Hamming distance from irreducible polynomials over $\mathbb {F}_2$ - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2007

Hamming distance from irreducible polynomials over $\mathbb {F}_2$


We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree $n$ polynomials of Hamming distance one from a randomly chosen set of $\lfloor 2^n/n \rfloor$ odd density polynomials, each of degree $n$ and each with non-zero constant term, is asymptotically $(1-e^{-4}) 2^{n-2}$, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant $c$ such that every polynomial has Hamming distance at most $c$ from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over $\mathbb{F}_2$ for degrees $1 ≤ n ≤ 32$, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this "empirical" study.
Fichier principal
Vignette du fichier
dmAH0113.pdf (211.21 Ko) Télécharger le fichier
Origin : Publisher files allowed on an open archive

Dates and versions

hal-01184798 , version 1 (17-08-2015)





Gilbert Lee, Frank Ruskey, Aaron Williams. Hamming distance from irreducible polynomials over $\mathbb {F}_2$. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. pp.183-196, ⟨10.46298/dmtcs.3550⟩. ⟨hal-01184798⟩


156 View
848 Download



Gmail Facebook X LinkedIn More