All one polynomial

An all one polynomial (AOP) is a polynomial used in finite fields, specifically GF(2) (binary). The AOP is a 1-equally spaced polynomial.

An AOP of degree "m" has all terms from "x"^"m" to "x"⁰ with coefficients of 1, and can be written as

:$AOP(x)\; =\; sum\_\{i=0\}^\{m\}\; x^i$

or

:$AOP(x)\; =\; x^m\; +\; x^\{m-1\}\; +\; cdots\; +\; x\; +\; 1$

or

:$(x-1)AOP(x)\; =\; x^\{m+1\}\; -\; 1$

thus the roots of the all one polynomial are all roots of unity.

Properties

Over GF(2) the AOP has many interesting properties, including:

*The Hamming weight of the AOP is "m" + 1
*The AOP is irreducible if and only if "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 implementations in areas such as coding theory and cryptography.

Over $mathbb\{Q\}$, the AOP is irreducible whenever "m + 1" is prime p, and therefore in these cases, the "p"th cyclotomic polynomial.

References