Second order Recurrences, quadratic number fields and cyclic codes
This work addresses theoretical problems in number theory and coding theory, focusing on prime properties and code constructions, but appears incremental as it builds on existing generalizations.
The paper investigates Wall-Sun-Sun primes generalized to second-order recurrences linked to quadratic number fields, analyzing weight distributions of cyclic codes over finite fields and rings, with some codes achieving MDS or NMDS properties.
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).