Bounds for discrepancies in the Hamming space

TL;DR

This paper establishes bounds for the discrepancies in the Hamming space, highlighting the fundamental difference in volume growth compared to Euclidean spaces, which impacts discrepancy estimates.

Contribution

It provides the first bounds for ball discrepancies in the Hamming space, emphasizing the exponential volume growth distinct from Riemannian manifolds.

Findings

Discrepancy bounds depend on exponential volume growth in Hamming space

Contrast with polynomial volume growth in Riemannian manifolds

Fundamental differences in discrepancy behavior across spaces

Abstract

We derive bounds for the ball $L_p$-discrepancies in the Hamming space for $0<p<\infty$ and $p=\infty$. Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres and more general compact Riemannian manifolds. In the present paper, we show that the behavior of discrepancies in the Hamming space differs fundamentally because the volume of the ball in this space depends on its radius exponentially while such a dependence for the Riemannian manifolds is polynomial.