One of the fundamental problems in the theory of quantum error correcting codes is that of finding good upper bounds on the minimum distance of a code of specified length and dimension. In part I, I will discuss this problem in the special case when the code is additive (i.e., is derived from a code over GF(4)). In this case, the problem is purely combinatorial, thus allowing standard coding-theory techniques to be adapted. In particular, the linear programming bound of classical coding theory can be adapted to give bounds for additive codes. I will also discuss the concept of the shadow of an additive code, the source of much of the power of the LP bound in the quantum setting.
This talk is based on joint work with A.R. Calderbank, P.W. Shor, and N.J.A. Sloane.