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

TL;DR

This paper derives closed-form formulas for the periods of the polynomial x^h + x + 1 over GF(2) for infinite h sets, aiding in algebraic code design and shift register analysis.

Contribution

It provides the first closed-form expressions for the periods of x^h + x + 1 over GF(2) for infinite h, extending beyond finite h cases.

Findings

Closed-form formulas for polynomial periods over GF(2)

Applicable to infinite h values, not just finite cases

Enhances understanding of algebraic structures in coding theory

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.