Equally spaced polynomial: Difference between revisions

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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:

ESP(x)=\sum _{i=0}^{m}x^{si}

for

i=0,1,\ldots ,m

or

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.^[1]

References

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

This polynomial-related article is a stub. You can help Wikipedia by expanding it.

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

[1]

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+<!-- {{more citations needed|date=March 2024}} -->
-An '''equally spaced polynomial''' (ESP) is a polynomial used in [[finite field]]s, specifically GF(2) ([[binamio merialsystem|[binam]]).
+{{one source |date=May 2024}}
+An '''equally spaced polynomial''' (ESP) is a [[polynomial]] used in [[finite field]]s, specifically [[GF(2)]] ([[binary numeral system|binary]]).
-An s-(ES of degreen 'sm'''can be written as:
+An ''s''-ESP of degree ''sm'' can be written as:
+:<math>ESP(x) = \sum_{i=0}^{m} x^{si}</math> for <math>i = 0, 1, \ldots, m</math>
-:<math>
-(ES(x) = \sum_{i=0}^{m} x^{si}</math> -t/}^{ES(0, 1, \ldotielm
-^{si}</
+or
-orn as:
+:<math>ESP(x) = x^{sm} + x^{s(m-1)} + \cdots + x^s + 1.</math>
-:<math>
-(ES(^{m}=0}+(^{m}(m-1)0}+(\cdoti}+(^{i}+(1
-^{si}</
 ==Properties==
+Over GF(2) the ESP - which then can be referred to as all one polynomial (AOP) - has many interesting properties, including:
-Over}eife }eifeife  manymiaterest p==Propert,miacludingitt*T }e[[Hamming weight]]An se }eifei'' ''gree{i}+
+* The [[Hamming weight]] of the ESP is ''m'' + 1.
-A 1-ifei'' knowe wr aal u sp oneESP) is a p]]Aandife  additionamip==Propertmiacludingse }eabove.
+A 1-ESP is known as an [[all one polynomial]] (AOP) and has additional properties including the above.<ref>{{Cite web |title=all one polynomial |url=https://planetmath.org/allonepolynomial |access-date=2024-03-07 |website=planetmath.org}}</ref>
-[[Category:F[finse }otem|
+==References==
+{{reflist}}
+[[Category:Field (mathematics)]]
+[[Category:Polynomials]]
+{{polynomial-stub}}