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arXiv · 2603.25343

Second order Recurrences, quadratic number fields and cyclic codes

Minjia Shi·Xuan Wang·Bouazzaoui Zakariae·Jon-Lark Kim·Patrick Solé
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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).

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Minjia Shi, Xuan Wang, Bouazzaoui Zakariae, Jon-Lark Kim, Patrick Solé. 2026-03-26. Second order Recurrences, quadratic number fields and cyclic codes. https://arxiv.org/abs/2603.25343

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