Second order Recurrences, quadratic number fields and cyclic codes

TL;DR

This paper explores the properties of primes related to second order recurrences in quadratic fields, analyzing their impact on the weight distributions of certain cyclic codes over finite fields and rings, including MDS and NMDS codes.

Contribution

It generalizes the concept of Wall-Sun-Sun primes to second order recurrences in quadratic fields and studies their influence on cyclic code weight distributions.

Findings

Identification of primes WSS(d) related to quadratic fields.

Analysis of weight distributions for cyclic codes over _p and _{p^2}.

Existence of MDS and NMDS codes in this context.

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).