Equally spaced polynomial explained

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 ^{m} _{i =0} x ^{si} for i = 0 , 1 , \ldots, m

or

ESP(x) = x ^{sm} + x ^{s(m-1)} + … + 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]

Notes and References

1. Web site: all one polynomial . 2024-03-07 . planetmath.org.