I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and after the reader is lost in abstract algebra for weeks, they start to apply abstract algebra concepts in defining codes such as BCH and RS codes. Therefore, for many weeks after the start of basic coding theory courses, students have no idea of why are they learning abstract algebra concepts, and all they have is a promise that they will use it in understanding coding theory sometime later. Can anyone please guide how can one explain to students why group theory is important in learning coding theory by giving a small example. Please keep in mind that the example you give should be understandable to students in the first two or three lectures of the course and use jargon only from undergraduate linear algebra.
My thought is as follows: We can give them the example of transmitting bits from the set $\{0,1\}$ through a communication channel and ask them to define a channel model that will change bits from '0' to '1' and '1' to '0', randomly to model the error in communication. For example, we can say that we transmitted the message $[ 0 1 1 1 0]$. The channel introduced two errors in the last two bits of the message, and the receiver gets the message as $[0 1 1 0 1]$. To mathematically model the error, we can say that a random number from $\{0, 1\}$ can be added to the transmitted message. The addition from $\{0, 1\}$ will produce a message $2$, which according to our modeling of the problem is not possible in the received message, i.e., we can confuse the transmitted message between 0 and 1, and not some third message we know does not exist. Therefore, since the set $\{0, 1\}$ is not closed under real addition, we will have to define an operation under which the set $\{0, 1\}$ is closed. From here, we can introduce groups and move onward, as the closure of the set is the first property of groups. My questions are as follows:
- Is the example mentioned above is correct?
Can we take one example (just like above) and build motivation for groups, rings and then fields (eventually Galois fields).
Can you please give another example for this purpose?
Thanks