Wireless Network Design: Optimization Models and Solution ...

24 Dinesh Rajan
2.4.1 Block Codes
In a simple block code, the information data stream is parsed into non-overlapping
groups of a fixed size, say k bits. Each group of k bits is then mapped to a group
of n > k bits independent of the bits in the other groups. This mapping into a higher
dimensional space essentially defines the code and is responsible for the error correction
and detection capabilities of the code. Since each group of n bits are essentially
independent of other groups, this type of coding is sometimes referred to as
memoryless. The rate, r, of the channel code is defined as r = k
n . The random coding
described in Section 2.3 and also depicted in Figure 2.6 is a block code (over a
non-binary alphabet) with k = log2 (M) and n coded symbols.
The channel encoding can be achieved using a generator matrix G of size k × n
which uniquely defines the code. Let x and y denote, respectively, the 1×k and 1×n
vectors of the information bits and coded bits. Then y = xG. For each code, there
is also a unique parity check matrix H of size n − k × n such that yHT = 0, for all
valid codewords y, which also implies that GH T = 0.
There are several decoding algorithms for block codes depending on the type
of code and the channel over which the code is used. These algorithms offer the
system designer several options to trade-off the complexity of the decoding process
Fig. 2.9 Illustration of capacity of a downlink channel