Equally spaced polynomial

An equally spaced polynomial (ESP) is a polynomial used in finite fields, specifically GF(2) (binary).

An s-ESP of degree sm can be written as:

[math]\displaystyle{ ESP(x) = \sum_{i=0}^{m} x^{si} }[/math] for [math]\displaystyle{ i = 0, 1, \ldots, m }[/math]

or

[math]\displaystyle{ ESP(x) = x^{sm} + x^{s(m-1)} + \cdots + x^s + 1. }[/math]

Properties

Over GF(2) the ESP has many interesting properties, including:

• The Hamming weight of the ESP is m + 1.

A 1-ESP is known as an all one polynomial and has additional properties including the above.