Second order Recurrences, quadratic number fields and cyclic codes

• URL: http://arxiv.org/abs/2603.25343v1
• Date: Thu, 26 Mar 2026 11:39:50 GMT
• Title: Second order Recurrences, quadratic number fields and cyclic codes
• Abstract summary: Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes $p$ such that the period of the Fibonacci recurrence is the same mod $p$ and mod $p2.
• Score: 25.427293165770767
• License: http://creativecommons.org/licenses/by/4.0/
• 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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