Number Theory–Arithmetic for Codes
The main outcome of the previous chapter was the explicit connection between the minimum distance of a code and its error-correcting and error–detecting capability (Theorems 2.1 and 2.2). So a code which is good at correcting errors should have a large minimum distance. Since codes with several thousand codewords are often required the job of designing such a code is daunting, and trial-and-error is really a non-starter. As always, mathematics comes to the rescue, for if we impose some mathematical structure on codes their properties are rather easier to sort out, and there is more hope of devising a feasible decoding procedure – that is, one which is not too expensive and which doesn’t take too long.