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This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Equally spaced polynomial" – news · newspapers · books · scholar · JSTOR (May 2024)

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.

References

1. "all one polynomial". planetmath.org. Retrieved 2024-03-07.

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