The Asymptotic Number of Anticodes in Hamming Spaces

In the Hamming space $\mathbb{F}_q^n$, a $d$-anticode is a subset in which every pair of points has Hamming distance at most $d$, or equivalently, a set of diameter at most $d$. Let $m_q(n,d)$ denote the maximum size of a $d$-anticode and let $\mathcal{A}_q(n,d)$ be the family of all $d$-anticodes.