Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions
Nuh Aydin, Peihan Liu, Bryan Yoshino
TL;DR
This paper explores polycyclic codes linked with trinomials, revealing their properties, constructing new quantum codes, and proposing open problems to advance understanding in coding theory.
Contribution
It provides a comprehensive analysis of polycyclic codes associated with trinomials, including new properties, quantum code constructions, and open conjectures.
Findings
New properties of trinomials and polycyclic codes
Construction of many new quantum codes
Identification of open problems and conjectures
Abstract
Polycyclic codes are a generalization of cyclic and constacyclic codes. Even though they have been known since 1972 and received some attention more recently, there have not been many studies on polycyclic codes. This paper presents an in-depth investigation of polycyclic codes associated with trinomials. Our results include a number of facts about trinomials, some properties of polycyclic codes, and many new quantum codes derived from polycyclic codes. We also state several conjectures about polynomials and polycyclic codes. Hence, we show useful features of polycyclic codes and present some open problems related to them.
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Taxonomy
TopicsCoding theory and cryptography · Quantum-Dot Cellular Automata · Quantum Computing Algorithms and Architecture