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

14.2 Redundancy for Error Correction 805
TABLE 14.1
Some Examples of Error Correcting Codes
n
k
Code
Code Efficiency
(or Code Rate)
Single-error correcting, t = I
Minimum code separation 3
3
4
5
6
7
1 5
31
1
2
3
4
11
26
(3, 1) 0.33
(4, 1) 0.25
(5, 2) 0.4
(6, 3) 0.5
(7, 4) 0.57
(15, 1 1) 0.73
(3 1 , 26) 0.838
Double-error correcting, t = 2
Minimum code separation 5
10
15
4
8
(10, 4) 0.4
(15, 8) 0.533
Triple-error correcting, t = 3
Minimum code separation 7
10
15
23
2
5
12
(10, 2) 0.2
(15, 5) 0.33
(23, 1 2) 0.52
and
n = 2 m - 1
Thus, Hamming codes are (n, k) codes with n = 2 m - 1 and k = 2 m - l - m and minimum
distance dmin = m. In general, we often write Hamming code as (2 m - 1, 2 m - l - m, m)
code. One of the most well-known Hamming codes is the (7, 4, 3) code.
Another way of correcting errors is to design a code to detect (not to correct) up to t errors.
When the receiver detects an error, it can request retransmission. This mechanism is known as
automatic repeat request ( or ARQ). Because error detection requires fewer check digits, these
codes operate at a higher rate (efficiency).
To detect t errors, codewords need to be separated by a Hamming distance of at least
t + 1. Otherwise, an erroneously received bit string with up to t error bits could be another
transmitted codeword. Suppose a transmitted codeword CJ contains ct bit errors (ct :S t) . Then
the received codeword c1 is at a distance of ct from CJ . Because ct :S t, however, c1 can never
be any other valid codeword, since all codewords are separated by at least t + l . Thus, the
reception of c1 immediately indicates that an error has been made. Thus, the minimum distance
d m in between t error detecting codewords is
d m in = t + l
In presenting coding theory, we shall use modulo-2 addition, defined by
1EB1=0EB0=0
0EB1=1EB0=1
This is also known as the exclusive OR (XOR) operation in digital logic. Note that the modulo-
2 sum of any binary digit with itself is always zero. All the additions in the mathematical
development of binary codes presented henceforth are modulo-2.