Pell Coding and Pell Decoding Methods with Some Applications

TL;DR

This paper introduces new coding and decoding methods based on generalized Pell numbers, establishing relations for error detection and correction, and proposing two blocking algorithms utilizing Pell and generalized Pell numbers.

Contribution

It presents novel Pell-based coding and decoding techniques, along with new algorithms for data blocking, expanding the applications of Pell numbers in coding theory.

Findings

New Pell coding and decoding methods developed.

Relations among code matrix elements for error detection and correction established.

Two new blocking algorithms using Pell numbers introduced.

Abstract

We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking algorithms using Pell numbers and generalized Pell $(p,i)$-numbers.

Equations

$$P_p^{(i)}(n)=2P_p^{(i)}(n-1)+P_p^{(i)}(n-p-1)\text{; }n>p+1,0\le i\le p,p=1,2,3,\dots ,$$

$$P_p^{(i)}(1)=\cdots =P_p^{(i)}(i)=0\text{, }{P}_p^{(i)}(i+1)=\cdots =P_p^{(i)}(p+1)=1\text{.}$$

$$P_{n+1}=2P_n+P_{n-1},n\in {\mathbb{Z}}^{+}$$

$$P_0=0\text{, }{P}_1=1.$$

$$P_n=\frac{{\gamma }^n-{\delta }^n}{\gamma -\delta }$$

$$\underset{n\to \infty }{\lim }\frac{P_{n+1}}{P_n}=\gamma \text{.}$$

$$P=\left[\begin{array}[]{cc}2&1\\ 1&0\end{array}\right]\text{.}$$

$$P^{n}=\left[\begin{array}[]{cc}P_{n+1}&P_{n}\\ P_{n}&P_{n-1}\end{array}\right]$$

$$Det(P^n)=P_{n+1}{P}_{n-1}-P_n^2=(-1{)}^n\text{.}$$

$$A=\left[\begin{array}[]{ccccc}2&0&\cdots&0&1\\ 1&0&\cdots&0&0\\ 0&1&\ddots&\vdots&0\\ \vdots&\cdots&\ddots&0&\vdots\\ 0&\cdots&0&1&0\end{array}\right]_{(p+1)\times(p+1)}\text{.}$$

$${\footnotesize A^{n}=G_{n}=\left[\begin{array}[]{ccccc}P_{p}^{(p)}(n+p+1)&P_{p}^{(p)}(n+1)&P_{p}^{(p)}(n+2)&\cdots&P_{p}^{(p)}(n+p)\\ P_{p}^{(p)}(n+p)&P_{p}^{(p)}(n)&P_{p}^{(p)}(n+1)&\cdots&P_{p}^{(p)}(n+p-1)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ P_{p}^{(p)}(n+2)&P_{p}^{(p)}(n-p+2)&P_{p}^{(p)}(n-p+3)&\cdots&P_{p}^{(p)}(n+1)\\ P_{p}^{(p)}(n+1)&P_{p}^{(p)}(n-p+1)&P_{p}^{(p)}(n-p+2)&\cdots&P_{p}^{(p)}(n)\end{array}\right]_{(p+1)\times(p+1)}\text{.}}$$

$$Det(A^n)=Det(G_n)=(-1{)}^{n(p+2)}\text{,}$$

$$M\times G_n=E$$

$$E\times {\left(G_n\right)}^{-1}=M\text{,}$$

$$M=\left[\begin{array}[]{cc}m_{1}&m_{2}\\ m_{3}&m_{4}\end{array}\right]\text{,}$$

$$G_{3}=\left[\begin{array}[]{cc}P_{1}^{(1)}(n+2)&P_{1}^{(1)}(n+1)\\ P_{1}^{(1)}(n+1)&P_{1}^{(1)}(n)\end{array}\right]=\left[\begin{array}[]{cc}P_{1}^{(1)}(5)&P_{1}^{(1)}(4)\\ P_{1}^{(1)}(4)&P_{1}^{(1)}(3)\end{array}\right]=\left[\begin{array}[]{cc}12&5\\ 5&2\end{array}\right]\text{.}$$

$$\left(G_{3}\right)^{-1}=\frac{1}{12.2-5.5}\left[\begin{array}[]{cc}2&-5\\ -5&12\end{array}\right]=\left[\begin{array}[]{cc}-2&5\\ 5&-12\end{array}\right]\text{.}$$

$$E=M\times G_{3}=\left[\begin{array}[]{cc}m_{1}&m_{2}\\ m_{3}&m_{4}\end{array}\right]\left[\begin{array}[]{cc}12&5\\ 5&2\end{array}\right]=\left[\begin{array}[]{cc}12m_{1}+5m_{2}&5m_{1}+2m_{2}\\ 12m_{3}+5m_{4}&5m_{3}+2m_{4}\end{array}\right]\text{,}$$

$$M$$

$$Det(E)=Det(M\times G_n)=Det(M)\times Det(G_n)=Det(M)\times (-1{)}^{n(p+2)}\text{.}$$

$$E=M\times G_{n}=\left[\begin{array}[]{cc}m_{1}&m_{2}\\ m_{3}&m_{4}\end{array}\right]\left[\begin{array}[]{cc}P_{1}^{(1)}(n+2)&P_{1}^{(1)}(n+1)\\ P_{1}^{(1)}(n+1)&P_{1}^{(1)}(n)\end{array}\right]=\left[\begin{array}[]{cc}e_{1}&e_{2}\\ e_{3}&e_{4}\end{array}\right]$$

$$M$$

$$m_1=-e_1{P}_1^{(1)}(n)+e_2{P}_1^{(1)}(n+1)>0\text{,}$$

$$m_2=e_1{P}_1^{(1)}(n+1)-e_2{P}_1^{(1)}(n+2)>0\text{,}$$

$$m_3=-e_3{P}_1^{(1)}(n)+e_4{P}_1^{(1)}(n+1)>0\text{,}$$

$$m_4=e_3{P}_1^{(1)}(n+1)-e_4{P}_1^{(1)}(n+2)>0\text{.}$$

$$\frac{P_1^{(1)}(n+2)}{P_1^{(1)}(n+1)}<\frac{e_1}{e_2}<\frac{P_1^{(1)}(n+1)}{P_1^{(1)}(n)}\text{.}$$

$$\frac{P_1^{(1)}(n+2)}{P_1^{(1)}(n+1)}<\frac{e_3}{e_4}<\frac{P_1^{(1)}(n+1)}{P_1^{(1)}(n)}\text{.}$$

$$\frac{e_1}{e_2}\approx \gamma \text{, }\frac{e_3}{e_4}\approx \gamma \text{.}$$

$$M=\left[\begin{array}[]{cc}m_{1}&m_{2}\\ m_{3}&m_{4}\end{array}\right].$$

Full text

*Corresponding author: N. TAŞ

Balıkesir University, Department of Mathematics, 10145 Balıkesir, TURKEY

e-mail: nihaltas@balikesir.edu.tr

1. Introduction

Recently, in [2], the generalized Pell $(p,i)$-numbers have been defined by the following recurrence relations

[TABLE]

with the initial terms

[TABLE]

For $i=p=1$, the generalized Pell $(1,1)$-number corresponds to the $(n+1)$-th classical Pell number defined as

[TABLE]

with the initial terms

[TABLE]

For the basic properties of these numbers, one can see [2] and [3].

It is known that $\gamma=1+\sqrt{2}$ and $\delta=1-\sqrt{2}$ are the roots of the characteristic equation of the Pell recurrence relation $t^{2}-2t-1=0$. Using these roots it can be given the Binet formula for the Pell number by

[TABLE]

with

[TABLE]

The Pell $P$-matrix of order $2$ is given by the following form:

[TABLE]

The $n$-th power of the $P$-matrix and its determinant are given by

[TABLE]

and

[TABLE]

In [2], it was introduced the matrix $A$ in the following form

[TABLE]

and computed

[TABLE]

Using the matrices given in the equations (1.3) and (1.4), we get

[TABLE]

for $n>0$.

Recently, Fibonacci coding theory has been introduced and studied in many aspects (see [1], [4], [6], [7], [8] and [11] for more details). For example, in [7], A. P. Stakhov presented a new coding theory using the generalization of the Cassini formula for Fibonacci $p$-numbers and $Q_{p}$-matrices. Later B. Prasad developed a new coding and decoding method followed from Lucas $p$ matrix [5]. More recently it has been obtained a new coding/decoding algorithm using Fibonacci $Q$-matrices called as “Fibonacci Blocking Algorithm” [9]. In [10], using $R$-matrices and Lucas numbers, it has been given “Lucas Blocking Algorithm” and the “Minesweeper Model” related to Fibonacci $Q^{n}$-matrices and $R$-matrices.

Motivated by the above studies, the main purpose of this paper is to develop a new coding and decoding method using the generalized Pell $(p,i)$-numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory with $p=i=1$. As an application, we give two new blocking algorithms using Pell and generalized Pell $(p,i)$-numbers.

2. Pell Coding and Decoding Method

In this section, we present a new coding and decoding method using the generalized Pell $(p,i)$-numbers. In our method, the nonsingular square matrix $M$ with order $(p+1)$, where $p=1,2,3,\ldots$ corresponds to our message matrix. We consider the matrix $G_{n}$ of order $(p+1)$ as coding matrix and its inverse matrix $\left(G_{n}\right)^{-1}$ as a decoding matrix. We introduce Pell coding and Pell decoding with transformations

[TABLE]

and

[TABLE]

respectively, where $E$ is a code matrix.

Now we give an example of Pell coding and decoding method. Let $M$ be a message matrix of the following form:

[TABLE]

where $m_{1},m_{2},m_{3},m_{4}$ are positive integers.

Let $p=i=1$ and $n=3$ in order to construct the coding matrix $G_{n}$:

[TABLE]

Then we find inverse of $G_{3}$:

[TABLE]

Now, we calculate the code matrix $E$.

[TABLE]

where $e_{1}=12m_{1}+5m_{2}$, $e_{2}=5m_{1}+2m_{2}$, $e_{3}=12m_{3}+5m_{4}$ and $e_{4}=5m_{3}+2m_{4}$.

Finally, the code matrix $E$ is sent to a channel, the message matrix $M$ can be obtained by decoding as the following way

[TABLE]

Notice that the relation between the code matrix $E$ and the message matrix $M$ is

[TABLE]

3. The Relationships between the Code Matrix Elements for $p=i=1$

We write $E$ and $M$ as follows:

[TABLE]

and

[TABLE]

for $n=2k+1$. Because $m_{1},m_{2},m_{3},m_{4}$ are positive integers, we get

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Using (3.3) and (3.4), we find

[TABLE]

Using (3.5) and (3.6), we get

[TABLE]

From the inequalities (3.7) and (3.8), we obtain

[TABLE]

For $n=2k$, we obtain the similar relations given in (3.9).

4. Error Detection/Correction for Pell Coding and Decoding Method

Because of the reasons arising in the channel, some errors may be occur in the code matrix $E.$ So we try to determine and correct these errors using the properties of determinant in this process. Let $p=i=1$ and the message matrix $M$ is given by

[TABLE]

We know that $Det(M)=m_{1}m_{4}-m_{2}m_{3},$ the code matrix $E=M\times G_{n}$ and $Det(E)=Det(M)\times\left(-1\right)^{n\left(p+2\right)}.$ From the relationship between determinants of $E$ and $M,$ if $n$ is an odd number, we have

[TABLE]

and if $n$ is an even number, we have

[TABLE]

The new method of the error detection is an application of the $G_{n}$ matrix. The basic idea of this method depends on calculating the determinants of $M$ and $E$. Comparing the determinants obtained from the channel, the receiver can decide whether the code message $E$ is true or not.

Actually, we cannot determine which element of the code message is damaged. In order to find damaged element, we suppose different cases such as single error, two errors etc. Now, we consider the first case with a single error in the code matrix $E$. We can easily obtain that there are four places where single error appear in $E:$

[TABLE]

where $t_{i}$ $\left(i\in\{1,2,3,4\}\right)$ is the damaged element. To check the above four different cases, we can use the following relations$:$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where a possible single error is the element $e_{1}$ $\left(\text{resp. }e_{2},e_{3}\text{ and }e_{4}\right)$ given in the relation $($4.5$)$ $\left(\text{resp. }(\text{\ref{eqn42}}),(\text{\ref{eqn43}})\text{ and }(\text{\ref{eqn44}})\right).$

Using the above relations, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since the elements of message matrix $M$ are positive integers, we should find integer solutions of the equations from $($4.9$)$ to $($4.12$)$. If there are not integer solutions of these equations, we find that our cases related to a single error is incorrect or an error can be occurred in the checking element “$Det(M)$”. If $Det(M)$ is incorrect, we use the relations given in $($3.9$)$ to check a correctness of the code matrix $E$.

Similarly, we can check the cases with double errors in the code matrix $E$. Let us consider the following case with double error in $E:$

[TABLE]

where $t_{1},t_{2}$ are the damaged elements of $E.$ Using the relation

[TABLE]

we can write following equation for the matrix given in $($4.13$)$

[TABLE]

Also, we know the following relation between $t_{1}$ and $t_{2}$

[TABLE]

It is clear that the equation $($4.14$)$ is a Diophantine equation. Because there are many solutions of Diophantine equations, we should choose the solutions $t_{1},t_{2}$ satisfying the above checking relation $($4.15$).$ Using the similar approach, we can correct the triple errors in the code matrix $E$ such that

[TABLE]

where $t_{1},t_{2}$ and $t_{3}$ are damaged elements of $E$.

Consequently, our method depends on confirmation of various cases about damaged elements of $E$ using the checking element $Det(M)$ and checking relation $t_{1}\approx\gamma t_{2}.$ Our correctness method permits us to correct $14$ cases among the $15$ cases, because our method is inadequate for the case with four errors. So we can say that correction ability of our method is $\frac{14}{15}=\%0,9333.$

5. Blocking Methods As an Application of Pell Numbers and Generalized

Pell $(p,i)$-Numbers

In this section we introduce new coding/decoding algorithms using Pell and generalized Pell $(p,i)$-numbers. We put our message in a matrix of even size adding zero between two words and end of the message until we obtain the size of the message matrix is even. Dividing the message square matrix $M$ of size $2m$ into the block matrices, named $B_{i}$ ($1\leq i\leq m^{2}$), of size $2\times 2$, from left to right, we can construct a new coding method.

Now we explain the symbols of our coding method. Assume that matrices $B_{i}$ and $E_{i}$ are of the following forms:

[TABLE]

We use the matrix $P^{n}$ given in (1.2) and rewrite the elements of this matrix as P^{n}=\left[\begin{array}[]{cc}p_{1}&p_{2}\\ p_{3}&p_{4}\end{array}\right]. The number of the block matrices $B_{i}$ is denoted by $b$. According to $b$, we choose the number $n$ as follows:

[TABLE]

Using the chosen $n$, we write the following character table according to $mod29$ (this table can be extended according to the used characters in the message matrix). We begin the “$n$” for the last character.

[TABLE]

Now we explain the following new coding and decoding algorithms.

Pell Blocking Algorithm

Coding Algorithm

Step 1. Divide the matrix $M$ into blocks $B_{i}$ $\left(1\leq i\leq m^{2}\right)$.

Step 2. Choose $n$.

**Step 3. **Determine $b_{j}^{i}$ $\left(1\leq j\leq 4\right)$.

Step 4. Compute $\det(B_{i})\rightarrow d_{i}$.

Step 5. Construct $K=\left[d_{i},b_{k}^{i}\right]_{k\in\{1,3,4\}}$.

**Step 6. **End of algorithm.

**Decoding Algorithm **

Step 1. Compute $P^{n}$.

Step 2. Determine $p_{j}$ $(1\leq j\leq 4)$.

**Step 3. **Compute $p_{1}b_{3}^{i}+p_{3}b_{4}^{i}\rightarrow e_{3}^{i}$ $\left(1\leq i\leq m^{2}\right)$.

Step 4. Compute $p_{2}b_{3}^{i}+p_{4}b_{4}^{i}\rightarrow e_{4}^{i}$.

Step 5. Solve $(-1)^{n}\times d_{i}=e_{4}^{i}(p_{1}b_{1}^{i}+p_{3}x_{i})-e_{3}^{i}(p_{2}b_{1}^{i}+p_{4}x_{i})$.

**Step 6. **Substitute for $x_{i}=b_{2}^{i}$.

Step 7. Construct $B_{i}$.

Step 8. Construct $M$.

Step 9. End of algorithm.

In the following example we give an application of the above algorithm for $b>3$.

Example 5.1**.**

Let us consider the message matrix for the following message text$:$

[TABLE]

Using the message text, we get the following message matrix $M:$

[TABLE]

Coding Algorithm:

Step 1. We can divide the message text $M$ of size $4\times 4$ into the matrices, named $B_{i}$ $\left(1\leq i\leq 4\right)$, from left to right, each of size is $2\times 2:$

[TABLE]

Step 2. Since $b=4>3$, we calculate $n=\left\lfloor\frac{b}{2}\right\rfloor=2$. For $n=2$, we use the following “letter table” for the message matrix $M:$

[TABLE]

Step 3. We have the elements of the blocks $B_{i}$ $\left(1\leq i\leq 4\right)$ as follows:

[TABLE]

Step 4. Now we calculate the determinants $d_{i}$ of the blocks $B_{i}:$

[TABLE]

Step 5. Using Step 3 and Step 4 we obtain the following matrix $K:$

[TABLE]

**Step 6. End of algorithm.

Decoding algorithm:**

Step 1. By $($1.2$)$ we know that

[TABLE]

Step 2. The elements of $P^{2}$ are denoted by

[TABLE]

Step 3. We compute the elements $e_{3}^{i}$ to construct the matrix $E_{i}:$

[TABLE]

Step 4. We compute the elements $e_{4}^{i}$ to construct the matrix $E_{i}:$

[TABLE]

Step 5. We calculate the elements $x_{i}:$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Step 6. We rename $x_{i}$ as follows$:$

[TABLE]

Step 7. We construct the block matrices $B_{i}:$

[TABLE]

Step 8. We obtain the message matrix $M:$

[TABLE]

Step 9. End of algorithm.

Now we give an application of generalized Pell $(p,i)$-numbers using blocking method. Assume that matrices $B_{i}$ and $E_{i}$ are of the following forms:

[TABLE]

We use the matrix $G_{n}$ defined in (1.4) and rewrite the elements of this matrix as G_{n}=\left[\begin{array}[]{cc}g_{1}&g_{2}\\ g_{3}&g_{4}\end{array}\right]. The number of the block matrices $B_{i}$ is denoted by $b$. According to $b$, we choose the number $n$ as follows:

[TABLE]

Generalized Pell Blocking Algorithm

Coding Algorithm

We consider coding algorithm like as the Pell Blocking coding algorithm.

**Decoding Algorithm **

Step 1. Compute $G_{n}$.

Step 2. Determine $g_{j}$ $(1\leq j\leq 4)$.

**Step 3. **Compute $g_{1}b_{3}^{i}+g_{3}b_{4}^{i}\rightarrow e_{3}^{i}$ $\left(1\leq i\leq m^{2}\right)$.

Step 4. Compute $g_{2}b_{3}^{i}+g_{4}b_{4}^{i}\rightarrow e_{4}^{i}$.

Step 5. Solve $(-1)^{n(p+2)}\times d_{i}=e_{4}^{i}(g_{1}b_{1}^{i}+g_{3}x_{i})-e_{3}^{i}(g_{2}b_{1}^{i}+g_{4}x_{i})$.

**Step 6. **Substitute for $x_{i}=b_{2}^{i}$.

Step 7. Construct $B_{i}$.

Step 8. Construct $M$.

Step 9. End of algorithm.

We choose $p=i=1$ in the following example.

Example 5.2**.**

Let us consider the message matrix for the following message text$:$

[TABLE]

Using the message text, we get the following message matrix

[TABLE]

Coding Algorithm:

Step 1. We can divide the message text $M$ of size $6\times 6$ into the matrices, named $B_{i}$ $\left(1\leq i\leq 9\right)$, from left to right, each of size is $2\times 2:$

[TABLE]

Step 2. Since $p=1$, we calculate $n=3$. For $n=3$, we use the following “letter table” for the message matrix $M:$

[TABLE]

Step 3. We have the elements of the blocks $B_{i}$ $\left(1\leq i\leq 9\right)$ as follows:

[TABLE]

Step 4. Now we calculate the determinants $d_{i}$ of the blocks $B_{i}:$

[TABLE]

Step 5. Using Step 3 and Step 4 we obtain the following matrix $K:$

[TABLE]

**Step 6. End of algorithm.

Decoding algorithm:**

Step 1. Using the equation $($1.4$)$ we have

[TABLE]

and

[TABLE]

Step 2. The elements of $G_{3}$ are denoted by

[TABLE]

Step 3. We compute the elements $e_{3}^{i}$ to construct the matrix $E_{i}:$

[TABLE]

Step 4. We compute the elements $e_{4}^{i}$ to construct the matrix $E_{i}:$

[TABLE]

Step 5. We calculate the elements $x_{i}:$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Step 6. We rename $x_{i}$ as follows$:$

[TABLE]

Step 10. We construct the block matrices $B_{i}:$

[TABLE]

Step 11. We obtain the following message matrix $M:$

[TABLE]

Step 9. End of algorithm.

6. Conclusion

We have presented two new coding and decoding algorithms using the generalized Pell $(p,i)$-numbers. Combining these algorithms together, we can generate a new mixed algorithm such as “Minesweeper model” introduced in [10]. Furthermore it is possible to develop new mixed models using Fibonacci and Lucas blocking algorithms given in [9] and [10].