Shattering bounds for tuple systems

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