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

• Published in: Discrete mathematics and theoretical computer science 2007-01, Vol.DMTCS Proceedings vol. AH,... (Proceedings), p.183-196
• Main Authors: Lee, Gilbert, Ruskey, Frank, Williams, Aaron
• Format: Article
• Language: English
• Subjects: [info.info-cg] computer science [cs]/computational geometry [cs.cg]; [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds]; [math.math-co] mathematics [math]/combinatorics [math.co]; asymptotics; binary strings; Combinatorics; Computational Geometry; Computer Science; Data Structures and Algorithms; Discrete Mathematics; exhaustive enumeration; finite fields; hamming distance; irreducible polynomials; Mathematics; random polynomials
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.3550

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.