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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:
$\operatorname{AOP}(x)=\sum_{{i=0}}^{{m}}x^{i}=x^{m}+x^{{m1}}+\ldots+x+1$ 
Over GF($2$) the AOP has many interesting properties, including:

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

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

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.
Related:
CyclotomicPolynomial, ProofThatTheCyclotomicPolynomialIsIrreducible, FactoringAllOnePolynomialsUsingTheGroupingMethod
Synonym:
allone polynomial, AOP
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Recent Activity
Jul 5
new correction: Error in proof of Proposition 2 by alex2907
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new question: binomial coefficients: is this a known relation? by pfb
new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Jun 13
new question: young tableau and young projectors by zmth
Jun 11
new question: binomial coefficients: is this a known relation? by pfb
Comments
name?
To be honest, the name "all one" does not parse grammatically. The closest thing I can think of "all ones". Google finds references to both, but for some reason "all one" gets more hits.
Is this already established terminology? What is its source?
Re: name?
I'm not sure where the term was first used, though I believe "all one" or "allone" to be the norm (just to be safe I have the appropriate synonym). I have never heard of "all ones" or "allones". Just a quick search through IEEE TECS turns up 7 results, while "all ones" gets 0. The oldest result is:
Hasan, M.A.; Wang, M.; Bhargava, V.K.; Modular construction of low complexity parallel multipliers for a class of finite fields GF(2^m); Computers, IEEE Transactions on, Volume: 41, Issue: 8, Aug. 1992, Pages:962  971