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