Polycyclic Codes over Serial Rings: Algebraic Structure and Annihilator CSS Construction

Kernkonzepte

This paper investigates the algebraic structure of polycyclic codes over a specific class of serial rings, defined as R = R[x1, ..., xs]/⟨t1(x1), ..., ts(xs)⟩, where R is a chain ring and each ti(xi) is a monic square-free polynomial. It provides necessary and sufficient conditions for the existence of various types of annihilator dual polycyclic codes over this class of rings and establishes an annihilator CSS construction to derive quantum codes from annihilator dual-preserving polycyclic codes.

Zusammenfassung

The paper starts by exploring the foundational structure of the serial ring R = R[x1, ..., xs]/⟨t1(x1), ..., ts(xs)⟩, where R is a finite commutative chain ring and each ti(xi) is a monic square-free polynomial. It is shown that R is a serial ring and a principal ideal Frobenius ring.

The authors then investigate polycyclic codes over R and establish that any polycyclic code C over R can be decomposed into a direct sum of polycyclic codes Ci over chain rings Ri. They define quasi-s-dimensional polycyclic (QsDP) codes and prove that polycyclic codes over R are equivalent to f(x)-polycyclic-QsDP codes.

The paper further explores the concept of annihilator dual for polycyclic codes over R and revises the existing statement about annihilator self-dual polycyclic codes. It is shown that a polycyclic code ⟨g(x)⟩ is annihilator self-dual if and only if f(x) = ag2(x) for some unit element a in R.

Finally, the authors present an annihilator CSS construction to derive quantum codes from annihilator dual-preserving f(x)-polycyclic codes over R, where g(x) divides f(x).

Statistiken

The ring R = R[x1, ..., xs]/⟨t1(x1), ..., ts(xs)⟩ is a serial ring and a principal ideal Frobenius ring.
If R is a field, then R is a semi-simple ring.
Any f(x)-polycyclic code over R is R-isomorphic to an f(x)-polycyclic-QsDP code.
A polycyclic code ⟨g(x)⟩ is annihilator self-dual if and only if f(x) = ag2(x) for some unit element a in R.

Zitate

"a polycyclic code ⟨g(x)⟩ is annihilator self-dual if and only if f(x) = ag2(x) for some unit element a in the ring R"

Wichtige Erkenntnisse aus

Polycyclic codes over serial rings and their annihilator CSS construction

by Maryam Bajal... um arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10452.pdf

Tiefere Fragen

How can the annihilator CSS construction be extended to other types of dual-preserving polycyclic codes over the serial ring R, beyond the case where g(x) divides f(x)

The annihilator CSS construction can be extended to other types of dual-preserving polycyclic codes over the serial ring R by considering cases where g(x) does not necessarily divide f(x). One approach is to explore the properties of polycyclic codes that are not generated by a single element g(x) dividing f(x). By studying the structure and properties of these codes, it may be possible to develop a generalized CSS construction that can handle a wider range of polycyclic codes over the serial ring R. This extension could involve investigating the relationships between different generators of the polycyclic codes and their corresponding dual-preserving properties under the annihilator duality.

What are the potential applications of the annihilator dual-preserving polycyclic codes and their associated quantum codes in areas such as cryptography or quantum computing

The potential applications of annihilator dual-preserving polycyclic codes and their associated quantum codes are significant in various fields such as cryptography and quantum computing. In cryptography, these codes can be utilized for secure communication protocols, error detection, and correction in data transmission, and enhancing the resilience of cryptographic systems against quantum attacks. The quantum codes derived from annihilator dual-preserving polycyclic codes can be employed in quantum error correction, quantum key distribution, and quantum secure communication. These codes play a crucial role in protecting quantum information from decoherence and errors, thereby improving the reliability and security of quantum communication systems.

Can the insights and techniques developed in this paper be applied to study polycyclic codes over other classes of rings beyond the serial ring R considered here

The insights and techniques developed in the study of polycyclic codes over the serial ring R can be applied to investigate polycyclic codes over other classes of rings beyond the serial ring R. By adapting the algebraic structures and properties of polycyclic codes to different ring structures, researchers can explore the behavior of codes in diverse mathematical settings. This can lead to the discovery of new coding techniques, error correction methods, and code optimization strategies applicable to a wide range of algebraic structures. The general principles and results obtained from studying polycyclic codes over the serial ring R can serve as a foundation for exploring coding theory in various algebraic contexts, contributing to advancements in both classical and quantum coding theory.