Block codes Basic concepts and definitions A source information is processed, first, in a channel encoder, next, in a discrete memoryless channel and finally, in a channel decoder, as presented in Fig.5.1.1.
The output sequence ; , is the result of mod-2 addition of a codeword ; , and error pattern ; :
; To reduce rate of erroneous transmissions, control codes are used. In such code a codeword consists of two parts:
information bits: and
(parity) check bits: ; .
Source encoder produces information bits (word).
Channel encoder determines check bits and append them to the information word, so giving a codeword of the length: A code that generates, at any given time unit, n outputs from k inputs is referred to as an block code , where: n is the block-length and k is the information part length. There are codewords in such code, i.e. such code enables encoding the maximum of M source messages., Non-block codes, the so called convolutional codes form another class of error control codes. Convolutional code differ from a block code in that the channel encoder output is constructed not only from k inputs at the given time unit, but also using some of the previous inputs. A code that generates, at any given time unit, n outputs from k inputs at that time and l previous inputs, is referred to as convolution code. Such codes are out of scope of this book. An block code totals of channel inputs, error patterns and outputs are:
total of (channel inputs) codewords: ,
total of error patterns: ,
total of nonzero error patterns: ,
total of output sequences: , with M codewords among them,
the remaining sequences are referred to as redundant sequences.
Presence of such sequences can only be due to the occurrence of errors. By adding, in channel encoder, redundancy to information, error detection can be carried out. Furthermore, it may be possible to carry out error correction, if there is sufficient redundancy. A useful measure of the redundancy is given by the ratio of information-part-length to the block-length: and is known as a code rate , . For information codes: R =1, there is no redundancy and therefore, no error detection ability.
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