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:

${\displaystyle ESP(x)=\sum _{i=0}^{m}x^{si}}$ for ${\displaystyle i=0,1,\ldots ,m}$

or

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

Properties

Over GF(2) the ESP - which then can be referred to as all one polynomial (AOP) - has many interesting properties, including:

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

A 1-ESP is known as an all one polynomial (AOP) and has additional properties including the above.^[1]