Perfect binary code 

An \((n,K,2t+1)\) binary code is perfect if parameters \(n\), \(K\), and \(t\) are such that the binary Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t} {n \choose j} \leq 2^{n}/K \tag*{(1)}\end{align} becomes an equality. For example, for a code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) binary strings.

Any perfect linear binary code is either a binary repetition code, a binary Hamming code, or the binary Golay code [1].

For codes with \(K=2^k\), one can work out an asymptotic Hamming bound in the large-\(n,k,t\) limit, \begin{align} \frac{k}{n}\leq 1-h(t/n), \tag*{(2)}\end{align} where \(h\) is the binary entropy function.