Error-correcting codes

TL;DRAbstract

We shall not go into details concerning all the technical applications of error-correcting codes. These include satellite pictures, telephone messages via glass fibre using light, compact disc audio system. The idea is as follows. We consider “information” presented as a very long sequence of symbols from a finite set called the “alphabet”. In this course the alphabet will be a finite field IF q . In the sequence each symbol occurs with equal probability. This information is sent to a receiver over a so-called “noisy channel”. In the model that we consider there is a fixed (small) probability p e that a symbol, that is sent over the channel, is changed into one of the other symbols (again, all equally likely). Such an event is called a “symbol-error” and p e is the symbol-error probability. As a result a fraction p e of the transmitted symbols arrives incorrectly at the receiver end of the channel. The aim of coding theory is to lower the probability of error (considerably) at the expe

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