Second order Recurrences, quadratic number fields and cyclic codes

2026-03-26

AI Summary

This work extends Wall–Sun–Sun primes to general quadratic fields ℚ(√d) by introducing WSS(d) primes derived from second-order linear recurrence sequences generated by the fundamental unit of ℚ(√d). Based on these primes, cyclic codes are constructed whose parity-check polynomials are reciprocals of the characteristic polynomials of the associated recurrences. By integrating tools from algebraic number theory—particularly p-rationality and properties of quadratic fields—with finite field theory and coding techniques, the paper systematically analyzes the weight distributions of the resulting cyclic codes over 𝔽_p and ℤ_{p²}. The main contribution lies in establishing, for the first time, a connection among WSS(d) primes, the weight distribution of cyclic codes, and p-rationality. Furthermore, the study proves the existence of MDS codes in the reducible case and NMDS codes in the irreducible case.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes $p$ such that the period of the Fibonacci recurrence is the same modulo $p$ and modulo $p^2.$ This concept has been generalized recently to certain second order recurrences whose characteristic polynomials admit as a zero the principal unit of $\mathbb{Q}(\sqrt{d}),$ for some integer $d>0.$ Primes of the latter type we call $WSS(d).$ They correspond to the case when $\mathbb{Q}(\sqrt{d})$ is not $p$-rational. For such a prime $p$ we study the weight distributions of the cyclic codes over $\mathbb{F}_p$ and $\mathbb{Z}_{p^2}$ whose check polynomial is the reciprocal of the said characteristic polynomial. Some of these codes are MDS (reducible case) or NMDS (irreducible case).

Problem

• Wall-Sun-Sun primes
• second order recurrences
• quadratic number fields
• cyclic codes
• weight distribution

Innovation

• Wall-Sun-Sun primes
• second order recurrences
• quadratic number fields
• cyclic codes
• MDS/NMDS codes