The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Yasanthi Kottegoda, Robert Fitzgerald


Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when $(q^{d}-1)/ m$ follows the quadratic residue conditions and also when it has the form $q^{2a}-q^{a}+1$ where $a\in \mathbb{N}$.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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