B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

804 ERROR CORRECTING CODES
Thus for each codeword, we must leave
vertices (words) unused. Because we have 2 k codewords, we must leave a total of
words unused. Therefore, the total number of words must at least be
But the total number of words, or vertices, available is 2 n . Thus, we require,
or
(14.3a)
Observe that n - k = mis the number of check digits. Hence, Eq. (14.3a) can be expressed as
(14.3b)
This is known as the Hamming bound. It should also be remembered that the Hamming bound
is a necessary but not a sufficient condition in general. However, for single-error correcting
codes, it is a necessary and sufficient condition. If some m satisfies the Hamming bound, it does
not necessarily mean that a t-error correcting code of n digits can be constructed. Table 14. J
shows some examples of error correction codes and their rates.
A code for which the inequalities in Eqs. (] 4.3) become equalities is known as a perfect
code. In such a code the Hamming spheres (about all the codewords) not only are nonoverlapping
but they exhaust all the 2 n vertices, leaving no vertex outside some sphere. An e-error
correcting perfect code satisfies the condition that every possible (received) sequence is at a
distance at most e from some codeword. Perfect codes exist in only a comparatively few cases.
Binary, single-error correcting, perfect codes are called Hamming codes. For a Hamming
code, t = l and dmin = 3. and from Eq. (14.3b) we have
1
2 m = L () = 1 + n
}=0 J
(14.4)