all one polynomial
An all one polynomial^{} (AOP) is a polynomial used in finite fields^{}, specifically GF($2$). The AOP is a 1equally spaced polynomial.
An AOP of degree $m$ can be written as follows:
$$\mathrm{AOP}(x)=\sum _{i=0}^{m}{x}^{i}={x}^{m}+{x}^{m1}+\mathrm{\dots}+x+1$$ 
Over GF($2$) the AOP has many interesting properties, including:

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The Hamming weight of the AOP is $m+1$.

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The AOP is irreducible polynomial^{} iff $m+1$ is prime and $2$ is a primitive root^{} modulo $m+1$.

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The only AOP that is a primitive polynomial^{} is ${x}^{2}+x+1$.
Despite the fact that the Hamming weight is large, because of the ease of representation and other improvements there are efficient hardware and software implementations for use in areas such as coding theory and cryptography.
Title  all one polynomial 

Canonical name  AllOnePolynomial 
Date of creation  20130322 15:00:26 
Last modified on  20130322 15:00:26 
Owner  Derk (34) 
Last modified by  Derk (34) 
Numerical id  7 
Author  Derk (34) 
Entry type  Definition 
Classification  msc 12E10 
Synonym  allone polynomial 
Synonym  AOP 
Related topic  CyclotomicPolynomial 
Related topic  ProofThatTheCyclotomicPolynomialIsIrreducible 
Related topic  FactoringAllOnePolynomialsUsingTheGroupingMethod 