7.8 Channel coding

Channel coding is a viable method to reduce information rate through the channel and increase reliability. This goal isachieved by adding redundancy to the information symbol vector resulting in a longer coded vector of symbols that aredistinguishable at the output of the channel. Another brief explanation of channel coding is offered in Channel Coding and the Repetition Code . We consider only two classes of codes, block codes and convolutional codes .

Block codes

The information sequence is divided into blocks of length $k$ . Each block is mapped into channel inputs of length $n$ . The mapping is independent from previous blocks, that is,there is no memory from one block to another.

A binary block code is completely defined by $2^k$ binary sequences of length $n$ called codewords.

• How can one find "good" codewords?
• How can one systematically map information sequences into codewords?
• How can one systematically find the corresponding information sequences from a codeword, i.e. , how can we decode?

A block code is linear if $c_i\in C$ and $c_j\in C$ implies $c_i\mathop{\mathrm{xor}}c_j\in C$ where $$ is an elementwise modulo 2 addition.

Hamming distance is a useful measure of codeword properties

Additional information about coding efficiency and error are provided in Block Channel Coding .

Examples of good linear codes include Hamming codes, BCH codes, Reed-Solomon codes, and many more. The rate of these codes is defined as $\frac{k}{n}$ and these codes have different error correction and error detection properties.