1Department of Computer Science [Victoria] (Department of Computer Science University of Victoria Engineering/ Computer Science Building (ECS), Room 504 PO Box 1700, STN CSC Victoria, BC Canada V8W 2Y2 - Canada)
Abstract : 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.
Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.183-196, 2007, DMTCS Proceedings
https://hal.inria.fr/hal-01184798
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Soumis le : lundi 17 août 2015 - 17:00:20
Dernière modification le : jeudi 11 mai 2017 - 01:02:52
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Gilbert Lee, Frank Ruskey, Aaron Williams. Hamming distance from irreducible polynomials over $\mathbb {F}_2$. Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.183-196, 2007, DMTCS Proceedings. 〈hal-01184798〉
