An equally spaced polynomial (ESP) is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

used in

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

s, specifically

GF(2) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field of two elements (GF is the initialism of ''Galois field'', another name for finite fields). Notations and \mathbb Z_2 may be encountered although they can be confused with ...

(

binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that t ...

). An ''s''-ESP of degree ''sm'' can be written as: :

ESP(x) = \sum_^ x^

for

i = 0, 1, \ldots, m

or :

ESP(x) = x^ + x^ + \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 In mathematics, an all one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are known, which allow this polynomial to be used to define efficient alg ...

and has additional properties including the above.

References

{{reflist Field (mathematics) Polynomials