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Error detection and correction for coding theory on k-order Gaussian Fibonacci matrices
Suleyman Aydinyuz1, Mustafa Asci1
1Pamukkale University, Science Faculty, Department of Mathematics, Kinikli, Denizli, Turkey.
Mathematical Biosciences and Engineering : MBE
|March 11, 2023
Summary
A new k-order Gaussian Fibonacci coding theory uses matrices for encryption. This method corrects infinite integers and shows high error correction capability, approaching zero decoding error for larger k.
Area of Science:
- Number Theory
- Coding Theory
- Cryptography
Background:
- Classical encryption methods rely on algebraic coding.
- Existing methods have limitations in handling certain types of errors or data.
Purpose of the Study:
- To introduce and define the k-order Gaussian Fibonacci coding theory.
- To adapt coding theory for k-order Gaussian Fibonacci polynomials with x=1.
- To explore its potential in data encryption and error correction.
Main Methods:
- Rearranging coding theory for k-order Gaussian Fibonacci polynomials by setting x=1.
- Utilizing Qk, Rk, and En(k) matrices for the coding method.
- Examining error detection for k=2 and generalizing to arbitrary k.
Main Results:
- The k-order Gaussian Fibonacci coding theory differs from classical encryption.
- The method theoretically allows correction of infinite integer matrix elements.
- For k=2, error correction capability reaches 93.33%, surpassing known codes.
- For large k, the probability of decoding error approaches zero.
Conclusions:
- The k-order Gaussian Fibonacci coding theory offers a novel approach to encryption.
- This method demonstrates superior error detection and correction capabilities.
- It presents a promising advancement in coding theory with potential applications in secure communication.
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