You are here
Homeall one polynomial
Primary tabs
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
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
12E10 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections
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