Some New Codes on the k –Fibonacci Sequence

In this paper, we deﬁne and study the k –Fibonacci sequence matrix for k ≥ 3. By these obtained results, we introduce some new codes on k –Fibonacci sequence.

In Section 2, we define the k-Fibonacci sequence matrix and calculate the determinant of it. Section 3 and Section 4 are devoted to obtain some codes on k-Fibonacci sequence.
where F 3 n is the element of the 3− Fibonacci sequence. For example, n � 10, and we have Q ( Now, we get the determinant of the 3− Fibonacci sequence matrix. By the definition of F 3 n , we have Theorem 1. e determinant of Q (n, 3) is equal to − 1.
Proof. For n � 3m + i where 0 ≤ i ≤ 2, we have We can perform specific column operations to reduce Q (3m+i, 3) to Q (i, 3) , where 0 ≤ i ≤ 2. e operations that we use in this process are as follows: (i) Add column 2 and column 3, then subtract this summation from column 1, and replace it to the result column 1(c 1 − (c 2 + c 3 ) ⟶ c 1 ). So, by relation (5), we have (ii) Add column 1 and column 3, then subtract it from column 2, and replace to the result column 2. en, by relation (5), we get (iii) Add column 1 and column 2, then subtract this summation from column 3, and replace to it. us, by relation (5), we have en, we have Continuing this process m − 1 steps on Q (3(m− 1)+i, 3) , we obtain the following: erefore, det Q (n,3) � det Q (i, 3) . It is sufficient that we get the determinant of Q (i, 3) where 0 ≤ i ≤ 2. If i � 0, then by Lemma 1, we obtain So, det Q (0,3) � − 1. If i � 1, then by Lemma 1, we have We get the following: Step 1: Step 2: Step 3: erefore, detdet Q (10,3) � det Q (1,3) � − 1. Now, we are ready to generalize the idea of the 3− Fibonacci sequence matrix to the k-Fibonacci sequence matrices (k > 3).

Definition 3.
e k-Fibonacci sequence matrices of size k × k (k ≥ 4), denoted by Q (n,k) , are defined as follows: where F k i is the element of the k-Fibonacci sequence.
Example 3. By the definition of Q (n,k) , we have In eorem 1, we get the determinant of the 3− Fibonacci sequence matrix. Now, we obtain the determinant of the k-Fibonacci sequence matrices for k ≥ 4. By the definition of F k n , we have We can perform specific column operations to reduce Q (mk+i,k) to Q (i,k) , where 0 ≤ i ≤ k − 1. e operations that we use in this process are as follows.
en, add column 1 until column k except column 2, then subtract it from column 2, and replace it to (c 2 − (c 1 + c 3 + · · · + c k ) ⟶ c 2 ). Continuing the process until column k and by relation (20), we have us, performing above the operations until m − 1 steps, we get Mathematical Problems in Engineering 3 By elementary matrix operations, we get det Q (0,k) � − 1. e other cases are proven similarly.

A Code on the k-Fibonacci Sequence
In this section, we introduce a coding method by using the k-Fibonacci sequence and obtain error detection and correction of it. First, we need the following definition.

Definition 4.
For k � 2, n ≥ 4 or k ≥ 3, n ≥ k + 1, let F k n and F k n+1 be two consecutive elements of the k-Fibonacci sequence. e sequence T s F k n+2 1 is defined as follows: For u � F k n+2 and 1 ≤ i, j ≤ u, by using definition T s , we define the matrix T u � (t ij ) u×u as follows: Example 3. Suppose k � 3 and n � 4. Since, F 3 5 � 4 and u � F 3 6 � 7, we have en, (6,4), (4,5), (2, 6), (7, 7), 0, otherwise. (28) at is, Now, by using above notations, we can explain a method coding, which is named T u − code. For where Let "mathematics is beautiful" be a message. en, by Table 1 and Hence, we get So, "htameiamcsi s uaebtluti" is the code matrix E.
Note that M � E × (T 5 ) − 1 is the T 5 − decoding method. By using the elementary operations on matrix T u , one can prove the following Lemma.
In particular, when u � 4, 7, we get det E � − det M, and for u � 5, we have det E � det M.
By using the above facts, we find the error detection and correction for T u − code where u � 4, 5, 7.
Suppose u � 4 and 7. For correction, by using the algebraic equations and the relation det E � − det M, we have Mathematical Problems in Engineering Mathematical Problems in Engineering By above relations, we can obtain "single" correction. Using an analogous argument, we can find the error correction for T 5 − code.

Coding and Decoding on the k-Fibonacci Sequence Matrix
Here, we find coding and decoding on the k-Fibonacci sequence matrix Q (n,k) and get its error detection and correction.For k ≥ 3 and an initial message M k×k , we will name a transformation E � M × Q (n,k) as the k-Fibonacci coding and a transformation M � E × Q − 1 (n,k) as the k-Fibonacci decoding. Also, the matrix E is as a code matrix. Now, we explain the above method by an example.
We have By the above notations, we have Mathematical Problems in Engineering e error-correction algorithm for the Fibonacci Q 2 -matrices is described in [16]. Now, we explain error detection and correction for the 3− Fibonacci coding. Also, Since m i ≥ 0, 1 ≤ i ≤ 9, we have From (44), we get By using (45), we have

Mathematical Problems in Engineering
From (46), we get (50) By relations (48)-(50), we have We calculate two cases and the rest of the cases get in similar ways. Case 1. Let A 1 > 0, A 2 > 0, and A 3 > 0. By using (51) and (53), we have en, we get So,
(68) erefore, we have In the following, by relations (52) and (53), we get en, we have We get So, By using relations (69) and (73), we have Similarly, we get Similarly, by the above argument, we get So, by above facts, we have (80) Now, we are in position that the above results are generalized to the k-Fibonacci coding.
Here, for k > 3, we get relations among entries the code matrix E. Similar to k � 3, we can obtain the following relations among the first-row entries code matrix E.
where μ k is the golden ratio of the k-Fibonacci sequence. In general, the following relations among the entries of each row of the code matrix E � (e ij ) k×k are obtained by where i, j � 1, 2, . . . , k, s � 1, 2, . . . , k − 1 and 2 ≤ j + s ≤ k, n > k. Now, we calculate the determinant of the code matrix E. For the coding matrix Q (n,k) , E � M × Q (n,k) and det Q (n,k) � − 1, and we have Here, we calculate the error detection and correction for the k-Fibonacci coding.
Let k � 3. According to the matrix E of the order 3 × 3, we have "single," "double,". . ., "nine-fold" errors. e first assumption is that there exists only one error in the matrix E received from the communication channel. It is clear that there are nine different cases for it, as follows: In a similar way, we will obtain a double error for the matrix E. For example, we consider a bivariate case for matrix E as follows: in which possible cases are 2C 9 � 36. Similarly, we obtain "triple," "four-fold," . . ., "nine-fold" errors, which the total number of cases is 1C 9 + 2C 9 + · · · + 9C 9 � 2 9 − 1 � 511 errors. By using detE � − detM and the relations (76)-(78), we can correct up to "single," "double," . . . , "eight" errors except "nine" errors. erefore, we get that the correctable possibility of the method is equal to 510/511 � 0.9980 � % 99.80.
erefore, there are 2 k 2 − 1 errors. By detE � − detM and relation (82), we can give that the correct ability of the k-Fibonacci sequence matrix coding is equal to en, for large value of k, the correct possibility of this method is

Conclusion
In this paper, we give some codes on the k-Fibonacci sequence. ese coding methods are the applications of this sequence. Also, we obtain the following results: (1) For u � 4, 5 and 7, we get error detection and correction.

Data Availability
ere are no applications, analysis, or generation during the study. e results are related to the Ph. D. thesis.

Conflicts of Interest
e authors declare that they have no conflicts of interest.