The last chapter
Most of the bounds on codes and orthogonal arrays that we encountered in this text can be deduced from the same source, the linear-programming bound. This is a very general bound, which works for general not necessarily linear codes. It turned out that the distance polynomial of the dual code (with respect to the dot product) is obtained by a simple polynomial transformation. In order to obtain general explicit bounds the conditions of applicability of the dual LP have to be expressed in an algebraic fashion. This is possible because of the orthogonality properties of the K-polynomials. The Singleton bound on codes is an example of this mechanism. Surprisingly its derivation is rather involved. Our direct proof was clear and short in comparison. Speaking of the Plotkin bound, it too is a consequence of the LP bound. Our proof was based on quadratic counting. In this case the derivation from the LP-bound is even easier.