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:

$$

ESP(x) = sum_{i=0}^{m} x^{si} for $i\; =\; 0,\; 1,\; ldots,\; m$

or

$$

ESP(x) = x^{sm} + x^{s(m-1)} + cdots + x^s + 1.

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.

References