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22 Introduction Chap. 1
detected and corrected if d Ú s + t + 1, where s is the number of errors that can be detected
and t is the number of errors that can be corrected (s Ú t). Thus, a pattern of t or fewer errors
can be both detected and corrected if d Ú 2t + 1.
A general code word can be expressed in the form
i 1 i 2 i 3 Á i k p 1 p 2 p 3 Á p r
where k is the number of information bits, r is the number of parity check bits, and n is the
total word length in the (n, k) block code, where n = k + r. This arrangement of the
information bits at the beginning of the code word followed by the parity bits is most
common. Such a block code is said to be systematic. Other arrangements with the parity bits
interleaved between the information bits are possible and are usually considered to be
equivalent codes.
Hamming has given a procedure for designing block codes that have single errorcorrection
capability [Hamming, 1950]. A Hamming code is a block code having a
Hamming distance of 3. Because d Ú 2t + 1, t = 1, and a single error can be detected and
corrected. However, only certain (n, k) codes are allowable. These allowable Hamming
codes are
(n, k) = (2 m - 1, 2 m - 1 - m)
(1–11)
where m is an integer and m Ú 3. Thus, some of the allowable codes are (7, 4), (15, 11), (31, 26),
(63, 57), and (127, 120). The code rate R approaches 1 as m becomes large.
In addition to Hamming codes, there are many other types of block codes. One popular
class consists of the cyclic codes. Cyclic codes are block codes, such that another code word
can be obtained by taking any one code word, shifting the bits to the right, and placing the
dropped-off bits on the left. These types of codes have the advantage of being easily encoded
from the message source by the use of inexpensive linear shift registers with feedback. This
structure also allows these codes to be easily decoded. Examples of cyclic and related codes
are Bose–Chaudhuri–Hocquenhem (BCH), Reed-Solomon, Hamming, maximal–length,
Reed–Müller, and Golay codes. Some properties of block codes are given in Table 1–3
[Bhargava, 1983].
TABLE 1–3
PROPERTIES OF BLOCK CODES
Code a
Property BCH Reed-Solomon Hamming Maximal Length
Block length
Number of
parity bits
Minimum
distance
Number of
information bits
n = 2 m = 1 n = m(2 m - 1) bits n = 2 m - 1 n = 2 m - 1
m = 3, 4, 5, Á
r = m2t bits
r = m
d Ú 2t + 1 d = m(2t + 1) bits d = 3 d = 2 m - 1
k Ú n - mt
k = m
a m is any positive integer unless otherwise indicated; n is the block length; k is the number of information bits.