Perfect code

An \((n,K,2t+1)_q\) code is perfect if parameters \(n\), \(K\), \(t\), and \(q\) are such that the Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t}(q-1)^{j}{n \choose j}\leq q^{n}/K \tag*{(1)}\end{align} becomes an equality. For example, for a binary \(q=2\) 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 \(n\) \(q\)-ary strings.

Any perfect linear code is either a repetition code, a Hamming code, or a binary or ternary Golay code [1]. If \(q\) is a prime power, any distance-three code is either a Hamming code or a nonlinear code with the same parameters; see Ref. [2], pg. 100, for more details.