyle \frac{e_{22}}{e_{31}}}=2.697\approx {\alpha }^2,{\displaystyle \frac{e_{22}}{e_{33}}}=102.793\not\approx {\alpha }^6$$

indicating that $e_{31}$ is correct and $e_{33}$ is erroneous.

Thus, $e_{12}$, $e_{22}$, and $e_{31}$ are correct, while the remaining entries are erroneous. Finally, we determine integer ranges for the erroneous entries to find integers that satisfy $det\left(E\right)=36·det\left(M\right)$.

$$2.5<{\displaystyle \frac{e_{11}}{e_{12}}}<2.7\Rightarrow 24,843,867<e_{11}<26,831,376$$

$$2.5<{\displaystyle \frac{e_{12}}{e_{13}}}<2.7\Rightarrow 3,680,572<e_{13}<3,975,018$$

$$2.5<{\displaystyle \frac{e_{21}}{e_{22}}}<2.7\Rightarrow 3,612,160<e_{21}<3,901,132$$

$$2.5<{\displaystyle \frac{e_{22}}{e_{23}}}<2.7\Rightarrow 535,134<e_{23}<214,221$$

$$6.7<{\displaystyle \frac{e_{31}}{e_{33}}}<7.1\Rightarrow 75,430<e_{33}<79,933$$

Using a simple program, the integers within these ranges are tested to satisfy the condition $det\left(E\right)=36·det\left(M\right)$, and the message matrix M is obtained as $S_8^{-1}\times E\times S_8^{-1}$.

In this method, it is observed that as long as there are at least two correct entries, the remaining entries can be determined. Naturally, the more erroneous entries there are, the more integers need to be tested.

5. Conclusions

Various matrices defined by Fibonacci numbers have been examined in terms of their properties. Some of these matrices have been applied in error detection and correction processes during information transfer, a crucial topic in information theory. The coding structures constructed with these matrices have enabled the development of more advanced structures beyond algebraic coding structures and those previously introduced by Stakhov.

In earlier studies, messages consisting of four characters were transmitted at once using 2 × 2 matrices. However, in this study, the developed coding structures enabled the transmission of messages with nine characters using 3 × 3 matrices and sixteen characters using 4 × 4 matrices in a single step. Consequently, a greater amount of information could be transmitted simultaneously compared to previous studies. Furthermore, the error detection and correction processes within these structures have been thoroughly explained, and illustrative examples demonstrating the process have been provided. Since the encoding process in the proposed methods is performed via matrix multiplication, it is highly efficient. The decoding process consists of two stages: first, multiplying the matrix by the inverse of the coding matrix, and second, identifying erroneous inputs and correcting them by testing integer values that satisfy the initially provided determinant and given ratios. The first step, being matrix multiplication, is completed very quickly, while the second step is also relatively fast since the number of integer values to be tested is significantly reduced due to the given ratios.

When analyzing the error correction capacity of the developed methods, it was observed that both approaches have high error correction capabilities, with the second method exhibiting a significantly higher capacity. In the first method, errors can be corrected for up to 50% of the inputs. In the second method, even if only two out of nine inputs are correct, the remaining erroneous inputs can still be corrected, indicating an error correction capacity of 78%, which is remarkably high.

In the provided examples, the messages completely fill the message matrix. If a message does not fully occupy the matrix, the issue can be resolved by inserting the ASCII code (42) of the ‘*’ symbol into the empty positions. Additionally, if the text to be transmitted is lengthy, it can be divided into multiple message matrices, thereby eliminating this limitation.

This study also reveals that the eigenvalues of matrices constructed using Fibonacci numbers and the ratios between elements in the powers of these matrices are related to the golden ratio. The existence of these ratios has been proven using the relationship between the largest eigenvalue of these matrices and their powers.

This research provides a novel perspective on error detection and correction while paving the way for more advanced studies in this field. In future research, larger Fibonacci-type matrices with a constant determinant could be explored to facilitate the transmission of greater amounts of information in a single step. Additionally, refining the range of integers tested during the decoding phase could further enhance the speed of the error correction process.