The period of $x^h + x + 1$ over GF(2)

Abstract

The periods of polynomials can be used to characterize discrete structures such as algebraic error control codes and feedback shift registers. We study trinomial $x^h+x+1$ over GF(2), which has the maximum number of consecutive zero coefficients and leads to efficient implementation. Existing results typically deal with finite values of $h$ and rely on computer computation methods for finding the periods. In contrast, here we derive closed-form expressions for the periods of this trinomial for infinite sets of $h$ values.