Error-Control Block Codes for Communications Engineers
An engineer s guide to the theory behind the common random-error-control block codes, and their applications in digital communications.
TABLE OF CONTENTS Error-Control Block Codes for Communications Engineers
3.2: Matrix Description of Linear Block Codes
3.2 Matrix Description of Linear Block Codes
From our earlier study of vector space theory and Definition 3.7, it is possible to find k linearly independent codewords G 0, G 1, . . ., G k ?1 in the q-ary code C such that
| (3.3) |  |
where
| (3.4) |  |
u i and v j are q-ary symbols for 0 ? i ? k ? 1 and 0 ? j ? n ? 1. V is a linear combination of the k linearly independent codewords. The k-by- n generator matrix G of the code C is
| (3.5) |  |
| (3.6) |  |
where G i = [ g i,0 g i,1 . . . g i, n ?1] with q-ary entries for 0 ? i ? k ? 1. The encoding operation, as shown in Figure 3.2, can be expressed as
| (3.7) |  |
where
| (3.8) |  |
Example 3.4
Consider a (7, 4) binary linear block code with 
. The information and code vectors are shown below.
-
U
V
0 0 0 0
0 0 0 0 0 0 0
0 0 0 1
0 0 0 1 0 1 1
0 0 1 0
0 0 1 0 1 1 0
0 0 1 1
0 0 1 1 1 0 1
0 1 0 0
0 1 0 0 1 1 1
0 1 0 1
0...
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