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On Optimal and Quantum Code Construction from Cyclic Codes over FqPQ with Applications
Shakir Ali1, Amal S Alali2, Pushpendra Sharma1
1Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India.
Entropy (Basel, Switzerland)
|August 26, 2023
Summary
This study explores cyclic codes over mixed alphabets in finite rings, leading to new quantum error-correcting (QEC) codes and optimal codes using the Gray image method.
Area of Science:
- Coding Theory
- Abstract Algebra
- Quantum Information Science
Background:
- Finite rings and their algebraic structures are crucial in coding theory.
- The development of efficient error-correcting codes is vital for reliable data transmission and storage.
- Quantum error-correcting (QEC) codes are essential for protecting quantum information.
Purpose of the Study:
- To investigate the structure of cyclic codes over mixed alphabets in the nonchain finite rings FqPQ.
- To derive new and improved quantum error-correcting (QEC) codes from these cyclic codes.
- To obtain optimal codes over the ring P using the Gray image of cyclic codes.
Main Methods:
- Studying the algebraic structure of the finite rings P=Fq[v]⟨v3-α22v⟩ and Q=Fq[u,v]⟨u2-α12,v3-α22v⟩.
- Applying the theory of cyclic codes over mixed alphabets to these ring structures.
- Utilizing the Gray image of codes for constructing optimal codes.
Main Results:
- Characterization of cyclic codes over the mixed alphabet rings.
- Development of novel quantum error-correcting (QEC) codes with potential for improved performance.
- Identification of several optimal codes over the ring P.
Conclusions:
- The study successfully establishes a framework for analyzing cyclic codes over specific nonchain finite rings.
- The derived QEC codes offer advancements in quantum information processing.
- The application of Gray images provides a practical method for obtaining optimal codes.
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