Hamming Distance from Irreducible Polynomials Over

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 b2n/nc odd density polynomials is asymptotically (1 − e−4)2n−1, 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 F2 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.