Shattering bounds for tuple systems

TL;DR

This paper establishes a general upper bound on the size of shattered sets within tuple systems constrained by unions of q-ary Hamming spheres, advancing understanding of combinatorial bounds in coding theory.

Contribution

It introduces a new upper bound for shattered set sizes in tuple systems formed from unions of Hamming spheres, extending previous combinatorial bounds.

Findings

Derived a general upper bound for shattered set sizes

Applied bounds to q-ary Hamming sphere unions

Enhanced understanding of combinatorial limits in tuple systems

Abstract

Let $\mbox{ V}(n,d,q)$ stand for the $q$--ary Hamming spheres. Let $\mbox{ C}\subseteq (q)^n$ denote a tuple system such that $\mbox{ C}\subseteq \cup_{i=0}^s \mbox{ V}(n,d_i,q)$, where $d_1<\ldots <d_s$. We give here a general upper bound on the size of a shattered sets of the tuple system $\mbox{ C}$.