ALGEBRAIC CODING THEORY 25
of choosing w out of n positions, the probability of w errors at arbitrary positions within
an n-dimensional binary received word follows the binomial distribution
Pr{w errors}=
n
w
ε
w
(1 − ε)
n−w
with mean nε. Because of the condition ε<
1
2
, the probability Pr{w errors} decreases with
increasing number of errors w, i.e. few errors are more likely than many errors.
The probability of error-free transmission is Pr{0 errors}=(1 − ε)
n
, whereas the prob-
ability of a disturbed transmission with r = b is given by
Pr{r = b}=
n
w=1
n
w
ε
w
(1 − ε)
n−w
= 1 −(1 − ε)
n
.
2.1.4 Error Detection and Error Correction
Based on the minimum distance decoding rule and the code space concept, we can assess
the error detection and error correction capabilities of a given channel code. To this end, let
b and b
be two code words of an (n, k) block code B(n,k,d). The distance of these code
words shall be equal to the minimum Hamming distance, i.e. dist(b, b
) = d. We are able to
detect errors as long as the erroneously received word r is not equal to a code word different
from the transmitted code word. This error detection capability is guaranteed as long as
the number of errors is smaller than the minimum Hamming distance d, because another
code word (e.g. b
) can be reached from a given code word (e.g. b) merely by changing at
least d components. For an (n, k) block code B(n,k,d) with minimum Hamming distance
d, the number of detectable errors is therefore given by (Bossert, 1999; Lin and Costello,
2004; Ling and Xing, 2004; van Lint, 1999)
e
det
= d − 1.
For the analysis of the error correction capabilities of the (n, k) block code B(n,k,d) we
define for each code word b the corresponding correction ball of radius as the subset
of all words that are closer to the code word b than to any other code word b
of the
block code B(n,k,d) (see Figure 2.10). As we have seen in the last section, for minimum
distance decoding, all received words within a particular correction ball are decoded into
the respective code word b. According to the radius of the correction balls, besides the
code word b, all words that differ in 1, 2,..., components from b are elements of the
corresponding correction ball. We can uniquely decode all elements of a correction ball
into the corresponding code word b as long as the correction balls do not intersect. This
condition is true if <
d
2
holds. Therefore, the number of correctable errors of a block
code B(n,k,d) with minimum Hamming distance d is given by (Bossert, 1999; Lin and
Costello, 2004; Ling and Xing, 2004; van Lint, 1999)
5
e
cor
=
d − 1
2
.
5
The term z denotes the largest integer number that is not larger than z.