In this paper, we study the quotients that arise when the Euclidean algorithm is applied to a primitive polynomial and x~s ―1. We analyze the asymptotic behavior of the the number of terms of the quotients as n→∞. This problem comes from the study of Low-Density Parity-Check codes. We obtain two characterizations of primitive polynomials over the field with two elements that are based on the number of nonzero terms in two polynomials obtained via division and the Euclidean algorithm with polynomials of the form x~s ―1. The analogous results do not hold for general finite fields but do restrict the order of the polynomials to a small set of positive integers with specific forms.
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