Computer Science Asked by dohmatob on December 7, 2020

Let $1 le r le n$ b e integer(with $n$ large) and let $mathscr X_n$ be the set set of all $2^n$ binary strings of length $n$. A binary $r$-coverage code is a subset $S$ of $mathscr X_n$ such that every $x in mathscr X_n$ is within Hamming distance $le r$ of some $y in S$. A minimal $r$-coverage code is one for which the cardinality $|S|$ is minimal. Let this minimal cardinality be $N_n(r)$. Note that we have the trivial bound $N_n(r) le 2^n$.

Question.What are good upper-bounds for $N_n(r)$ in terms of $N$ and $r$ ?

So, according to **Theorem 4.3** of this monograph by Réné Struik, if we consider $q$-ary binary codes of length $n$, such that $n to infty$ with $r/n to p in [0,(q-1)/q]$, then the minimal code size has the following asymptotic limit
$$
(1/n)log_q N_{n,q}(r) to 1-H_q(p),
$$
where $H_q(p)$ is the $q$-ary entropy of $p$. This answers my specific problem which corresponds to $q=2$.

Answered by dohmatob on December 7, 2020

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