Perfect codes have been introduced in §6.5 in connection with the Hamming bound (6.5.1) which for a *q*-ary code *C* = (*n,k,d*) with *d* ≤ 2*t* + 1 implies that

$\left|C\right|{\displaystyle \sum _{i=0}^{t}\left(\begin{array}{l}n\\ i\end{array}\right){(q-1)}^{i}\le {q}^{n},}$ |
(7.1.1) |

where |*C*| (= *M*) denotes the size of the code *C* (see §6.1) A code is said to be *perfect* if equality holds in (7.1.1). Since a terror correcting code can correct up to *t* errors, a perfect *t*-error correcting code must be such that every codeword lies within a distance of *t* to exactly one codeword. In other words, if the code has *d*_{min} = 2*t* + 1 that covers radius *t*, where the covering radius has the smallest number, then every codeword lies within a distance of the Hamming radius *t* to a codeword. In the case of a perfect code, the ...

Get *Algebraic and Stochastic Coding Theory* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.