chapter  Chapter 3
29 Pages

General linear codes

ByJuergen Bierbrauer

Linear algebra works over any field. In the binary case the underlying field is F2={0,1}, the smallest of all. Many fruitful code constructions rely on codes defined over a large field and involve "going down" to codes over a small field. Also, there are constructions that work best over larger fields. There is simply no reasonable way to get around linear codes over general finite fields. If it is the case one speaks of a primitive polynomial and calls the corresponding field elements primitive elements. Every finite field can be described by a primitive polynomial. The principle of duality, although rather elementary, is very important. One could almost say it distinguishes coding theorists from the rest of the world. If really understand it may (almost) call ourself a coding theorist. We saw that duality motivates the definition of orthogonal arrays.