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Related Experiment Video

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Generation and Coherent Control of Pulsed Quantum Frequency Combs

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On Optimal and Quantum Code Construction from Cyclic Codes over F_qPQ 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.

Keywords:

QEC code cyclic code dual code mixed alphabet code

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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.