ALGEBRAIC CODING THEORY 27
2.2 Linear Block Codes
In the foregoing discussion of the fundamentals of general block codes we exclusively
focused on the distance properties between code words and received words in the code
space F
n
q
. If we consider the code words to be vectors in the finite vector space F
n
q
,
taking into account the respective algebraic properties of vector spaces, we gain efficiency
especially with regard to the encoding scheme.
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2.2.1 Definition of Linear Block Codes
The (n, k) block code B(n,k,d) with minimum Hamming distance d over the finite field
F
q
is called linear,ifB(n,k,d) is a subspace of the vector space F
n
q
of dimension k (Lin
and Costello, 2004; Ling and Xing, 2004). The number of code words is then given by
M = q
k
according to the code rate
R =
k
n
.
Because of the linearity property, an arbitrary superposition of code words again leads to
a valid code word of the linear block code B(n,k,d), i.e.
α
2
b
1
+ α
2
b
2
+···+α
l
b
l
∈ B(n,k,d)
with α
1
,α
2
,...,α
l
∈ F
q
and b
1
, b
2
,...,b
l
∈ B(n,k,d). Owing to the linearity, the n-
dimensional zero row vector 0 = (0, 0,...,0) consisting of n zeros is always a valid
code word. It can be shown that the minimum Hamming distance of a linear block code
B(n,k,d) is equal to the minimum weight of all non-zero code words, i.e.
d = min
∀b=b
dist(b, b
) = min
∀b=0
wt(b).
These properties are summarised in Figure 2.11. As a simple example of a linear block
code, the binary parity-check code is described in Figure 2.12 (Bossert, 1999).
For each linear block code an equivalent code can be found by rearranging the code
word symbols.
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This equivalent code is characterised by the same code parameters as the
original code, i.e. the equivalent code has the same dimension k and the same minimum
Hamming distance d.
2.2.2 Generator Matrix
The linearity property of a linear block code B(n,k,d) can be exploited for efficiently
encoding a given information word u = (u
0
,u
1
,...,u
k−1
). To this end, a basis {g
0
, g
1
,...,
g
k−1
} of the subspace spanned by the linear block code is chosen, consisting of k linearly
independent n-dimensional vectors
g
i
= (g
i,0
,g
i,1
, ···,g
i,n−1
)
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Finite fields and vector spaces are briefly reviewed in Sections A.1 and A.2 in Appendix A.
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In general, an equivalent code is obtained by suitable operations on the rows and columns of the generator
matrix G which is defined in Section 2.2.2.